Piotr Dyraga
12b3dadef7
For an odd n-bit number, we can't find two n-1-bit numbers which multiplied gives an n-bit number. That's why public key length n must be an even number. I have also renamed AreSafePrimes to IsSafePrime since only the first function argument is considered to be a safe prime according to a safe prime definition.
274 lines
7.5 KiB
Go
274 lines
7.5 KiB
Go
package paillier
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import (
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"crypto/rand"
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"errors"
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"io"
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"math/big"
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"time"
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)
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// Generates a threshold Paillier key with an algorithm based on [DJN 10],
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// section 5.1, "Key generation".
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//
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// Bear in mind that the algorithm assumes an existence of a trusted dealer
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// to generate and distribute the keys.
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//
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//
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// [DJN 10]: Ivan Damgard, Mads Jurik, Jesper Buus Nielsen, (2010)
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// A Generalization of Paillier’s Public-Key System
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// with Applications to Electronic Voting
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// Aarhus University, Dept. of Computer Science, BRICS
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type ThresholdKeyGenerator struct {
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publicKeyBitLength int
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TotalNumberOfDecryptionServers int
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Threshold int
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Random io.Reader
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// Both p1 and q1 are primes of length nbits - 1
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p1 *big.Int
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q1 *big.Int
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p *big.Int // p is prime and p=2*p1+1
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q *big.Int // q is prime and q=2*q1+1
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n *big.Int // n=p*q
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m *big.Int // m = p1*q1
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nSquare *big.Int // nSquare = n*n
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nm *big.Int // nm = n*m
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// As specified in the paper, d must satify d=1 mod n and d=0 mod m
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d *big.Int
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// A generator of QR in Z_{n^2}
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v *big.Int
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// The polynomial coefficients to hide a secret. See Shamir.
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polynomialCoefficients []*big.Int
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}
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// GetThresholdKeyGenerator is a preferable way to construct the
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// ThresholdKeyGenerator.
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// Due to the various properties that must be met for the threshold key to be
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// considered valid, the minimum public key `N` bit length is 18 bits and the
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// public key bit length should be an even number.
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// The plaintext space for the key will be `Z_N`.
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func GetThresholdKeyGenerator(
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publicKeyBitLength int,
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totalNumberOfDecryptionServers int,
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threshold int,
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random io.Reader,
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) (*ThresholdKeyGenerator, error) {
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if publicKeyBitLength%2 == 1 {
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// For an odd n-bit number, we can't find two n-1-bit numbers which
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// multiplied gives an n-bit number.
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return nil, errors.New("Public key bit length must be an even number")
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}
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if publicKeyBitLength < 18 {
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// We need to find two n-bit safe primes, P and Q which are not equal.
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// This is not possible for n<18.
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return nil, errors.New("Public key bit length must be at least 18 bits")
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}
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generator := new(ThresholdKeyGenerator)
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generator.publicKeyBitLength = publicKeyBitLength
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generator.TotalNumberOfDecryptionServers = totalNumberOfDecryptionServers
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generator.Threshold = threshold
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generator.Random = random
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return generator, nil
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}
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func (tkg *ThresholdKeyGenerator) generateSafePrimes() (*big.Int, *big.Int, error) {
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concurrencyLevel := 4
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timeout := 120 * time.Second
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safePrimeBitLength := tkg.publicKeyBitLength / 2
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return GenerateSafePrime(safePrimeBitLength, concurrencyLevel, timeout, tkg.Random)
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}
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func (tkg *ThresholdKeyGenerator) initPandP1() error {
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var err error
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tkg.p, tkg.p1, err = tkg.generateSafePrimes()
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return err
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}
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func (tkg *ThresholdKeyGenerator) initQandQ1() error {
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var err error
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tkg.q, tkg.q1, err = tkg.generateSafePrimes()
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return err
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}
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func (tkg *ThresholdKeyGenerator) initShortcuts() {
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tkg.n = new(big.Int).Mul(tkg.p, tkg.q)
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tkg.m = new(big.Int).Mul(tkg.p1, tkg.q1)
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tkg.nSquare = new(big.Int).Mul(tkg.n, tkg.n)
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tkg.nm = new(big.Int).Mul(tkg.n, tkg.m)
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}
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func (tkg *ThresholdKeyGenerator) arePsAndQsGood() bool {
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if tkg.p.Cmp(tkg.q) == 0 {
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return false
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}
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if tkg.p.Cmp(tkg.q1) == 0 {
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return false
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}
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if tkg.p1.Cmp(tkg.q) == 0 {
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return false
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}
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return true
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}
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func (tkg *ThresholdKeyGenerator) initPsAndQs() error {
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if err := tkg.initPandP1(); err != nil {
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return err
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}
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if err := tkg.initQandQ1(); err != nil {
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return err
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}
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if !tkg.arePsAndQsGood() {
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return tkg.initPsAndQs()
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}
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return nil
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}
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// v generates a cyclic group of squares in Zn^2.
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func (tkg *ThresholdKeyGenerator) computeV() error {
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var err error
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tkg.v, err = GetRandomGeneratorOfTheQuadraticResidue(tkg.nSquare, tkg.Random)
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return err
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}
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// Choose d such that d=0 (mod m) and d=1 (mod n).
