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shamir/cryptographie_visuelle.slides.html
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2014-11-28 20:29:06 -08:00

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.form-search .input-prepend .btn{border-radius:14px 0 0 14px;-webkit-border-radius:14px 0 0 14px;-moz-border-radius:14px 0 0 14px;border-radius:14px 0 0 14px}
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.control-group{margin-bottom:10px}
legend+.control-group{margin-top:20px;-webkit-margin-top-collapse:separate}
.form-horizontal .control-group{margin-bottom:20px;*zoom:1}.form-horizontal .control-group:before,.form-horizontal .control-group:after{display:table;content:"";line-height:0}
.form-horizontal .control-group:after{clear:both}
.form-horizontal .control-label{float:left;width:160px;padding-top:5px;text-align:right}
.form-horizontal .controls{*display:inline-block;*padding-left:20px;margin-left:180px;*margin-left:0}.form-horizontal .controls:first-child{*padding-left:180px}
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.form-horizontal .form-actions{padding-left:180px}
table{max-width:100%;background-color:transparent;border-collapse:collapse;border-spacing:0}
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.table th{font-weight:bold}
.table thead th{vertical-align:bottom}
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.table-bordered caption+thead tr:first-child th,.table-bordered caption+tbody tr:first-child th,.table-bordered caption+tbody tr:first-child td,.table-bordered colgroup+thead tr:first-child th,.table-bordered colgroup+tbody tr:first-child th,.table-bordered colgroup+tbody tr:first-child td,.table-bordered thead:first-child tr:first-child th,.table-bordered tbody:first-child tr:first-child th,.table-bordered tbody:first-child tr:first-child td{border-top:0}
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.table-bordered thead:first-child tr:first-child>th:last-child,.table-bordered tbody:first-child tr:first-child>td:last-child,.table-bordered tbody:first-child tr:first-child>th:last-child{-webkit-border-top-right-radius:4px;-moz-border-radius-topright:4px;border-top-right-radius:4px}
.table-bordered thead:last-child tr:last-child>th:first-child,.table-bordered tbody:last-child tr:last-child>td:first-child,.table-bordered tbody:last-child tr:last-child>th:first-child,.table-bordered tfoot:last-child tr:last-child>td:first-child,.table-bordered tfoot:last-child tr:last-child>th:first-child{-webkit-border-bottom-left-radius:4px;-moz-border-radius-bottomleft:4px;border-bottom-left-radius:4px}
.table-bordered thead:last-child tr:last-child>th:last-child,.table-bordered tbody:last-child tr:last-child>td:last-child,.table-bordered tbody:last-child tr:last-child>th:last-child,.table-bordered tfoot:last-child tr:last-child>td:last-child,.table-bordered tfoot:last-child tr:last-child>th:last-child{-webkit-border-bottom-right-radius:4px;-moz-border-radius-bottomright:4px;border-bottom-right-radius:4px}
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.table-bordered caption+thead tr:first-child th:first-child,.table-bordered caption+tbody tr:first-child td:first-child,.table-bordered colgroup+thead tr:first-child th:first-child,.table-bordered colgroup+tbody tr:first-child td:first-child{-webkit-border-top-left-radius:4px;-moz-border-radius-topleft:4px;border-top-left-radius:4px}
.table-bordered caption+thead tr:first-child th:last-child,.table-bordered caption+tbody tr:first-child td:last-child,.table-bordered colgroup+thead tr:first-child th:last-child,.table-bordered colgroup+tbody tr:first-child td:last-child{-webkit-border-top-right-radius:4px;-moz-border-radius-topright:4px;border-top-right-radius:4px}
.table-striped tbody>tr:nth-child(odd)>td,.table-striped tbody>tr:nth-child(odd)>th{background-color:#f9f9f9}
.table-hover tbody tr:hover>td,.table-hover tbody tr:hover>th{background-color:#f5f5f5}
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.table td.span1,.table th.span1{float:none;width:44px;margin-left:0}
.table td.span2,.table th.span2{float:none;width:124px;margin-left:0}
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.table td.span8,.table th.span8{float:none;width:604px;margin-left:0}
.table td.span9,.table th.span9{float:none;width:684px;margin-left:0}
.table td.span10,.table th.span10{float:none;width:764px;margin-left:0}
.table td.span11,.table th.span11{float:none;width:844px;margin-left:0}
.table td.span12,.table th.span12{float:none;width:924px;margin-left:0}
.table tbody tr.success>td{background-color:#dff0d8}
.table tbody tr.error>td{background-color:#f2dede}
.table tbody tr.warning>td{background-color:#fcf8e3}
.table tbody tr.info>td{background-color:#d9edf7}
.table-hover tbody tr.success:hover>td{background-color:#d0e9c6}
.table-hover tbody tr.error:hover>td{background-color:#ebcccc}
.table-hover tbody tr.warning:hover>td{background-color:#faf2cc}
.table-hover tbody tr.info:hover>td{background-color:#c4e3f3}
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.icon-white,.nav-pills>.active>a>[class^="icon-"],.nav-pills>.active>a>[class*=" icon-"],.nav-list>.active>a>[class^="icon-"],.nav-list>.active>a>[class*=" icon-"],.navbar-inverse .nav>.active>a>[class^="icon-"],.navbar-inverse .nav>.active>a>[class*=" icon-"],.dropdown-menu>li>a:hover>[class^="icon-"],.dropdown-menu>li>a:focus>[class^="icon-"],.dropdown-menu>li>a:hover>[class*=" icon-"],.dropdown-menu>li>a:focus>[class*=" icon-"],.dropdown-menu>.active>a>[class^="icon-"],.dropdown-menu>.active>a>[class*=" icon-"],.dropdown-submenu:hover>a>[class^="icon-"],.dropdown-submenu:focus>a>[class^="icon-"],.dropdown-submenu:hover>a>[class*=" icon-"],.dropdown-submenu:focus>a>[class*=" icon-"]{background-image:url("../img/glyphicons-halflings-white.png")}
.icon-glass{background-position:0 0}
.icon-music{background-position:-24px 0}
.icon-search{background-position:-48px 0}
.icon-envelope{background-position:-72px 0}
.icon-heart{background-position:-96px 0}
.icon-star{background-position:-120px 0}
.icon-star-empty{background-position:-144px 0}
.icon-user{background-position:-168px 0}
.icon-film{background-position:-192px 0}
.icon-th-large{background-position:-216px 0}
.icon-th{background-position:-240px 0}
.icon-th-list{background-position:-264px 0}
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.icon-zoom-in{background-position:-336px 0}
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.icon-off{background-position:-384px 0}
.icon-signal{background-position:-408px 0}
.icon-cog{background-position:-432px 0}
.icon-trash{background-position:-456px 0}
.icon-home{background-position:0 -24px}
.icon-file{background-position:-24px -24px}
.icon-time{background-position:-48px -24px}
.icon-road{background-position:-72px -24px}
.icon-download-alt{background-position:-96px -24px}
.icon-download{background-position:-120px -24px}
.icon-upload{background-position:-144px -24px}
.icon-inbox{background-position:-168px -24px}
.icon-play-circle{background-position:-192px -24px}
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.icon-lock{background-position:-287px -24px}
.icon-flag{background-position:-312px -24px}
.icon-headphones{background-position:-336px -24px}
.icon-volume-off{background-position:-360px -24px}
.icon-volume-down{background-position:-384px -24px}
.icon-volume-up{background-position:-408px -24px}
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.icon-barcode{background-position:-456px -24px}
.icon-tag{background-position:0 -48px}
.icon-tags{background-position:-25px -48px}
.icon-book{background-position:-48px -48px}
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.icon-exclamation-sign{background-position:0 -120px}
.icon-gift{background-position:-24px -120px}
.icon-leaf{background-position:-48px -120px}
.icon-fire{background-position:-72px -120px}
.icon-eye-open{background-position:-96px -120px}
.icon-eye-close{background-position:-120px -120px}
.icon-warning-sign{background-position:-144px -120px}
.icon-plane{background-position:-168px -120px}
.icon-calendar{background-position:-192px -120px}
.icon-random{background-position:-216px -120px;width:16px}
.icon-comment{background-position:-240px -120px}
.icon-magnet{background-position:-264px -120px}
.icon-chevron-up{background-position:-288px -120px}
.icon-chevron-down{background-position:-313px -119px}
.icon-retweet{background-position:-336px -120px}
.icon-shopping-cart{background-position:-360px -120px}
.icon-folder-close{background-position:-384px -120px;width:16px}
.icon-folder-open{background-position:-408px -120px;width:16px}
.icon-resize-vertical{background-position:-432px -119px}
.icon-resize-horizontal{background-position:-456px -118px}
.icon-hdd{background-position:0 -144px}
.icon-bullhorn{background-position:-24px -144px}
.icon-bell{background-position:-48px -144px}
.icon-certificate{background-position:-72px -144px}
.icon-thumbs-up{background-position:-96px -144px}
.icon-thumbs-down{background-position:-120px -144px}
.icon-hand-right{background-position:-144px -144px}
.icon-hand-left{background-position:-168px -144px}
.icon-hand-up{background-position:-192px -144px}
.icon-hand-down{background-position:-216px -144px}
.icon-circle-arrow-right{background-position:-240px -144px}
.icon-circle-arrow-left{background-position:-264px -144px}
.icon-circle-arrow-up{background-position:-288px -144px}
.icon-circle-arrow-down{background-position:-312px -144px}
.icon-globe{background-position:-336px -144px}
.icon-wrench{background-position:-360px -144px}
.icon-tasks{background-position:-384px -144px}
.icon-filter{background-position:-408px -144px}
.icon-briefcase{background-position:-432px -144px}
.icon-fullscreen{background-position:-456px -144px}
.dropup,.dropdown{position:relative}
.dropdown-toggle{*margin-bottom:-3px}
.dropdown-toggle:active,.open .dropdown-toggle{outline:0}
.caret{display:inline-block;width:0;height:0;vertical-align:top;border-top:4px solid #000;border-right:4px solid transparent;border-left:4px solid transparent;content:""}
.dropdown .caret{margin-top:8px;margin-left:2px}
.dropdown-menu{position:absolute;top:100%;left:0;z-index:1000;display:none;float:left;min-width:160px;padding:5px 0;margin:2px 0 0;list-style:none;background-color:#fff;border:1px solid #ccc;border:1px solid rgba(0,0,0,0.2);*border-right-width:2px;*border-bottom-width:2px;-webkit-border-radius:6px;-moz-border-radius:6px;border-radius:6px;-webkit-box-shadow:0 5px 10px rgba(0,0,0,0.2);-moz-box-shadow:0 5px 10px rgba(0,0,0,0.2);box-shadow:0 5px 10px rgba(0,0,0,0.2);-webkit-background-clip:padding-box;-moz-background-clip:padding;background-clip:padding-box}.dropdown-menu.pull-right{right:0;left:auto}
.dropdown-menu .divider{*width:100%;height:1px;margin:9px 1px;*margin:-5px 0 5px;overflow:hidden;background-color:#e5e5e5;border-bottom:1px solid #fff}
.dropdown-menu>li>a{display:block;padding:3px 20px;clear:both;font-weight:normal;line-height:20px;color:#333;white-space:nowrap}
.dropdown-menu>li>a:hover,.dropdown-menu>li>a:focus,.dropdown-submenu:hover>a,.dropdown-submenu:focus>a{text-decoration:none;color:#fff;background-color:#0081c2;background-image:-moz-linear-gradient(top, #08c, #0077b3);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#08c), to(#0077b3));background-image:-webkit-linear-gradient(top, #08c, #0077b3);background-image:-o-linear-gradient(top, #08c, #0077b3);background-image:linear-gradient(to bottom, #08c, #0077b3);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff0088cc', endColorstr='#ff0077b3', GradientType=0)}
.dropdown-menu>.active>a,.dropdown-menu>.active>a:hover,.dropdown-menu>.active>a:focus{color:#fff;text-decoration:none;outline:0;background-color:#0081c2;background-image:-moz-linear-gradient(top, #08c, #0077b3);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#08c), to(#0077b3));background-image:-webkit-linear-gradient(top, #08c, #0077b3);background-image:-o-linear-gradient(top, #08c, #0077b3);background-image:linear-gradient(to bottom, #08c, #0077b3);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff0088cc', endColorstr='#ff0077b3', GradientType=0)}
.dropdown-menu>.disabled>a,.dropdown-menu>.disabled>a:hover,.dropdown-menu>.disabled>a:focus{color:#999}
.dropdown-menu>.disabled>a:hover,.dropdown-menu>.disabled>a:focus{text-decoration:none;background-color:transparent;background-image:none;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false);cursor:default}
.open{*z-index:1000}.open>.dropdown-menu{display:block}
.dropdown-backdrop{position:fixed;left:0;right:0;bottom:0;top:0;z-index:990}
.pull-right>.dropdown-menu{right:0;left:auto}
.dropup .caret,.navbar-fixed-bottom .dropdown .caret{border-top:0;border-bottom:4px solid #000;content:""}
.dropup .dropdown-menu,.navbar-fixed-bottom .dropdown .dropdown-menu{top:auto;bottom:100%;margin-bottom:1px}
.dropdown-submenu{position:relative}
.dropdown-submenu>.dropdown-menu{top:0;left:100%;margin-top:-6px;margin-left:-1px;border-radius:0 6px 6px 6px;-webkit-border-radius:0 6px 6px 6px;-moz-border-radius:0 6px 6px 6px;border-radius:0 6px 6px 6px}
.dropdown-submenu:hover>.dropdown-menu{display:block}
.dropup .dropdown-submenu>.dropdown-menu{top:auto;bottom:0;margin-top:0;margin-bottom:-2px;border-radius:5px 5px 5px 0;-webkit-border-radius:5px 5px 5px 0;-moz-border-radius:5px 5px 5px 0;border-radius:5px 5px 5px 0}
.dropdown-submenu>a:after{display:block;content:" ";float:right;width:0;height:0;border-color:transparent;border-style:solid;border-width:5px 0 5px 5px;border-left-color:#ccc;margin-top:5px;margin-right:-10px}
.dropdown-submenu:hover>a:after{border-left-color:#fff}
.dropdown-submenu.pull-left{float:none}.dropdown-submenu.pull-left>.dropdown-menu{left:-100%;margin-left:10px;border-radius:6px 0 6px 6px;-webkit-border-radius:6px 0 6px 6px;-moz-border-radius:6px 0 6px 6px;border-radius:6px 0 6px 6px}
.dropdown .dropdown-menu .nav-header{padding-left:20px;padding-right:20px}
.typeahead{z-index:1051;margin-top:2px;border-radius:4px;-webkit-border-radius:4px;-moz-border-radius:4px;border-radius:4px}
