In the spirit of earlier encryption schemes suggested by Goldwasser–Micali, Benaloh, Naccache–Stern, and Okamoto–Uchiyama, Paillier proposed in 1999 a public-key cryptosystem [4] (see publickey cryptograhy) based on the properties of nth powers modulo n2 where n is an RSA modulus (see modular arithmetic and RSA public key encryption). The original observation is that the function ε(x, y)=g x y n mod n 2 is a one-way trapdoor permutation (see trapdoor one-way function) over the group \(\mathbb{Z}_n\times\mathbb{Z}^{\ast}_n\simeq \mathbb{Z}^{\ast}_{n^2}\) where the trapdoor information is the factorization of n. The group \(\mathbb{Z}^{\ast}_{n^2}\) is of order nϕ where \(\phi = \phi(n)\) is Euler's totient function of n and the base \(g\in\mathbb{Z}^{\ast}_{n^2}\) is an element of order α · n for some divisor α of ϕ (for instance n + 1 for which α = 1). Noting \(L(u)=(u-1)/n\) when u = 1 mod n, x is recovered from \(w = \mathcal{E}(x, y)\) as \(x = L(w^ \phi \bmod{n^2}) /L(g^ \phi...
References
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Paillier, P. (2005). Paillier Encryption and Signature Schemes. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_293
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