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//
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// From Chinese Remainder Theorem:
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// x = a1 (mod n1)
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// x = a2 (mod n2)
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//
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// N = n1*n2
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// y1 = N/n1
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// y2 = N/n2
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// z1 = y1^-1 mod n1
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// z2 = y2^-1 mod n2
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// Solution is x = a1*y1*z1 + a2*y2*z2
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//
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// In our case:
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// x = 0 (mod m)
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// x = 1 (mod n)
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//
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// Since a1 = 0, it's enough to compute a2*y2*z2 to get x.
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//
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// a2 = 1
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// y2 = mn/n = m
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// z2 = m^-1 mod n
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//
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// x = a2*y2*z2 = 1 * m * [m^-1 mod n]
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func (tkg *ThresholdKeyGenerator) initD() {
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mInverse := new(big.Int).ModInverse(tkg.m, tkg.n)
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tkg.d = new(big.Int).Mul(mInverse, tkg.m)
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}
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func (tkg *ThresholdKeyGenerator) initNumerialValues() error {
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if err := tkg.initPsAndQs(); err != nil {
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return err
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}
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tkg.initShortcuts()
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tkg.initD()
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return tkg.computeV()
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}
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// f(X) = a_0 X^0 + a_1 X^1 + ... + a_(w-1) X^(w-1)
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//
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// where:
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// `w` - threshold
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// `a_i` - random value from {0, ... nm - 1} for 0<i<w
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// `a_0` is always equal `d`
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func (tkg *ThresholdKeyGenerator) generateHidingPolynomial() error {
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tkg.polynomialCoefficients = make([]*big.Int, tkg.Threshold)
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tkg.polynomialCoefficients[0] = tkg.d
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var err error
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for i := 1; i < tkg.Threshold; i++ {
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tkg.polynomialCoefficients[i], err = rand.Int(tkg.Random, tkg.nm)
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if err != nil {
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return err
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}
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}
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return nil
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}
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// The secred share of the i'th authority is `f(i) mod nm`, where `f` is
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// the polynomial we generated in `GenerateHidingPolynomial` function.
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func (tkg *ThresholdKeyGenerator) computeShare(index int) *big.Int {
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share := big.NewInt(0)
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for i := 0; i < tkg.Threshold; i++ {
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a := tkg.polynomialCoefficients[i]
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// we index authorities from 1, that's why we do index+1 here
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b := new(big.Int).Exp(big.NewInt(int64(index+1)), big.NewInt(int64(i)), nil)
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tmp := new(big.Int).Mul(a, b)
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share = new(big.Int).Add(share, tmp)
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}
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return new(big.Int).Mod(share, tkg.nm)
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}
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func (tkg *ThresholdKeyGenerator) createShares() []*big.Int {
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shares := make([]*big.Int, tkg.TotalNumberOfDecryptionServers)
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for i := 0; i < tkg.TotalNumberOfDecryptionServers; i++ {
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shares[i] = tkg.computeShare(i)
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}
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return shares
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}
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func (tkg *ThresholdKeyGenerator) delta() *big.Int {
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return Factorial(tkg.TotalNumberOfDecryptionServers)
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}
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// Generates verification keys for actions of decryption servers.
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//
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// For each decryption server `i`, we generate
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// v_i = v^(l! s_i) mod n^2
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//
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// where:
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// `l` is the number of decryption servers
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// `s_i` is a secret share for server `i`.
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// Secret shares were previously generated in the `CrateShares` function.
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func (tkg *ThresholdKeyGenerator) createViArray(shares []*big.Int) (viArray []*big.Int) {
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viArray = make([]*big.Int, len(shares))
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delta := tkg.delta()
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for i, share := range shares {
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tmp := new(big.Int).Mul(share, delta)
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viArray[i] = new(big.Int).Exp(tkg.v, tmp, tkg.nSquare)
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}
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return viArray
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}
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func (tkg *ThresholdKeyGenerator) createPrivateKey(i int, share *big.Int, viArray []*big.Int) *ThresholdPrivateKey {
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ret := new(ThresholdPrivateKey)
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ret.N = tkg.n
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ret.V = tkg.v
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ret.TotalNumberOfDecryptionServers = tkg.TotalNumberOfDecryptionServers
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ret.Threshold = tkg.Threshold
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ret.Share = share
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ret.Id = i + 1
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ret.Vi = viArray
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return ret
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}
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func (tkg *ThresholdKeyGenerator) createPrivateKeys() []*ThresholdPrivateKey {
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shares := tkg.createShares()
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viArray := tkg.createViArray(shares)
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ret := make([]*ThresholdPrivateKey, tkg.TotalNumberOfDecryptionServers)
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for i := 0; i < tkg.TotalNumberOfDecryptionServers; i++ {
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ret[i] = tkg.createPrivateKey(i, shares[i], viArray)
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}
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return ret
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}
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func (tkg *ThresholdKeyGenerator) Generate() ([]*ThresholdPrivateKey, error) {
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if err := tkg.initNumerialValues(); err != nil {
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return nil, err
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}
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if err := tkg.generateHidingPolynomial(); err != nil {
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return nil, err
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}
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return tkg.createPrivateKeys(), nil
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}
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