.well{min-height:20px;padding:19px;margin-bottom:20px;background-color:#f5f5f5;border:1px solid #e3e3e3;-webkit-border-radius:4px;-moz-border-radius:4px;border-radius:4px;-webkit-box-shadow:inset 0 1px 1px rgba(0,0,0,0.05);-moz-box-shadow:inset 0 1px 1px rgba(0,0,0,0.05);box-shadow:inset 0 1px 1px rgba(0,0,0,0.05)}.well blockquote{border-color:#ddd;border-color:rgba(0,0,0,0.15)}
.well-large{padding:24px;border-radius:6px;-webkit-border-radius:6px;-moz-border-radius:6px;border-radius:6px}
.well-small{padding:9px;border-radius:3px;-webkit-border-radius:3px;-moz-border-radius:3px;border-radius:3px}
.fade{opacity:0;-webkit-transition:opacity .15s linear;-moz-transition:opacity .15s linear;-o-transition:opacity .15s linear;transition:opacity .15s linear}.fade.in{opacity:1}
.collapse{position:relative;height:0;overflow:hidden;-webkit-transition:height .35s ease;-moz-transition:height .35s ease;-o-transition:height .35s ease;transition:height .35s ease}.collapse.in{height:auto}
.close{float:right;font-size:20px;font-weight:bold;line-height:20px;color:#000;text-shadow:0 1px 0 #fff;opacity:.2;filter:alpha(opacity=20)}.close:hover,.close:focus{color:#000;text-decoration:none;cursor:pointer;opacity:.4;filter:alpha(opacity=40)}
button.close{padding:0;cursor:pointer;background:transparent;border:0;-webkit-appearance:none}
.btn{display:inline-block;*display:inline;*zoom:1;padding:4px 12px;margin-bottom:0;font-size:13px;line-height:20px;text-align:center;vertical-align:middle;cursor:pointer;color:#333;text-shadow:0 1px 1px rgba(255,255,255,0.75);background-color:#f5f5f5;background-image:-moz-linear-gradient(top, #fff, #e6e6e6);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#fff), to(#e6e6e6));background-image:-webkit-linear-gradient(top, #fff, #e6e6e6);background-image:-o-linear-gradient(top, #fff, #e6e6e6);background-image:linear-gradient(to bottom, #fff, #e6e6e6);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffffffff', endColorstr='#ffe6e6e6', GradientType=0);border-color:#e6e6e6 #e6e6e6 #bfbfbf;border-color:rgba(0,0,0,0.1) rgba(0,0,0,0.1) rgba(0,0,0,0.25);*background-color:#e6e6e6;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false);border:1px solid #ccc;*border:0;border-bottom-color:#b3b3b3;-webkit-border-radius:4px;-moz-border-radius:4px;border-radius:4px;*margin-left:.3em;-webkit-box-shadow:inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);-moz-box-shadow:inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);box-shadow:inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05)}.btn:hover,.btn:focus,.btn:active,.btn.active,.btn.disabled,.btn[disabled]{color:#333;background-color:#e6e6e6;*background-color:#d9d9d9}
.btn:active,.btn.active{background-color:#ccc \9}
.btn:first-child{*margin-left:0}
.btn:hover,.btn:focus{color:#333;text-decoration:none;background-position:0 -15px;-webkit-transition:background-position .1s linear;-moz-transition:background-position .1s linear;-o-transition:background-position .1s linear;transition:background-position .1s linear}
.btn:focus{outline:thin dotted #333;outline:5px auto -webkit-focus-ring-color;outline-offset:-2px}
.btn.active,.btn:active{background-image:none;outline:0;-webkit-box-shadow:inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05);-moz-box-shadow:inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05);box-shadow:inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05)}
.btn.disabled,.btn[disabled]{cursor:default;background-image:none;opacity:.65;filter:alpha(opacity=65);-webkit-box-shadow:none;-moz-box-shadow:none;box-shadow:none}
.btn-large{padding:11px 19px;font-size:16.25px;border-radius:6px;-webkit-border-radius:6px;-moz-border-radius:6px;border-radius:6px}
.btn-large [class^="icon-"],.btn-large [class*=" icon-"]{margin-top:4px}
.btn-small{padding:2px 10px;font-size:11.049999999999999px;border-radius:3px;-webkit-border-radius:3px;-moz-border-radius:3px;border-radius:3px}
.btn-small [class^="icon-"],.btn-small [class*=" icon-"]{margin-top:0}
.btn-mini [class^="icon-"],.btn-mini [class*=" icon-"]{margin-top:-1px}
.btn-mini{padding:0 6px;font-size:9.75px;border-radius:3px;-webkit-border-radius:3px;-moz-border-radius:3px;border-radius:3px}
.btn-block{display:block;width:100%;padding-left:0;padding-right:0;-webkit-box-sizing:border-box;-moz-box-sizing:border-box;box-sizing:border-box}
.btn-block+.btn-block{margin-top:5px}
input[type="submit"].btn-block,input[type="reset"].btn-block,input[type="button"].btn-block{width:100%}
.btn-primary.active,.btn-warning.active,.btn-danger.active,.btn-success.active,.btn-info.active,.btn-inverse.active{color:rgba(255,255,255,0.75)}
.btn-primary{color:#fff;text-shadow:0 -1px 0 rgba(0,0,0,0.25);background-color:#006dcc;background-image:-moz-linear-gradient(top, #08c, #04c);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#08c), to(#04c));background-image:-webkit-linear-gradient(top, #08c, #04c);background-image:-o-linear-gradient(top, #08c, #04c);background-image:linear-gradient(to bottom, #08c, #04c);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff0088cc', endColorstr='#ff0044cc', GradientType=0);border-color:#04c #04c #002a80;border-color:rgba(0,0,0,0.1) rgba(0,0,0,0.1) rgba(0,0,0,0.25);*background-color:#04c;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false)}.btn-primary:hover,.btn-primary:focus,.btn-primary:active,.btn-primary.active,.btn-primary.disabled,.btn-primary[disabled]{color:#fff;background-color:#04c;*background-color:#003bb3}
.btn-primary:active,.btn-primary.active{background-color:#039 \9}
.btn-warning{color:#fff;text-shadow:0 -1px 0 rgba(0,0,0,0.25);background-color:#faa732;background-image:-moz-linear-gradient(top, #fbb450, #f89406);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#fbb450), to(#f89406));background-image:-webkit-linear-gradient(top, #fbb450, #f89406);background-image:-o-linear-gradient(top, #fbb450, #f89406);background-image:linear-gradient(to bottom, #fbb450, #f89406);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#fffbb450', endColorstr='#fff89406', GradientType=0);border-color:#f89406 #f89406 #ad6704;border-color:rgba(0,0,0,0.1) rgba(0,0,0,0.1) rgba(0,0,0,0.25);*background-color:#f89406;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false)}.btn-warning:hover,.btn-warning:focus,.btn-warning:active,.btn-warning.active,.btn-warning.disabled,.btn-warning[disabled]{color:#fff;background-color:#f89406;*background-color:#df8505}
.btn-warning:active,.btn-warning.active{background-color:#c67605 \9}
.btn-danger{color:#fff;text-shadow:0 -1px 0 rgba(0,0,0,0.25);background-color:#da4f49;background-image:-moz-linear-gradient(top, #ee5f5b, #bd362f);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#ee5f5b), to(#bd362f));background-image:-webkit-linear-gradient(top, #ee5f5b, #bd362f);background-image:-o-linear-gradient(top, #ee5f5b, #bd362f);background-image:linear-gradient(to bottom, #ee5f5b, #bd362f);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffee5f5b', endColorstr='#ffbd362f', GradientType=0);border-color:#bd362f #bd362f #802420;border-color:rgba(0,0,0,0.1) rgba(0,0,0,0.1) rgba(0,0,0,0.25);*background-color:#bd362f;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false)}.btn-danger:hover,.btn-danger:focus,.btn-danger:active,.btn-danger.active,.btn-danger.disabled,.btn-danger[disabled]{color:#fff;background-color:#bd362f;*background-color:#a9302a}
.btn-danger:active,.btn-danger.active{background-color:#942a25 \9}
.btn-success{color:#fff;text-shadow:0 -1px 0 rgba(0,0,0,0.25);background-color:#5bb75b;background-image:-moz-linear-gradient(top, #62c462, #51a351);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#62c462), to(#51a351));background-image:-webkit-linear-gradient(top, #62c462, #51a351);background-image:-o-linear-gradient(top, #62c462, #51a351);background-image:linear-gradient(to bottom, #62c462, #51a351);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff62c462', endColorstr='#ff51a351', GradientType=0);border-color:#51a351 #51a351 #387038;border-color:rgba(0,0,0,0.1) rgba(0,0,0,0.1) rgba(0,0,0,0.25);*background-color:#51a351;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false)}.btn-success:hover,.btn-success:focus,.btn-success:active,.btn-success.active,.btn-success.disabled,.btn-success[disabled]{color:#fff;background-color:#51a351;*background-color:#499249}
.btn-success:active,.btn-success.active{background-color:#408140 \9}
.btn-info{color:#fff;text-shadow:0 -1px 0 rgba(0,0,0,0.25);background-color:#49afcd;background-image:-moz-linear-gradient(top, #5bc0de, #2f96b4);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#5bc0de), to(#2f96b4));background-image:-webkit-linear-gradient(top, #5bc0de, #2f96b4);background-image:-o-linear-gradient(top, #5bc0de, #2f96b4);background-image:linear-gradient(to bottom, #5bc0de, #2f96b4);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff5bc0de', endColorstr='#ff2f96b4', GradientType=0);border-color:#2f96b4 #2f96b4 #1f6377;border-color:rgba(0,0,0,0.1) rgba(0,0,0,0.1) rgba(0,0,0,0.25);*background-color:#2f96b4;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false)}.btn-info:hover,.btn-info:focus,.btn-info:active,.btn-info.active,.btn-info.disabled,.btn-info[disabled]{color:#fff;background-color:#2f96b4;*background-color:#2a85a0}
.btn-info:active,.btn-info.active{background-color:#24748c \9}
.btn-inverse{color:#fff;text-shadow:0 -1px 0 rgba(0,0,0,0.25);background-color:#363636;background-image:-moz-linear-gradient(top, #444, #222);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#444), to(#222));background-image:-webkit-linear-gradient(top, #444, #222);background-image:-o-linear-gradient(top, #444, #222);background-image:linear-gradient(to bottom, #444, #222);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff444444', endColorstr='#ff222222', GradientType=0);border-color:#222 #222 #000;border-color:rgba(0,0,0,0.1) rgba(0,0,0,0.1) rgba(0,0,0,0.25);*background-color:#222;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false)}.btn-inverse:hover,.btn-inverse:focus,.btn-inverse:active,.btn-inverse.active,.btn-inverse.disabled,.btn-inverse[disabled]{color:#fff;background-color:#222;*background-color:#151515}
.btn-inverse:active,.btn-inverse.active{background-color:#080808 \9}
button.btn,input[type="submit"].btn{*padding-top:3px;*padding-bottom:3px}button.btn::-moz-focus-inner,input[type="submit"].btn::-moz-focus-inner{padding:0;border:0}
button.btn.btn-large,input[type="submit"].btn.btn-large{*padding-top:7px;*padding-bottom:7px}
button.btn.btn-small,input[type="submit"].btn.btn-small{*padding-top:3px;*padding-bottom:3px}
button.btn.btn-mini,input[type="submit"].btn.btn-mini{*padding-top:1px;*padding-bottom:1px}
.btn-link,.btn-link:active,.btn-link[disabled]{background-color:transparent;background-image:none;-webkit-box-shadow:none;-moz-box-shadow:none;box-shadow:none}
.btn-link{border-color:transparent;cursor:pointer;color:#08c;border-radius:0;-webkit-border-radius:0;-moz-border-radius:0;border-radius:0}
.btn-link:hover,.btn-link:focus{color:#005580;text-decoration:underline;background-color:transparent}
.btn-link[disabled]:hover,.btn-link[disabled]:focus{color:#333;text-decoration:none}
.btn-group{position:relative;display:inline-block;*display:inline;*zoom:1;font-size:0;vertical-align:middle;white-space:nowrap;*margin-left:.3em}.btn-group:first-child{*margin-left:0}
.btn-group+.btn-group{margin-left:5px}
.btn-toolbar{font-size:0;margin-top:10px;margin-bottom:10px}.btn-toolbar>.btn+.btn,.btn-toolbar>.btn-group+.btn,.btn-toolbar>.btn+.btn-group{margin-left:5px}
.btn-group>.btn{position:relative;border-radius:0;-webkit-border-radius:0;-moz-border-radius:0;border-radius:0}
.btn-group>.btn+.btn{margin-left:-1px}
.btn-group>.btn,.btn-group>.dropdown-menu,.btn-group>.popover{font-size:13px}
.btn-group>.btn-mini{font-size:9.75px}
.btn-group>.btn-small{font-size:11.049999999999999px}
.btn-group>.btn-large{font-size:16.25px}
.btn-group>.btn:first-child{margin-left:0;-webkit-border-top-left-radius:4px;-moz-border-radius-topleft:4px;border-top-left-radius:4px;-webkit-border-bottom-left-radius:4px;-moz-border-radius-bottomleft:4px;border-bottom-left-radius:4px}
.btn-group>.btn:last-child,.btn-group>.dropdown-toggle{-webkit-border-top-right-radius:4px;-moz-border-radius-topright:4px;border-top-right-radius:4px;-webkit-border-bottom-right-radius:4px;-moz-border-radius-bottomright:4px;border-bottom-right-radius:4px}
.btn-group>.btn.large:first-child{margin-left:0;-webkit-border-top-left-radius:6px;-moz-border-radius-topleft:6px;border-top-left-radius:6px;-webkit-border-bottom-left-radius:6px;-moz-border-radius-bottomleft:6px;border-bottom-left-radius:6px}
.btn-group>.btn.large:last-child,.btn-group>.large.dropdown-toggle{-webkit-border-top-right-radius:6px;-moz-border-radius-topright:6px;border-top-right-radius:6px;-webkit-border-bottom-right-radius:6px;-moz-border-radius-bottomright:6px;border-bottom-right-radius:6px}
.btn-group>.btn:hover,.btn-group>.btn:focus,.btn-group>.btn:active,.btn-group>.btn.active{z-index:2}
.btn-group .dropdown-toggle:active,.btn-group.open .dropdown-toggle{outline:0}
.btn-group>.btn+.dropdown-toggle{padding-left:8px;padding-right:8px;-webkit-box-shadow:inset 1px 0 0 rgba(255,255,255,.125), inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);-moz-box-shadow:inset 1px 0 0 rgba(255,255,255,.125), inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);box-shadow:inset 1px 0 0 rgba(255,255,255,.125), inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);*padding-top:5px;*padding-bottom:5px}
.btn-group>.btn-mini+.dropdown-toggle{padding-left:5px;padding-right:5px;*padding-top:2px;*padding-bottom:2px}
.btn-group>.btn-small+.dropdown-toggle{*padding-top:5px;*padding-bottom:4px}
.btn-group>.btn-large+.dropdown-toggle{padding-left:12px;padding-right:12px;*padding-top:7px;*padding-bottom:7px}
.btn-group.open .dropdown-toggle{background-image:none;-webkit-box-shadow:inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05);-moz-box-shadow:inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05);box-shadow:inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05)}
.btn-group.open .btn.dropdown-toggle{background-color:#e6e6e6}
.btn-group.open .btn-primary.dropdown-toggle{background-color:#04c}
.btn-group.open .btn-warning.dropdown-toggle{background-color:#f89406}
.btn-group.open .btn-danger.dropdown-toggle{background-color:#bd362f}
.btn-group.open .btn-success.dropdown-toggle{background-color:#51a351}
