According to [DJN 10], the plaintext space is `Z_n`. We had this
information in the `Encrypt` and `EncryptWithR` documentation but
sometimes, when encrypting result of more complex equations, it's easy
to miss the fact the result is out of the allowed range.
Here we have just a safety check making sure that if plaintext is out of
allowed space, we report an error.
It's now possible to multiply Cypher by an integer.
Paillier is a homomorphic encryption scheme, where an encrypted
plaintext `m` raised to an integer `k` will decrypt to the product of
the plaintext `m` and `k`.
`PublicKey.Add` accepts now an arbitrary number of `*Cypher`s.
Previously, it was possible to add only two `*Cypher*` instances and it
was quite unwieldy for a lot of use cases.
The current implementation is backward compatible. It's still possible
to pass just two `*Cypher` arguments, so every project referencing the
library will observe no compilation issues.
The code for generating private key is based on approach described in [KL 08], construction 11.32 which is compatible with the one described in [DJN 10], section 3.2 except that instead of generating Lamda private key compontent from LCM of `p` and `q` we use Euler's totient function as suggested in [KL 08]. I cleared up naming as well as removed G and Mu parameters from Paillier key. For the modification of Paillier scheme we use, where g = n + 1, N is the only one parameter needed for a public key. Also, Mu is needed only for Decrypting and can be computed there from Lambda and N so there is no need to keep it in the private key.
I added documentation to <Key, Enc, Dec> functions with an important note that `p` and `q` supplied to `CreatePrivateKey` must be either of equal length or `gcd(pq, (p-1)(q-1)) = 1` which is assured if both primes are of equal length. This requirement is mentioned both in [KL 08] and [DJN 10].
[KL 08]: Jonathan Katz, Yehuda Lindell, (2008)
Introduction to Modern Cryptography: Principles and Protocols,
Chapman & Hall/CRC
[DJN 10]: Ivan Damgard, Mads Jurik, Jesper Buus Nielsen, (2010)
A Generalization of Paillier’s Public-Key System
with Applications to Electronic Voting
Aarhus University, Dept. of Computer Science, BRICS
Piotr Dyraga