.btn-group.open .btn-info.dropdown-toggle{background-color:#2f96b4}
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.btn-large .caret{border-left-width:5px;border-right-width:5px;border-top-width:5px}
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.alert{padding:8px 35px 8px 14px;margin-bottom:20px;text-shadow:0 1px 0 rgba(255,255,255,0.5);background-color:#fcf8e3;border:1px solid #fbeed5;border-radius:4px;-webkit-border-radius:4px;-moz-border-radius:4px;border-radius:4px}
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.navbar .btn-navbar{display:none;float:right;padding:7px 10px;margin-left:5px;margin-right:5px;color:#fff;text-shadow:0 -1px 0 rgba(0,0,0,0.25);background-color:#ededed;background-image:-moz-linear-gradient(top, #f2f2f2, #e5e5e5);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#f2f2f2), to(#e5e5e5));background-image:-webkit-linear-gradient(top, #f2f2f2, #e5e5e5);background-image:-o-linear-gradient(top, #f2f2f2, #e5e5e5);background-image:linear-gradient(to bottom, #f2f2f2, #e5e5e5);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#fff2f2f2', endColorstr='#ffe5e5e5', GradientType=0);border-color:#e5e5e5 #e5e5e5 #bfbfbf;border-color:rgba(0,0,0,0.1) rgba(0,0,0,0.1) rgba(0,0,0,0.25);*background-color:#e5e5e5;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false);-webkit-box-shadow:inset 0 1px 0 rgba(255,255,255,.1), 0 1px 0 rgba(255,255,255,.075);-moz-box-shadow:inset 0 1px 0 rgba(255,255,255,.1), 0 1px 0 rgba(255,255,255,.075);box-shadow:inset 0 1px 0 rgba(255,255,255,.1), 0 1px 0 rgba(255,255,255,.075)}.navbar .btn-navbar:hover,.navbar .btn-navbar:focus,.navbar .btn-navbar:active,.navbar .btn-navbar.active,.navbar .btn-navbar.disabled,.navbar .btn-navbar[disabled]{color:#fff;background-color:#e5e5e5;*background-color:#d9d9d9}
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.btn-navbar .icon-bar+.icon-bar{margin-top:3px}
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.navbar-inverse .navbar-search .search-query::-webkit-input-placeholder{color:#ccc}
.navbar-inverse .navbar-search .search-query:focus,.navbar-inverse .navbar-search .search-query.focused{padding:5px 15px;color:#333;text-shadow:0 1px 0 #fff;background-color:#fff;border:0;-webkit-box-shadow:0 0 3px rgba(0,0,0,0.15);-moz-box-shadow:0 0 3px rgba(0,0,0,0.15);box-shadow:0 0 3px rgba(0,0,0,0.15);outline:0}
.navbar-inverse .btn-navbar{color:#fff;text-shadow:0 -1px 0 rgba(0,0,0,0.25);background-color:#0e0e0e;background-image:-moz-linear-gradient(top, #151515, #040404);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#151515), to(#040404));background-image:-webkit-linear-gradient(top, #151515, #040404);background-image:-o-linear-gradient(top, #151515, #040404);background-image:linear-gradient(to bottom, #151515, #040404);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff151515', endColorstr='#ff040404', GradientType=0);border-color:#040404 #040404 #000;border-color:rgba(0,0,0,0.1) rgba(0,0,0,0.1) rgba(0,0,0,0.25);*background-color:#040404;filter:progid:DXImageTransform.Microsoft.gradient(enabled = false)}.navbar-inverse .btn-navbar:hover,.navbar-inverse .btn-navbar:focus,.navbar-inverse .btn-navbar:active,.navbar-inverse .btn-navbar.active,.navbar-inverse .btn-navbar.disabled,.navbar-inverse .btn-navbar[disabled]{color:#fff;background-color:#040404;*background-color:#000}
.navbar-inverse .btn-navbar:active,.navbar-inverse .btn-navbar.active{background-color:#000 \9}
.breadcrumb{padding:8px 15px;margin:0 0 20px;list-style:none;background-color:#f5f5f5;border-radius:4px;-webkit-border-radius:4px;-moz-border-radius:4px;border-radius:4px}.breadcrumb>li{display:inline-block;*display:inline;*zoom:1;text-shadow:0 1px 0 #fff}.breadcrumb>li>.divider{padding:0 5px;color:#ccc}
.breadcrumb>.active{color:#999}
.pagination{margin:20px 0}
.pagination ul{display:inline-block;*display:inline;*zoom:1;margin-left:0;margin-bottom:0;-webkit-border-radius:4px;-moz-border-radius:4px;border-radius:4px;-webkit-box-shadow:0 1px 2px rgba(0,0,0,0.05);-moz-box-shadow:0 1px 2px rgba(0,0,0,0.05);box-shadow:0 1px 2px rgba(0,0,0,0.05)}
.pagination ul>li{display:inline}
.pagination ul>li>a,.pagination ul>li>span{float:left;padding:4px 12px;line-height:20px;text-decoration:none;background-color:#fff;border:1px solid #ddd;border-left-width:0}
.pagination ul>li>a:hover,.pagination ul>li>a:focus,.pagination ul>.active>a,.pagination ul>.active>span{background-color:#f5f5f5}
.pagination ul>.active>a,.pagination ul>.active>span{color:#999;cursor:default}
.pagination ul>.disabled>span,.pagination ul>.disabled>a,.pagination ul>.disabled>a:hover,.pagination ul>.disabled>a:focus{color:#999;background-color:transparent;cursor:default}
.pagination ul>li:first-child>a,.pagination ul>li:first-child>span{border-left-width:1px;-webkit-border-top-left-radius:4px;-moz-border-radius-topleft:4px;border-top-left-radius:4px;-webkit-border-bottom-left-radius:4px;-moz-border-radius-bottomleft:4px;border-bottom-left-radius:4px}
.pagination ul>li:last-child>a,.pagination ul>li:last-child>span{-webkit-border-top-right-radius:4px;-moz-border-radius-topright:4px;border-top-right-radius:4px;-webkit-border-bottom-right-radius:4px;-moz-border-radius-bottomright:4px;border-bottom-right-radius:4px}
.pagination-centered{text-align:center}
.pagination-right{text-align:right}
.pagination-large ul>li>a,.pagination-large ul>li>span{padding:11px 19px;font-size:16.25px}
.pagination-large ul>li:first-child>a,.pagination-large ul>li:first-child>span{-webkit-border-top-left-radius:6px;-moz-border-radius-topleft:6px;border-top-left-radius:6px;-webkit-border-bottom-left-radius:6px;-moz-border-radius-bottomleft:6px;border-bottom-left-radius:6px}
.pagination-large ul>li:last-child>a,.pagination-large ul>li:last-child>span{-webkit-border-top-right-radius:6px;-moz-border-radius-topright:6px;border-top-right-radius:6px;-webkit-border-bottom-right-radius:6px;-moz-border-radius-bottomright:6px;border-bottom-right-radius:6px}
.pagination-mini ul>li:first-child>a,.pagination-small ul>li:first-child>a,.pagination-mini ul>li:first-child>span,.pagination-small ul>li:first-child>span{-webkit-border-top-left-radius:3px;-moz-border-radius-topleft:3px;border-top-left-radius:3px;-webkit-border-bottom-left-radius:3px;-moz-border-radius-bottomleft:3px;border-bottom-left-radius:3px}
.pagination-mini ul>li:last-child>a,.pagination-small ul>li:last-child>a,.pagination-mini ul>li:last-child>span,.pagination-small ul>li:last-child>span{-webkit-border-top-right-radius:3px;-moz-border-radius-topright:3px;border-top-right-radius:3px;-webkit-border-bottom-right-radius:3px;-moz-border-radius-bottomright:3px;border-bottom-right-radius:3px}
.pagination-small ul>li>a,.pagination-small ul>li>span{padding:2px 10px;font-size:11.049999999999999px}
.pagination-mini ul>li>a,.pagination-mini ul>li>span{padding:0 6px;font-size:9.75px}
.pager{margin:20px 0;list-style:none;text-align:center;*zoom:1}.pager:before,.pager:after{display:table;content:"";line-height:0}
.pager:after{clear:both}
.pager li{display:inline}
.pager li>a,.pager li>span{display:inline-block;padding:5px 14px;background-color:#fff;border:1px solid #ddd;border-radius:15px;-webkit-border-radius:15px;-moz-border-radius:15px;border-radius:15px}
.pager li>a:hover,.pager li>a:focus{text-decoration:none;background-color:#f5f5f5}
.pager .next>a,.pager .next>span{float:right}
.pager .previous>a,.pager .previous>span{float:left}
.pager .disabled>a,.pager .disabled>a:hover,.pager .disabled>a:focus,.pager .disabled>span{color:#999;background-color:#fff;cursor:default}
.modal-backdrop{position:fixed;top:0;right:0;bottom:0;left:0;z-index:1040;background-color:#000}.modal-backdrop.fade{opacity:0}
.modal-backdrop,.modal-backdrop.fade.in{opacity:.8;filter:alpha(opacity=80)}
.modal{position:fixed;top:10%;left:50%;z-index:1050;width:560px;margin-left:-280px;background-color:#fff;border:1px solid #999;border:1px solid rgba(0,0,0,0.3);*border:1px solid #999;-webkit-border-radius:6px;-moz-border-radius:6px;border-radius:6px;-webkit-box-shadow:0 3px 7px rgba(0,0,0,0.3);-moz-box-shadow:0 3px 7px rgba(0,0,0,0.3);box-shadow:0 3px 7px rgba(0,0,0,0.3);-webkit-background-clip:padding-box;-moz-background-clip:padding-box;background-clip:padding-box;outline:none}.modal.fade{-webkit-transition:opacity .3s linear, top .3s ease-out;-moz-transition:opacity .3s linear, top .3s ease-out;-o-transition:opacity .3s linear, top .3s ease-out;transition:opacity .3s linear, top .3s ease-out;top:-25%}
.modal.fade.in{top:10%}
.modal-header{padding:9px 15px;border-bottom:1px solid #eee}.modal-header .close{margin-top:2px}
.modal-header h3{margin:0;line-height:30px}
.modal-body{position:relative;overflow-y:auto;max-height:400px;padding:15px}
.modal-form{margin-bottom:0}
.modal-footer{padding:14px 15px 15px;margin-bottom:0;text-align:right;background-color:#f5f5f5;border-top:1px solid #ddd;-webkit-border-radius:0 0 6px 6px;-moz-border-radius:0 0 6px 6px;border-radius:0 0 6px 6px;-webkit-box-shadow:inset 0 1px 0 #fff;-moz-box-shadow:inset 0 1px 0 #fff;box-shadow:inset 0 1px 0 #fff;*zoom:1}.modal-footer:before,.modal-footer:after{display:table;content:"";line-height:0}
.modal-footer:after{clear:both}
.modal-footer .btn+.btn{margin-left:5px;margin-bottom:0}
.modal-footer .btn-group .btn+.btn{margin-left:-1px}
.modal-footer .btn-block+.btn-block{margin-left:0}
.tooltip{position:absolute;z-index:1030;display:block;visibility:visible;font-size:11px;line-height:1.4;opacity:0;filter:alpha(opacity=0)}.tooltip.in{opacity:.8;filter:alpha(opacity=80)}
.tooltip.top{margin-top:-3px;padding:5px 0}
.tooltip.right{margin-left:3px;padding:0 5px}
.tooltip.bottom{margin-top:3px;padding:5px 0}
.tooltip.left{margin-left:-3px;padding:0 5px}
.tooltip-inner{max-width:200px;padding:8px;color:#fff;text-align:center;text-decoration:none;background-color:#000;border-radius:4px;-webkit-border-radius:4px;-moz-border-radius:4px;border-radius:4px}
.tooltip-arrow{position:absolute;width:0;height:0;border-color:transparent;border-style:solid}
.tooltip.top .tooltip-arrow{bottom:0;left:50%;margin-left:-5px;border-width:5px 5px 0;border-top-color:#000}
.tooltip.right .tooltip-arrow{top:50%;left:0;margin-top:-5px;border-width:5px 5px 5px 0;border-right-color:#000}
.tooltip.left .tooltip-arrow{top:50%;right:0;margin-top:-5px;border-width:5px 0 5px 5px;border-left-color:#000}
.tooltip.bottom .tooltip-arrow{top:0;left:50%;margin-left:-5px;border-width:0 5px 5px;border-bottom-color:#000}
.popover{position:absolute;top:0;left:0;z-index:1010;display:none;max-width:276px;padding:1px;text-align:left;background-color:#fff;-webkit-background-clip:padding-box;-moz-background-clip:padding;background-clip:padding-box;border:1px solid #ccc;border:1px solid rgba(0,0,0,0.2);-webkit-border-radius:6px;-moz-border-radius:6px;border-radius:6px;-webkit-box-shadow:0 5px 10px rgba(0,0,0,0.2);-moz-box-shadow:0 5px 10px rgba(0,0,0,0.2);box-shadow:0 5px 10px rgba(0,0,0,0.2);white-space:normal}.popover.top{margin-top:-10px}
.popover.right{margin-left:10px}
.popover.bottom{margin-top:10px}
.popover.left{margin-left:-10px}
.popover-title{margin:0;padding:8px 14px;font-size:14px;font-weight:normal;line-height:18px;background-color:#f7f7f7;border-bottom:1px solid #ebebeb;border-radius:5px 5px 0 0;-webkit-border-radius:5px 5px 0 0;-moz-border-radius:5px 5px 0 0;border-radius:5px 5px 0 0}.popover-title:empty{display:none}
.popover-content{padding:9px 14px}
.popover .arrow,.popover .arrow:after{position:absolute;display:block;width:0;height:0;border-color:transparent;border-style:solid}
.popover .arrow{border-width:11px}
.popover .arrow:after{border-width:10px;content:""}
.popover.top .arrow{left:50%;margin-left:-11px;border-bottom-width:0;border-top-color:#999;border-top-color:rgba(0,0,0,0.25);bottom:-11px}.popover.top .arrow:after{bottom:1px;margin-left:-10px;border-bottom-width:0;border-top-color:#fff}
.popover.right .arrow{top:50%;left:-11px;margin-top:-11px;border-left-width:0;border-right-color:#999;border-right-color:rgba(0,0,0,0.25)}.popover.right .arrow:after{left:1px;bottom:-10px;border-left-width:0;border-right-color:#fff}
.popover.bottom .arrow{left:50%;margin-left:-11px;border-top-width:0;border-bottom-color:#999;border-bottom-color:rgba(0,0,0,0.25);top:-11px}.popover.bottom .arrow:after{top:1px;margin-left:-10px;border-top-width:0;border-bottom-color:#fff}
.popover.left .arrow{top:50%;right:-11px;margin-top:-11px;border-right-width:0;border-left-color:#999;border-left-color:rgba(0,0,0,0.25)}.popover.left .arrow:after{right:1px;border-right-width:0;border-left-color:#fff;bottom:-10px}
.thumbnails{margin-left:-20px;list-style:none;*zoom:1}.thumbnails:before,.thumbnails:after{display:table;content:"";line-height:0}
.thumbnails:after{clear:both}
.row-fluid .thumbnails{margin-left:0}
.thumbnails>li{float:left;margin-bottom:20px;margin-left:20px}
.thumbnail{display:block;padding:4px;line-height:20px;border:1px solid #ddd;-webkit-border-radius:4px;-moz-border-radius:4px;border-radius:4px;-webkit-box-shadow:0 1px 3px rgba(0,0,0,0.055);-moz-box-shadow:0 1px 3px rgba(0,0,0,0.055);box-shadow:0 1px 3px rgba(0,0,0,0.055);-webkit-transition:all .2s ease-in-out;-moz-transition:all .2s ease-in-out;-o-transition:all .2s ease-in-out;transition:all .2s ease-in-out}
a.thumbnail:hover,a.thumbnail:focus{border-color:#08c;-webkit-box-shadow:0 1px 4px rgba(0,105,214,0.25);-moz-box-shadow:0 1px 4px rgba(0,105,214,0.25);box-shadow:0 1px 4px rgba(0,105,214,0.25)}
.thumbnail>img{display:block;max-width:100%;margin-left:auto;margin-right:auto}
.thumbnail .caption{padding:9px;color:#555}
.media,.media-body{overflow:hidden;*overflow:visible;zoom:1}
.media,.media .media{margin-top:15px}
.media:first-child{margin-top:0}
.media-object{display:block}
.media-heading{margin:0 0 5px}
.media>.pull-left{margin-right:10px}
.media>.pull-right{margin-left:10px}
.media-list{margin-left:0;list-style:none}
.label,.badge{display:inline-block;padding:2px 4px;font-size:10.998px;font-weight:bold;line-height:14px;color:#fff;vertical-align:baseline;white-space:nowrap;text-shadow:0 -1px 0 rgba(0,0,0,0.25);background-color:#999}
.label{border-radius:3px;-webkit-border-radius:3px;-moz-border-radius:3px;border-radius:3px}
.badge{padding-left:9px;padding-right:9px;border-radius:9px;-webkit-border-radius:9px;-moz-border-radius:9px;border-radius:9px}
.label:empty,.badge:empty{display:none}
a.label:hover,a.label:focus,a.badge:hover,a.badge:focus{color:#fff;text-decoration:none;cursor:pointer}
.label-important,.badge-important{background-color:#b94a48}
.label-important[href],.badge-important[href]{background-color:#953b39}
.label-warning,.badge-warning{background-color:#f89406}
.label-warning[href],.badge-warning[href]{background-color:#c67605}
.label-success,.badge-success{background-color:#468847}
.label-success[href],.badge-success[href]{background-color:#356635}
.label-info,.badge-info{background-color:#3a87ad}
.label-info[href],.badge-info[href]{background-color:#2d6987}
.label-inverse,.badge-inverse{background-color:#333}
.label-inverse[href],.badge-inverse[href]{background-color:#1a1a1a}
.btn .label,.btn .badge{position:relative;top:-1px}
.btn-mini .label,.btn-mini .badge{top:0}
@-webkit-keyframes progress-bar-stripes{from{background-position:40px 0} to{background-position:0 0}}@-moz-keyframes progress-bar-stripes{from{background-position:40px 0} to{background-position:0 0}}@-ms-keyframes progress-bar-stripes{from{background-position:40px 0} to{background-position:0 0}}@-o-keyframes progress-bar-stripes{from{background-position:0 0} to{background-position:40px 0}}@keyframes progress-bar-stripes{from{background-position:40px 0} to{background-position:0 0}}.progress{overflow:hidden;height:20px;margin-bottom:20px;background-color:#f7f7f7;background-image:-moz-linear-gradient(top, #f5f5f5, #f9f9f9);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#f5f5f5), to(#f9f9f9));background-image:-webkit-linear-gradient(top, #f5f5f5, #f9f9f9);background-image:-o-linear-gradient(top, #f5f5f5, #f9f9f9);background-image:linear-gradient(to bottom, #f5f5f5, #f9f9f9);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#fff5f5f5', endColorstr='#fff9f9f9', GradientType=0);-webkit-box-shadow:inset 0 1px 2px rgba(0,0,0,0.1);-moz-box-shadow:inset 0 1px 2px rgba(0,0,0,0.1);box-shadow:inset 0 1px 2px rgba(0,0,0,0.1);border-radius:4px;-webkit-border-radius:4px;-moz-border-radius:4px;border-radius:4px}
.progress .bar{width:0;height:100%;color:#fff;float:left;font-size:12px;text-align:center;text-shadow:0 -1px 0 rgba(0,0,0,0.25);background-color:#0e90d2;background-image:-moz-linear-gradient(top, #149bdf, #0480be);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#149bdf), to(#0480be));background-image:-webkit-linear-gradient(top, #149bdf, #0480be);background-image:-o-linear-gradient(top, #149bdf, #0480be);background-image:linear-gradient(to bottom, #149bdf, #0480be);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff149bdf', endColorstr='#ff0480be', GradientType=0);-webkit-box-shadow:inset 0 -1px 0 rgba(0,0,0,0.15);-moz-box-shadow:inset 0 -1px 0 rgba(0,0,0,0.15);box-shadow:inset 0 -1px 0 rgba(0,0,0,0.15);-webkit-box-sizing:border-box;-moz-box-sizing:border-box;box-sizing:border-box;-webkit-transition:width .6s ease;-moz-transition:width .6s ease;-o-transition:width .6s ease;transition:width .6s ease}
.progress .bar+.bar{-webkit-box-shadow:inset 1px 0 0 rgba(0,0,0,.15), inset 0 -1px 0 rgba(0,0,0,.15);-moz-box-shadow:inset 1px 0 0 rgba(0,0,0,.15), inset 0 -1px 0 rgba(0,0,0,.15);box-shadow:inset 1px 0 0 rgba(0,0,0,.15), inset 0 -1px 0 rgba(0,0,0,.15)}
.progress-striped .bar{background-color:#149bdf;background-image:-webkit-gradient(linear, 0 100%, 100% 0, color-stop(.25, rgba(255,255,255,0.15)), color-stop(.25, transparent), color-stop(.5, transparent), color-stop(.5, rgba(255,255,255,0.15)), color-stop(.75, rgba(255,255,255,0.15)), color-stop(.75, transparent), to(transparent));background-image:-webkit-linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);background-image:-moz-linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);background-image:-o-linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);background-image:linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);-webkit-background-size:40px 40px;-moz-background-size:40px 40px;-o-background-size:40px 40px;background-size:40px 40px}
.progress.active .bar{-webkit-animation:progress-bar-stripes 2s linear infinite;-moz-animation:progress-bar-stripes 2s linear infinite;-ms-animation:progress-bar-stripes 2s linear infinite;-o-animation:progress-bar-stripes 2s linear infinite;animation:progress-bar-stripes 2s linear infinite}
.progress-danger .bar,.progress .bar-danger{background-color:#dd514c;background-image:-moz-linear-gradient(top, #ee5f5b, #c43c35);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#ee5f5b), to(#c43c35));background-image:-webkit-linear-gradient(top, #ee5f5b, #c43c35);background-image:-o-linear-gradient(top, #ee5f5b, #c43c35);background-image:linear-gradient(to bottom, #ee5f5b, #c43c35);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffee5f5b', endColorstr='#ffc43c35', GradientType=0)}
.progress-danger.progress-striped .bar,.progress-striped .bar-danger{background-color:#ee5f5b;background-image:-webkit-gradient(linear, 0 100%, 100% 0, color-stop(.25, rgba(255,255,255,0.15)), color-stop(.25, transparent), color-stop(.5, transparent), color-stop(.5, rgba(255,255,255,0.15)), color-stop(.75, rgba(255,255,255,0.15)), color-stop(.75, transparent), to(transparent));background-image:-webkit-linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);background-image:-moz-linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);background-image:-o-linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);background-image:linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent)}
.progress-success .bar,.progress .bar-success{background-color:#5eb95e;background-image:-moz-linear-gradient(top, #62c462, #57a957);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#62c462), to(#57a957));background-image:-webkit-linear-gradient(top, #62c462, #57a957);background-image:-o-linear-gradient(top, #62c462, #57a957);background-image:linear-gradient(to bottom, #62c462, #57a957);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff62c462', endColorstr='#ff57a957', GradientType=0)}
.progress-success.progress-striped .bar,.progress-striped .bar-success{background-color:#62c462;background-image:-webkit-gradient(linear, 0 100%, 100% 0, color-stop(.25, rgba(255,255,255,0.15)), color-stop(.25, transparent), color-stop(.5, transparent), color-stop(.5, rgba(255,255,255,0.15)), color-stop(.75, rgba(255,255,255,0.15)), color-stop(.75, transparent), to(transparent));background-image:-webkit-linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);background-image:-moz-linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);background-image:-o-linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent);background-image:linear-gradient(45deg, rgba(255,255,255,0.15) 25%, transparent 25%, transparent 50%, rgba(255,255,255,0.15) 50%, rgba(255,255,255,0.15) 75%, transparent 75%, transparent)}
.progress-info .bar,.progress .bar-info{background-color:#4bb1cf;background-image:-moz-linear-gradient(top, #5bc0de, #339bb9);background-image:-webkit-gradient(linear, 0 0, 0 100%, from(#5bc0de), to(#339bb9));background-image:-webkit-linear-gradient(top, #5bc0de, #339bb9);background-image:-o-linear-gradient(top, #5bc0de, #339bb9);background-image:linear-gradient(to bottom, #5bc0de, #339bb9);background-repeat:repeat-x;filter:progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff5bc0de', endColorstr='#ff339bb9', GradientType=0)}
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.accordion-group{margin-bottom:2px;border:1px solid #e5e5e5;border-radius:4px;-webkit-border-radius:4px;-moz-border-radius:4px;border-radius:4px}
.accordion-heading{border-bottom:0}
.accordion-heading .accordion-toggle{display:block;padding:8px 15px}
.accordion-toggle{cursor:pointer}
.accordion-inner{padding:9px 15px;border-top:1px solid #e5e5e5}
.carousel{position:relative;margin-bottom:20px;line-height:1}
.carousel-inner{overflow:hidden;width:100%;position:relative}
.carousel-inner>.item{display:none;position:relative;-webkit-transition:.6s ease-in-out left;-moz-transition:.6s ease-in-out left;-o-transition:.6s ease-in-out left;transition:.6s ease-in-out left}.carousel-inner>.item>img,.carousel-inner>.item>a>img{display:block;line-height:1}
.carousel-inner>.active,.carousel-inner>.next,.carousel-inner>.prev{display:block}
.carousel-inner>.active{left:0}
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.carousel-inner>.active.left{left:-100%}
.carousel-inner>.active.right{left:100%}
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.carousel-indicators .active{background-color:#fff}
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.carousel-caption h4,.carousel-caption p{color:#fff;line-height:20px}
.carousel-caption h4{margin:0 0 5px}
.carousel-caption p{margin-bottom:0}
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.hero-unit li{line-height:30px}
.pull-right{float:right}
.pull-left{float:left}
.hide{display:none}
.show{display:block}
.invisible{visibility:hidden}
.affix{position:fixed}
@-ms-viewport{width:device-width}.hidden{display:none;visibility:hidden}
.visible-phone{display:none !important}
.visible-tablet{display:none !important}
.hidden-desktop{display:none !important}
.visible-desktop{display:inherit !important}
@media (min-width:768px) and (max-width:979px){.hidden-desktop{display:inherit !important} .visible-desktop{display:none !important} .visible-tablet{display:inherit !important} .hidden-tablet{display:none !important}}@media (max-width:767px){.hidden-desktop{display:inherit !important} .visible-desktop{display:none !important} .visible-phone{display:inherit !important} .hidden-phone{display:none !important}}.visible-print{display:none !important}
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span.raw_input_prompt{}
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input.raw_input:focus{box-shadow:none}
p.p-space{margin-bottom:10px}
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.rendered_html pre{margin:1em 2em}
.rendered_html pre,.rendered_html code{border:0;background-color:#fff;color:#000;font-size:100%;padding:0}
.rendered_html blockquote{margin:1em 2em}
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.rendered_html td,.rendered_html th{text-align:left;vertical-align:middle;padding:4px}
.rendered_html th{font-weight:bold}
.rendered_html *+table{margin-top:1em}
.rendered_html p{text-align:justify}
.rendered_html *+p{margin-top:1em}
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div.text_cell{padding:5px 5px 5px 0;display:-webkit-box;-webkit-box-orient:horizontal;-webkit-box-align:stretch;display:-moz-box;-moz-box-orient:horizontal;-moz-box-align:stretch;display:box;box-orient:horizontal;box-align:stretch;display:flex;flex-direction:row;align-items:stretch}
@media (max-width:480px){div.text_cell>div.prompt{display:none}}div.text_cell_render{outline:none;resize:none;width:inherit;border-style:none;padding:.5em .5em .5em .4em;color:#000}
a.anchor-link:link{text-decoration:none;padding:0 20px;visibility:hidden}
h1:hover .anchor-link,h2:hover .anchor-link,h3:hover .anchor-link,h4:hover .anchor-link,h5:hover .anchor-link,h6:hover .anchor-link{visibility:visible}
div.cell.text_cell.rendered{padding:0}
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.widget-vlabel{padding-bottom:5px;text-align:center;vertical-align:text-bottom}
.widget-hreadout{padding-left:8px;padding-top:3px;text-align:left;vertical-align:text-top}
.widget-vreadout{padding-top:5px;text-align:center;vertical-align:text-top}
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body{background-color:#fff}
body.notebook_app{overflow:hidden}
@media (max-width:767px){body.notebook_app{padding-left:0;padding-right:0}}span#notebook_name{height:1em;line-height:1em;padding:3px;border:none;font-size:146.5%}
div#notebook_panel{margin:0 0 0 0;padding:0;-webkit-box-shadow:0 -1px 10px rgba(0,0,0,0.1);-moz-box-shadow:0 -1px 10px rgba(0,0,0,0.1);box-shadow:0 -1px 10px rgba(0,0,0,0.1)}
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div.ui-widget-content{border:1px solid #ababab;outline:none}
pre.dialog{background-color:#f7f7f7;border:1px solid #ddd;border-radius:4px;padding:.4em;padding-left:2em}
p.dialog{padding:.2em}
pre,code,kbd,samp{white-space:pre-wrap}
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p{margin-bottom:0}
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#menubar .navbar{margin-bottom:8px}
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#menubar li.dropdown{line-height:12px}
i.menu-icon{padding-top:4px}
ul#help_menu li a{overflow:hidden;padding-right:2.2em}ul#help_menu li a i{margin-right:-1.2em}
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div#pager_splitter{height:8px}
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div#pager{font-size:14px;line-height:20px;overflow:auto;display:none}div#pager pre{line-height:1.21429em;color:#000;background-color:#f7f7f7;padding:.4em}
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span#save_widget{padding:0 5px;margin-top:12px}
span#checkpoint_status,span#autosave_status{font-size:small}
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.pretooltiparrow:before{background-color:#f7f7f7;border:1px #ababab solid;z-index:11;content:"";position:absolute;left:15px;top:10px;width:25px;height:25px;-webkit-transform:rotate(45deg);-moz-transform:rotate(45deg);-ms-transform:rotate(45deg);-o-transform:rotate(45deg)}
</style>
<style type="text/css">
.highlight .hll { background-color: #ffffcc }
.highlight { background: #f8f8f8; }
.highlight .c { color: #408080; font-style: italic } /* Comment */
.highlight .err { border: 1px solid #FF0000 } /* Error */
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</style>
<style type="text/css">
/* Overrides of notebook CSS for static HTML export */
html {
overflow-y: auto;
}
.reveal {
font-size: 160%;
}
.reveal pre {
width: inherit;
padding: 0.4em;
margin: 0px;
font-family: monospace, sans-serif;
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<div class="highlight"><pre><span class="c"># all the imports</span>
<span class="o">%</span><span class="k">pylab</span> <span class="n">inline</span>
<span class="kn">from</span> <span class="nn">visual_cryptography.visual_cryptography</span> <span class="kn">import</span> <span class="p">(</span><span class="n">ImageLayer</span><span class="p">,</span>
<span class="n">produce_image_layer_from_real_image</span><span class="p">)</span>
<span class="n">pylab</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s">&#39;figure.figsize&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mf">18.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">)</span>
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<pre>
Populating the interactive namespace from numpy and matplotlib
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<div class="highlight"><pre><span class="n">noir</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">blanc</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">def</span> <span class="nf">plot_matrix</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="k">if</span> <span class="n">ax</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">pylab</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s">&#39;nearest&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="n">title</span><span class="p">:</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s">&#39;off&#39;</span><span class="p">)</span>
<span class="c">#pylab.show()</span>
<span class="k">def</span> <span class="nf">do_not_show_axis</span><span class="p">(</span><span class="n">ax</span><span class="p">):</span>
<span class="n">ax</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_major_locator</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">NullLocator</span><span class="p">())</span>
<span class="n">ax</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_major_locator</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">NullLocator</span><span class="p">())</span>
<span class="k">def</span> <span class="nf">plot_function</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">xmin</span><span class="o">=-</span><span class="mi">10</span><span class="p">,</span> <span class="n">xmax</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">):</span>
<span class="k">if</span> <span class="n">ax</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">pylab</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">arange</span><span class="p">(</span><span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
<span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">X</span><span class="p">]</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
<span class="k">if</span> <span class="n">title</span><span class="p">:</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">plot_function_in_finite_field</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">xmin</span><span class="o">=-</span><span class="mi">10</span><span class="p">,</span> <span class="n">xmax</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">):</span>
<span class="k">if</span> <span class="n">ax</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">pylab</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">arange</span><span class="p">(</span><span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">X</span><span class="p">]</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="s">&#39;r--&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.75</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="s">&#39;bo&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="n">title</span><span class="p">:</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">show_sub_pixel</span><span class="p">(</span><span class="n">matrix</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Draw Hinton diagram for visualizing a weight matrix.&quot;&quot;&quot;</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">ax</span> <span class="ow">or</span> <span class="n">pylab</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span>
<span class="k">if</span> <span class="n">title</span><span class="p">:</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
<span class="n">max_weight</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">matrix</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">())</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
<span class="n">ax</span><span class="o">.</span><span class="n">patch</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s">&#39;white&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s">&#39;equal&#39;</span><span class="p">,</span> <span class="s">&#39;box&#39;</span><span class="p">)</span>
<span class="n">do_not_show_axis</span><span class="p">(</span><span class="n">ax</span><span class="p">)</span>
<span class="k">for</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),</span><span class="n">w</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">ndenumerate</span><span class="p">(</span><span class="n">matrix</span><span class="p">):</span>
<span class="k">if</span> <span class="n">w</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">color</span> <span class="o">=</span> <span class="s">&quot;white&quot;</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">color</span> <span class="o">=</span> <span class="s">&quot;gray&quot;</span>
<span class="n">size</span> <span class="o">=</span> <span class="mf">0.8</span>
<span class="n">rect</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">Rectangle</span><span class="p">([</span><span class="n">y</span> <span class="o">-</span> <span class="n">size</span> <span class="o">/</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="n">size</span> <span class="o">/</span> <span class="mi">2</span><span class="p">],</span> <span class="n">size</span><span class="p">,</span> <span class="n">size</span><span class="p">,</span>
<span class="n">facecolor</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s">&quot;black&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">autoscale_view</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">invert_yaxis</span><span class="p">()</span>
<span class="c">#pylab.show()</span>
<span class="k">def</span> <span class="nf">show_text</span><span class="p">(</span><span class="n">text</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">ax</span> <span class="ow">or</span> <span class="n">pylab</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_major_locator</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">NullLocator</span><span class="p">())</span>
<span class="n">ax</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_major_locator</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">NullLocator</span><span class="p">())</span>
<span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span><span class="n">text</span> <span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#39;black&#39;</span><span class="p">,</span>
<span class="n">fontsize</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span>
<span class="c">#fontname=&#39;Courier&#39;,</span>
<span class="n">horizontalalignment</span><span class="o">=</span><span class="s">&#39;center&#39;</span><span class="p">,</span>
<span class="n">verticalalignment</span><span class="o">=</span><span class="s">&#39;center&#39;</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s">&#39;off&#39;</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel_1</span><span class="p">,</span> <span class="n">pixel_2</span><span class="p">,</span> <span class="n">axs</span><span class="p">):</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;(&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">show_sub_pixel</span><span class="p">(</span><span class="n">pixel_1</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;,&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
<span class="n">show_sub_pixel</span><span class="p">(</span><span class="n">pixel_2</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axs</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;)&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">4</span><span class="p">])</span>
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<h1 id="presentation-overview">presentation overview</h1>
<h3 id="-1-visual-cryptography">[1] Visual cryptography</h3>
<p>Encrypting information in a 100% visual way.</p>
<h3 id="-2-sharing-a-secret">[2] Sharing a secret</h3>
<p>Sharing a secret between $n$ persons such that any subset of
size $k$ can recover it.</p>
<h3 id="-3-password-alternative">[3] Password alternative</h3>
<p>Using a zero knowledge proof to authenticate ourself to a third
party.</p>
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<h1 id="-1-visual-cryptography">[1] Visual cryptography</h1>
<h2 id="objectives">objectives</h2>
<p>Let&#39;s present the <a href="http://goo.gl/bxMw33">Shamir and Naor</a> ideas to encrypt a message
in a <em>secure</em> way such that the decrypting does not require any computers or calculations.</p>
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<h2 id="idea">Idea</h2>
<p>The big idea is to encrypt an <strong>image</strong> on two separate transparents.
When the two transparents are stack togheter, the original image
can be recover.</p>
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<h1 id="first-transparent">First transparent</h1>
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In&nbsp;[3]:
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<div class="highlight"><pre><span class="n">img</span> <span class="o">=</span> <span class="n">produce_image_layer_from_real_image</span><span class="p">(</span><span class="s">&quot;image.png&quot;</span><span class="p">)</span>
<span class="n">shares</span> <span class="o">=</span> <span class="n">img</span><span class="o">.</span><span class="n">produce_two_shares_from_image</span><span class="p">()</span>
<span class="n">plot_matrix</span><span class="p">(</span><span class="n">shares</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">title</span><span class="o">=</span><span class="s">&#39;first transparent&#39;</span><span class="p">)</span>
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<h1 id="second-transparent">Second transparent</h1>
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In&nbsp;[4]:
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<div class="highlight"><pre><span class="n">plot_matrix</span><span class="p">(</span><span class="n">shares</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">title</span><span class="o">=</span><span class="s">&#39;second transparent&#39;</span><span class="p">)</span>
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<h1 id="stacking-the-two-transparents-togheter">Stacking the two transparents togheter</h1>
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In&nbsp;[5]:
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<div class="highlight"><pre><span class="n">sum_of_the_shares</span> <span class="o">=</span> <span class="n">shares</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">shares</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="c"># note that matplotlib associates 0 to black and 1 to white</span>
<span class="n">plot_matrix</span><span class="p">(</span><span class="n">sum_of_the_shares</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;both transparents&quot;</span><span class="p">)</span>
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>
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<h3 id="remarks">remarks</h3>
<ul>
<li><p>The person that creates encrypt the message is called the <em>dealer</em></p>
</li>
<li><p>One of the transparent can be called the <em>cypher</em> and the other the <em>key</em>.</p>
</li>
<li><p>A crytosystem is said to be <em>secure</em> if, without the key, it impossible to
know the original message from the cypher.</p>
</li>
</ul>
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<section>
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<h1 id="let-s-look-at-what-happend">Let&#39;s look at what happend</h1>
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<h2 id="we-started-by-an-image">We started by an image</h2>
<p>This image is said as the <em>message</em></p>
<p><img src="image.png" alt="image initiale"></p>
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<h2 id="we-splitted-the-image-into-many-pixels">We splitted the image into many pixels</h2>
<p><img src="image_pixelized.png" alt="pixel decomposition of the original image"></p>
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<h2 id="constructe-transparent-methodology">constructe transparent methodology</h2>
<p>We construct each transparent in a pixel by pixel fashion.</p>
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In&nbsp;[6]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span> <span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">img</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">image</span><span class="o">.</span><span class="n">imread</span><span class="p">(</span><span class="s">&#39;image.png&#39;</span><span class="p">)</span>
<span class="n">do_not_show_axis</span><span class="p">(</span><span class="n">ax0</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">img</span><span class="p">)</span>
<span class="n">show_sub_pixel</span><span class="p">([[</span><span class="n">blanc</span><span class="p">]</span><span class="o">*</span><span class="mi">10</span><span class="p">]</span><span class="o">*</span><span class="mi">10</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;transparent 1&quot;</span><span class="p">)</span>
<span class="n">show_sub_pixel</span><span class="p">([[</span><span class="n">blanc</span><span class="p">]</span><span class="o">*</span><span class="mi">10</span><span class="p">]</span><span class="o">*</span><span class="mi">10</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;transparent 2&quot;</span><span class="p">)</span>
</pre></div>
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<pre>
-c:43: RuntimeWarning: divide by zero encountered in log
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<h2 id="encrypting-each-pixels">Encrypting each pixels</h2>
<p>To encrypt a pixel, we&#39;ll decompose it in $m$ sub-pixel. Each sub-pixel can be black or white.</p>
<ul>
<li><p>The humain eye will approximate the $m$ sub-pixel in a pixel</p>
</li>
<li><p>If there is enought black sub-pixel, the eye will see a black pixel</p>
</li>
<li><p>If there is not enought of black sub-pixel, the eye will see a white pixel</p>
</li>
</ul>
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In&nbsp;[7]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span> <span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">show_sub_pixel</span><span class="p">([[</span><span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">],</span>
<span class="p">[</span><span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">],</span>
<span class="p">[</span><span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">]],</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax0</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="s">&quot;a black pixel&quot;</span><span class="p">)</span>
<span class="n">show_sub_pixel</span><span class="p">([[</span><span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">],</span>
<span class="p">[</span><span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">,</span> <span class="n">blanc</span><span class="p">],</span>
<span class="p">[</span><span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">]],</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="s">&quot;a black pixel&quot;</span><span class="p">)</span>
<span class="n">show_sub_pixel</span><span class="p">([[</span><span class="n">blanc</span><span class="p">,</span> <span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span><span class="p">],</span>
<span class="p">[</span><span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span><span class="p">,</span> <span class="n">blanc</span><span class="p">],</span>
<span class="p">[</span><span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span><span class="p">,</span> <span class="n">blanc</span><span class="p">]],</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="s">&quot;a white pixel&quot;</span><span class="p">)</span>
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<h2 id="techinical-objectives">techinical objectives</h2>
<p>To help the humain, we need the black to be as dark as possible and the white to be
as bright as possible</p>
<h4 id="black-pixel">black pixel</h4>
<p>A pixel is said to be <em>black</em> if it is made of at least $d \leq m$ black sub-pixels</p>
<h4 id="white-pixel">white pixel</h4>
<p>A pixel is said to be <em>white</em> if it is made of no more than $d - \alpha m$ black sub-pixels</p>
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<h2 id="remarks">remarks</h2>
<ul>
<li><p>$m$ is the number of sub-pixels in a pixel</p>
</li>
<li><p>Bigger $d$ is, darker the blacks are</p>
</li>
<li><p>Bigger $\alpha$ is, brighter the whites are.</p>
</li>
<li><p>$\alpha$ represent here a &quot;distance&quot; between the white and black. Greather this distance
will be easier it will be to distinguist between the blacks and whites</p>
</li>
</ul>
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In&nbsp;[8]:
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<div class="highlight"><pre><span class="kn">import</span> <span class="nn">matplotlib</span> <span class="kn">as</span> <span class="nn">mpl</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">pylab</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">cmap</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray_r</span>
<span class="n">norm</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
<span class="n">cb</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">colorbar</span><span class="o">.</span><span class="n">ColorbarBase</span><span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">,</span>
<span class="n">norm</span><span class="o">=</span><span class="n">norm</span><span class="p">,</span>
<span class="n">orientation</span><span class="o">=</span><span class="s">&#39;horizontal&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_axis_off</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="s">&#39;black&#39;</span><span class="p">,</span>
<span class="n">horizontalalignment</span><span class="o">=</span><span class="s">&#39;center&#39;</span><span class="p">,</span>
<span class="n">verticalalignment</span><span class="o">=</span><span class="s">&#39;center&#39;</span><span class="p">,</span>
<span class="n">rotation</span><span class="o">=</span><span class="s">&#39;vertical&#39;</span><span class="p">,</span>
<span class="n">size</span><span class="o">=</span><span class="s">&quot;20&quot;</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s">&quot;white&quot;</span><span class="p">,</span>
<span class="n">transform</span><span class="o">=</span><span class="n">ax</span><span class="o">.</span><span class="n">transAxes</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="s">&#39;white&#39;</span><span class="p">,</span>
<span class="n">horizontalalignment</span><span class="o">=</span><span class="s">&#39;center&#39;</span><span class="p">,</span>
<span class="n">verticalalignment</span><span class="o">=</span><span class="s">&#39;center&#39;</span><span class="p">,</span>
<span class="n">rotation</span><span class="o">=</span><span class="s">&#39;vertical&#39;</span><span class="p">,</span>
<span class="n">size</span><span class="o">=</span><span class="s">&quot;20&quot;</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s">&quot;black&quot;</span><span class="p">,</span>
<span class="n">transform</span><span class="o">=</span><span class="n">ax</span><span class="o">.</span><span class="n">transAxes</span><span class="p">)</span>
<span class="n">pylab</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<h2 id="encryption">encryption</h2>
<p>In a way or an other, for each pixel we figured out sub-pixel to place on each transparent
such that.</p>
<ul>
<li><p>when stacked, they form the the original image</p>
</li>
<li><p>with only one transparent, it is impossible to know what is the original image</p>
</li>
</ul>
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In&nbsp;[9]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span> <span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">plot_matrix</span><span class="p">(</span><span class="n">shares</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">title</span><span class="o">=</span><span class="s">&#39;first transparent&#39;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax0</span><span class="p">)</span>
<span class="n">plot_matrix</span><span class="p">(</span><span class="n">shares</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">title</span><span class="o">=</span><span class="s">&#39;second transparent&#39;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">)</span>
<span class="n">plot_matrix</span><span class="p">(</span><span class="n">shares</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">shares</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">title</span><span class="o">=</span><span class="s">&#39;stacked transparents&#39;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">)</span>
</pre></div>
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<h1 id="some-formalization">Some formalization</h1>
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<h2 id="definition-of-a-pixel-structure">Definition of a <em>pixel</em> structure</h2>
<p>We encode a pixel in an array of size $m$, $[ p_1, p_2, \dots, p_m ]$</p>
<ul>
<li><p>where $i$ represents a specific sub-pixel position</p>
</li>
<li><p>the $i^th$ sub-pixel is black is $p_i$ is $1$ and white if $p_i$ is $0$</p>
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In&nbsp;[10]:
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<div class="highlight"><pre><span class="n">show_sub_pixel</span><span class="p">([[</span><span class="n">noir</span><span class="p">,</span> <span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span><span class="p">],</span>
<span class="p">[</span><span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span><span class="p">,</span> <span class="n">blanc</span><span class="p">],</span>
<span class="p">[</span><span class="n">blanc</span><span class="p">,</span> <span class="n">blanc</span><span class="p">,</span> <span class="n">blanc</span><span class="p">]],</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;[1 0 1 0 1 0 0 0 0]&quot;</span><span class="p">)</span>
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<h2 id="transparent-superposition">transparent superposition</h2>
<p>Consequently the transparent superposition can be seen as componentwise <em>or</em> operation</p>
<pre><code> [1 0 1 0 ]
+ [1 1 0 0 ]
-----------------------
[1 1 1 0 ]
</code></pre>
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In&nbsp;[11]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span><span class="p">,</span> <span class="n">ax3</span><span class="p">,</span> <span class="n">ax4</span> <span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">show_sub_pixel</span><span class="p">([[</span><span class="n">noir</span><span class="p">,</span> <span class="n">blanc</span><span class="p">],</span>
<span class="p">[</span><span class="n">noir</span><span class="p">,</span> <span class="n">blanc</span><span class="p">]],</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;[1 0 1 0]&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax0</span><span class="p">)</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;+&quot;</span><span class="p">,</span> <span class="n">ax1</span><span class="p">)</span>
<span class="n">show_sub_pixel</span><span class="p">([[</span><span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">],</span>
<span class="p">[</span><span class="n">blanc</span><span class="p">,</span> <span class="n">blanc</span><span class="p">]],</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;[1 1 0 0]&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">)</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;=&quot;</span><span class="p">,</span> <span class="n">ax3</span><span class="p">)</span>
<span class="n">show_sub_pixel</span><span class="p">([[</span><span class="n">noir</span><span class="p">,</span> <span class="n">noir</span><span class="p">],</span>
<span class="p">[</span><span class="n">noir</span><span class="p">,</span> <span class="n">blanc</span><span class="p">]],</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;[1 1 1 0]&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax4</span><span class="p">)</span>
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<h1 id="encryption-">encryption...</h1>
<p>How can we encode a pixel $P$ (black or white) in two pixels $Q_1$ $Q_2$ such that...</p>
<ol>
<li><p>$Q_1 + Q_2 \sim P$</p>
</li>
<li><p>Without having access to $Q_1$ <strong>and</strong> $Q_2$, we grain no knowledge on $P$</p>
</li>
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<p>$$
Prob(\text{P is white} | Q_i) = Prob(\text{P is black} | Q_i)
$$</p>
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In&nbsp;[11]:
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<p>$A \sim B$ if $A$ is the same colour as $B$</p>
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<h2 id="solution">solution</h2>
<p>We create two sets</p>
<p>$W = \{ (r_1, l_1) , (r_2, l_2), \dots, (r_k, l_k) \}$</p>
<p>$K = \{ (g_1, d_1) , (g_2, d_2), \dots, (g_k, d_k) \}$</p>
<p>such that...</p>
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<li><p>$r_i + l_i$ is white</p>
</li>
<li><p>$g_i + d_i$ is black</p>
</li>
<li><p>The cardinality of $x$ in the multiset $\{r_1, l_1, \dots r_k, l_k\}$ is
the same in the multiset $\{g_1, d_1, \dots g_k, d_k\}$.</p>
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<h1 id="a-working-example">A working example</h1>
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In&nbsp;[12]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
<span class="n">pixel</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="p">:</span> <span class="p">[[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="p">[</span><span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">]]</span>
<span class="n">w</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;white={ &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;,&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">7</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">8</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;}&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">13</span><span class="p">])</span>
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In&nbsp;[13]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
<span class="n">pixel</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="p">:</span> <span class="p">[[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="p">[</span><span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">]]</span>
<span class="n">w</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;black={ &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;,&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">7</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">8</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;}&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">13</span><span class="p">])</span>
</pre></div>
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<h1 id="an-other-working-example">An other working example</h1>
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In&nbsp;[14]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
<span class="n">pixel</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="p">:</span> <span class="p">[[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="p">[</span><span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">]]</span>
<span class="n">w</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;white={ &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;,&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">7</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">8</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;}&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">13</span><span class="p">])</span>
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In&nbsp;[15]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
<span class="n">pixel</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="p">:</span> <span class="p">[[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="p">[</span><span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">]]</span>
<span class="n">w</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;black={ &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;,&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">7</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">8</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;}&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">13</span><span class="p">])</span>
</pre></div>
</div>
</div>
</div>
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<h1 id="a-simpler-working-example">A simpler working example</h1>
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In&nbsp;[16]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
<span class="n">pixel</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="p">[[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">]]</span>
<span class="n">w</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;white={ &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;,&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">7</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">8</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;}&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">13</span><span class="p">])</span>
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In&nbsp;[17]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
<span class="n">pixel</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="p">:</span> <span class="p">[[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">]]</span>
<span class="n">w</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">blanc</span><span class="p">,</span> <span class="n">noir</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;black={ &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;,&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">7</span><span class="p">])</span>
<span class="n">plot_tuple_of_pixels</span><span class="p">(</span><span class="n">pixel</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">w</span><span class="p">),</span> <span class="n">pixel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">axs</span><span class="p">[</span><span class="mi">8</span><span class="p">:])</span>
<span class="n">show_text</span><span class="p">(</span><span class="s">&quot;}&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">[</span><span class="mi">13</span><span class="p">])</span>
</pre></div>
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<h1 id="a-possible-attack-">A possible attack?</h1>
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<h1 id="yes-">Yes!</h1>
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<h1 id="the-last-person-that-shows-its-transparent-can-cheat-">the last person that shows its transparent can cheat.</h1>
<p>By crafting its own transparent, knowing the other transparent,
a cheater could show whatever he wants as a results</p>
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In&nbsp;[18]:
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<div class="highlight"><pre><span class="n">cheated</span> <span class="o">=</span> <span class="n">produce_image_layer_from_real_image</span><span class="p">(</span><span class="s">&quot;cheatedimage.png&quot;</span><span class="p">)</span>
<span class="n">cheated_share</span> <span class="o">=</span> <span class="n">cheated</span><span class="o">.</span><span class="n">produce_cheated_image_from_other_share</span><span class="p">(</span><span class="n">shares</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">plot_matrix</span><span class="p">(</span><span class="n">cheated_share</span> <span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&#39;crafted transparent&#39;</span><span class="p">)</span>
<span class="n">savefig</span><span class="p">(</span><span class="s">&quot;crafted_transparen.pdf&quot;</span><span class="p">)</span>
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In&nbsp;[19]:
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<div class="highlight"><pre><span class="n">plot_matrix</span><span class="p">(</span><span class="n">cheated_share</span> <span class="o">*</span> <span class="n">shares</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">title</span><span class="o">=</span><span class="s">&#39;forge results&#39;</span><span class="p">)</span>
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<h3 id="-2-sharing-a-secret">[2] Sharing a secret</h3>
<h2 id="objectives">objectives</h2>
<p>Let&#39;s present <a href="http://en.wikipedia.org/wiki/Shamir&#39;s_Secret_Sharing">Shamir&#39;s</a> ideas of secret sharing.</p>
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<h3 id="the-problem">the problem</h3>
<p><em>Alice</em> wants to share his swiss bank account number with <em>Bob TheFirst</em>, <em>Bob TheSecond</em>, <em>Bob TheThird</em>,... <em>Bob TheNth</em> such that...</p>
<ol>
<li><p>If any $k$ <em>Bob</em>s comes togheter they need to be able to recover the bank account number</p>
</li>
<li><p>If any less that $k$ <em>Bob</em>s comes togheter they need to have no idea whatsoever about the bank account number</p>
</li>
</ol>
<pre><code> ***********
***** ***********
** ****** *** ********
**** ****** ** *******
*** ******* ** ******
*** ** * **
*|/------ -------\ ** *
| |=| :===**
| O | | O | }|*
|---- | ---- | |*
| |___ |\/
| |
\ ----- |
\ |
-__ -- -/
</code></pre>
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<h1 id="shamir-solution-use-polynomials-">shamir solution: use polynomials!</h1>
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<h3 id="definition">definition</h3>
<p>A polynomial is a function that can be written as</p>
<p>$$
a_0 + a_1 \cdot x^1 + a_2 \cdot x^2 + \dots + a_n \cdot x^n
$$</p>
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In&nbsp;[20]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">plot_function</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;$2 \cdot x + 1$&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax0</span><span class="p">)</span>
<span class="n">plot_function</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;$2 \cdot x^2 + 1$&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">)</span>
<span class="n">plot_function</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;$2 \cdot x^3 - 2 \cdot x^2 + 1$&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">)</span>
</pre></div>
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<h3 id="important-feature-of-polynomials">important feature of polynomials</h3>
<p>Any $n + 1$ points define an unique polynomials of degre $n$.</p>
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In&nbsp;[21]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">plot_function</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;$2 \cdot x + 1$&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax0</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="mi">7</span><span class="p">]</span> <span class="p">,</span> <span class="p">[</span><span class="n">p</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">),</span> <span class="n">p</span><span class="p">(</span><span class="mi">7</span><span class="p">)],</span> <span class="s">&quot;ro&quot;</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">plot_function</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;$2 \cdot x^2 + 1$&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="n">p</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">),</span> <span class="n">p</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">p</span><span class="p">(</span><span class="mi">7</span><span class="p">)],</span> <span class="s">&quot;ro&quot;</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">plot_function</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;$2 \cdot x^3 - 2 \cdot x^2 + 1$&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="n">p</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">),</span> <span class="n">p</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">),</span> <span class="n">p</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">p</span><span class="p">(</span><span class="mi">7</span><span class="p">)],</span> <span class="s">&quot;ro&quot;</span><span class="p">)</span>
<span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
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<h3 id="similarly-">Similarly,</h3>
<p>by $n$ points pass an infinite number of polynomials of degree $n$</p>
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In&nbsp;[28]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">b</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">d</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">c</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span> <span class="p">:</span> <span class="o">-</span><span class="mi">100</span> <span class="o">-</span> <span class="mi">7</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">a</span>
<span class="n">P</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span> <span class="p">:</span> <span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">a</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">b</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">*</span> <span class="n">c</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">d</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
<span class="n">plot_function</span><span class="p">(</span><span class="n">P</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">P</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">7</span><span class="p">]</span> <span class="p">,</span> <span class="p">[</span><span class="n">p</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">),</span> <span class="n">p</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">p</span><span class="p">(</span><span class="mi">7</span><span class="p">)],</span> <span class="s">&quot;ro&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&quot;Polynomials passing through $(-7, 701), (0, 1), (7, -699)$&quot;</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
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f5ckSZIkSd02GqdsSJIkSZKkUc6EhCRJkiRJ6joTEpIkSZIkqetMSEiSJEmSpK4zISFJkiRJkrrO
hIQkSZIkSeo6ExKSJEmSJKnr/j8c845mpUoLZgAAAABJRU5ErkJggg==
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<h3 id="shamir-s-secret-sharing">Shamir&#39;s secret sharing</h3>
<ul>
<li>Alice needs to create a random polynomial of degre $k-1$ such that $a_0$ is the
secret to share.</li>
</ul>
<p>$$ P(x) = a_0 + a_1 \cdot x^1 + a_2 \cdot x^2 + \dots + a_{k-1}\cdot x^{x-1} $$</p>
<ul>
<li>Alice gives to each <em>Bob TheIth</em> the couple $(i, P(i))$</li>
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<h3 id="secret-recovery">secret recovery</h3>
<p>If $K$ bob comes togheter, they can figure out the secret polynomial $P(x)$ and compute $P(0)$</p>
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<h1 id="interesting-facts">interesting facts</h1>
<ul>
<li>the number $n$ needs not to be known at first. (Additional bobs can
come and join the party!)</li>
</ul>
<ul>
<li>the number $k$ is fixed forever</li>
</ul>
<ul>
<li>the fact that we evaluate $P(x)$ at $0$ helps tremendously in
the algebraic resolution of the problems</li>
</ul>
<ul>
<li>the Shamir&#39;s secret sharing is at the base of almost every &quot;threshold&quot;
cryptography</li>
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<h1 id="small-example-to-see-if-everything-works">small example to see if everything works</h1>
<p><em>Peter</em> wants to share between <em>Wolf</em> and <em>Didier</em> the
ACS&#39;s credit card: <code>5243 9021 9129 1828</code>.</p>
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<div class="highlight"><pre><span class="n">p</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="mi">5243902191291828</span> <span class="o">+</span> <span class="mi">7873787349</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">7893478934</span> <span class="o">*</span> <span class="n">x</span> <span class="o">**</span> <span class="mi">2</span>
<span class="n">credit_card_number</span> <span class="o">=</span> <span class="mi">5243902191291828</span>
<span class="n">wolf_secret</span> <span class="o">=</span> <span class="n">p</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="c"># keyboard random number</span>
<span class="n">didier_secret</span> <span class="o">=</span> <span class="n">p</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">wolf_secret</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">didier_secret</span><span class="p">)</span>
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<pre>
5243917958558111
5243949512782262
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<h3 id="step-1">step 1</h3>
<p>Peter choose the bellow polynomials because he likes it very much</p>
<p>$$P(X) = 5243902191291828 + 7873787349\cdot x + 7893478934\cdot x^2$$</p>
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<h4 id="step-2">step 2</h4>
<ul>
<li>Peter gives $(1, 5243917958558111)$ to Wolf</li>
<li>Peter gives $(2, 5243949512782262)$ to Didier</li>
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<h2 id="as-all-canadian-can-expect-">As all Canadian can expect...</h2>
<p>Didier will try to figure out the credit card number to buy a
truck load of maple sirup.</p>
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<h3 id="can-didier-acheve-its-goal-">Can Didier acheve its goal?</h3>
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<h2 id="let-s-resume-what-didier-knows">Let&#39;s resume what Didier knows</h2>
<ol>
<li><p>He knows that he is looking for a number $a_0$ of 12 digits</p>
</li>
<li><p>$a_0$ is positive</p>
</li>
<li><p>$a_0$ is an integer</p>
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<h3 id="but-if-didier-does-a-little-bit-of-algebra-">But if didier does a little bit of algebra...</h3>
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<p>...
$$P(x) = a_0 + a_1 \cdot x + a_2 \cdot x^2 $$
$$5243949512782262 = a_0 + a_1 \cdot 2 + a_2 \cdot 4 $$
$$a_0 = 5243949512782262 - a_1 \cdot 2 - a_2 \cdot 4$$
$$a_0 = 5243949512782262 - 2\cdot(a_1 \cdot - 2 \cdot a_2)$$
$$a_0 = 5243949512782262 - \text{even number}$$</p>
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<h2 id="hence-">Hence...</h2>
<ul>
<li><p>Didier knows the credit card number is even</p>
</li>
<li><p>With the help of <em>EC2</em> Didier can find the solution in a relative small amount of time.</p>
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<h1 id="what-can-we-do-to-avoid-such-situation-">What can we do to avoid such situation?</h1>
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<ol>
<li><p>We could process the original message before wharing it</p>
<p>$$\underbrace{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}_\text{very random number}\underbrace{5243902191291828}_\text{the credit card number}$$</p>
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<ol>
<li><p>We could work in a more appropriate field</p>
<p>$$\mathbb{Z}/p\mathbb{Z}$$</p>
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<h1 id="what-is-0-">What is $\mathbb{Z}/p\mathbb{Z}$ ?</h1>
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<ul>
<li>$p$ is (very large) prime</li>
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<ul>
<li>$\mathbb{Z}/p\mathbb{Z}$ is the set of all the integers between $0$ and $p-1$</li>
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<li>we can add two elements of $\mathbb{Z}/p\mathbb{Z}$ as usual. If the result
is larger than $p$ we <em>loop</em> using the modulo operation</li>
</ul>
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<li>we can multiply two elements of $\mathbb{Z}/p\mathbb{Z}$ as usual. If the result
is larger than $p$ we <em>loop</em> using the modulo operation</li>
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<h1 id="uber-important-things">Uber important things</h1>
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<li>Almost everything you know about the &quot;numbers&quot; is still
true in $\mathbb{Z}/p\mathbb{Z}$ </li>
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<li>In particular, each element has a multiplicative inverse</li>
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<h2 id="example-0-">example $\mathbb{Z}/7\mathbb{Z}$</h2>
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<h1 id="back-to-polyominals-but-int-0-">back to polyominals... (but int $\mathbb{Z}/p\mathbb{Z}$)</h1>
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">plot_function_in_finite_field</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">7</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;$2 \cdot x + 1$ mod 7&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax0</span><span class="p">)</span>
<span class="n">plot_function_in_finite_field</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">7</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;$2 \cdot x^2 + 1$ mod 7&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">)</span>
<span class="n">plot_function_in_finite_field</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">7</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;$2 \cdot x^3 - 2 \cdot x^2 + 1$ mod 7&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">)</span>
</pre></div>
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<h2 id="the-main-two-polyominals-facts-are-still-almost-true">The main two polyominals facts are still (almost) true</h2>
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<h3 id="any-0-points-define-an-unique-polynomial-of-degre-1-">Any $n + 1$ points define an unique polynomial of degre $n$.</h3>
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In&nbsp;[25]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">7</span>
<span class="n">plot_function_in_finite_field</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&quot;$2 \cdot x^3 - 2 \cdot x^2 + 1 mod 7$&quot;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="mi">7</span><span class="p">)],</span> <span class="s">&#39;rD&#39;</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<h3 id="by-0-points-pass-1-polyomials-of-degree-2-">By $n$ points pass $p$ polyomials of degree $n$.</h3>
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In&nbsp;[29]:
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<div class="highlight"><pre><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">b</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">d</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">c</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span> <span class="p">:</span> <span class="o">-</span><span class="mi">100</span> <span class="o">-</span> <span class="mi">7</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">a</span>
<span class="n">P</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span> <span class="p">:</span> <span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="n">a</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">b</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">*</span> <span class="n">c</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">d</span><span class="p">)</span> <span class="o">%</span> <span class="mi">7</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
<span class="n">plot_function_in_finite_field</span><span class="p">(</span><span class="n">P</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">P</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">7</span><span class="p">]</span> <span class="p">,</span> <span class="p">[</span><span class="n">p</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">),</span> <span class="n">p</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">p</span><span class="p">(</span><span class="mi">7</span><span class="p">)],</span> <span class="s">&quot;ro&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&quot;Polynomials passing through $(-7, 1), (0, 1), (7, 1)$&quot;</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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SUVORK5CYII=
"
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<h2 id="back-to-shamir-s-secret-sharing">back to Shamir&#39;s secret sharing</h2>
</div>
</div>
</div><div class="fragment">
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</div>
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<p>If peter would have choosen polynomials in $\mathbb{Z}/p\mathbb{Z}$ with $p$ big enough...</p>
<ol>
<li><p>Didier could not have easely discarted some numbers</p>
</li>
<li><p>For Didier, a large $p$ in infinite even with the help of $aws$</p>
</li>
</ol>
</div>
</div>
</div>
</div></section>
</section><section>
<section>
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<h3 id="-3-password-alternative">[3] Password alternative</h3>
<p>The objective of this sections is to present the idea of the
article <a href="http://www.cs.rit.edu/~jjk8346/FiatShamir.pdf">how to proof yourself</a>,
an article publish by Fiat and Shamir in the year of grace 1987.</p>
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</div>
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<h2 id="objective">Objective</h2>
<p>The objective is to find a way to allow <em>Alice</em> to authenticate to <em>Bob</em> such that...</p>
<ol>
<li><p>Alice can authenticate</p>
</li>
<li><p>Bob is sure that the <em>Alice</em> is really Alice</p>
</li>
<li><p>Bob cannot copy the credentials of alice in any way.</p>
</li>
</ol>
</div>
</div>
</div><div class="fragment">
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<h3 id="in-the-contect-of-a-zero-knoledge-proof-">in the contect of a zero-knoledge proof...</h3>
<ol>
<li><p>Completness</p>
</li>
<li><p>Soundness</p>
</li>
<li><p>Zero-knowledge</p>
</li>
</ol>
</div>
</div>
</div>
</div></section><section>
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<h3 id="how-can-we-do-that-">how can we do that?</h3>
<ol>
<li><p>Bob and Alice exchange some information while they know they are talking to the each other (sign up)</p>
</li>
<li><p>In that exchange, Alice will tell Bob that she knows something, proof without giving Bob or anyone else
any factual proof.</p>
</li>
<li><p>Each time Alice will authenticate, Alice will provide enough information to Bob to demonstrate that she knows the
information but without giving any factual proofs.</p>
</li>
</ol>
</div>
</div>
</div></section>
</section><section>
<section>
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<h1 id="quadratic-residue">quadratic residue</h1>
</div>
</div>
</div><div class="fragment">
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<p>There are some problems that are easy to solve in $\mathbb{R}$ but very hard to solve in $\mathbb{Z}/n\mathbb{Z}$</p>
</div>
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<h1 id="quadratic-residue">quadratic residue</h1>
<ul>
<li><p>The quandratric residue of $x$ in $\mathbb{Z}/n\mathbb{Z}$ is an
element $y$ such that $x^2 = y$</p>
</li>
<li><p>We say that $y$ is a <em>quadratic residue</em> if there exists $x$ such
that $x^2 = y$</p>
</li>
</ul>
</div>
</div>
</div>
</div><div class="fragment">
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<h3 id="facts">facts</h3>
<ul>
<li><p>not every elements are quadratic residue</p>
</li>
<li><p>It is <strong>hard</strong> to know if a number is a quadratic residue</p>
</li>
<li><p>It is easy to find a quadratic residue: just take a random
$r$ and compute $r^2$</p>
</li>
</ul>
</div>
</div>
</div>
</div></section>
</section><section>
<section>
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<h1 id="proving-ourselves-with-quadratic-residues-">Proving ourselves with quadratic residues...</h1>
</div>
</div>
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<h3 id="idea">idea</h3>
<p>The idea is that Alice will demonstrate to Bob that she knows that a certain element $x$ is a quadratic residue
without giving $\omega$ such that $\omega ^2 = x$</p>
</div>
</div>
</div></section><section>
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<h1 id="protocol-">protocol...</h1>
<table>
<thead>
<tr>
<th style="text-align:center">Alice</th>
<th style="text-align:center">Bob</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">chose a random $r$</td>
<td style="text-align:center"></td>
</tr>
<tr>
<td style="text-align:center">sends $a = r^2$</td>
<td style="text-align:center"></td>
</tr>
<tr>
<td style="text-align:center"></td>
<td style="text-align:center">send challenge 0 or 1 at random</td>
</tr>
<tr>
<td style="text-align:center">if challenge is 1 returns r</td>
<td style="text-align:center">accept if $r^2 = a$</td>
</tr>
<tr>
<td style="text-align:center">if challenge is 0 returns $r\cdot\omega$</td>
<td style="text-align:center">accept is $(r\cdot\omega)^2 = a \cdot x$</td>
</tr>
</tbody>
</table>
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<ul>
<li><p>completeness:
If Alice plays the game, she will be to proof that she is herself with probability 1</p>
</li>
<li><p>soundness:
Bob as a probablylity $1/2$ to catch a cheeter</p>
</li>
<li><p>zero-knowledge:
Proof by intimidation</p>
</li>
</ul>
</div>
</div>
</div>
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In&nbsp;[]:
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<div class="highlight"><pre>
</pre></div>
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