Paillier scheme Sample Clauses

Paillier scheme.ย 1โ€Œ In Scheme 1 are the public key ๐‘๐‘˜ and secret key ๐‘ ๐‘˜ generated by the following algorithm. First are chosen two large prime numbers ๐‘ and ๐‘ž at random such that they are independent of each other and that GCD(๐‘๐‘ž, (๐‘ โˆ’ 1)(๐‘ž โˆ’ 1)) = 1 (this property is assured if both primes are of equal length). Than is computed the parameter ๐‘› = ๐‘๐‘ž and parameter ๐œ† as LCM(๐‘ โˆ’ 1, ๐‘ž โˆ’ 1) followed by the generation of parameter ๐‘” โˆˆ Z๐‘›* 2 at random. At this point must be ensured that ๐‘› divides the ๐‘” by checking the existence of modular multiplicative inverse ๐œ‡ as ๐œ‡ = (๐ฟ(๐‘”๐œ† mod ๐‘›2))โˆ’1 mod ๐‘›, (2.5) where function ๐ฟ(๐‘ฅ) = ๐‘ฅโˆ’1 where the notation ๐‘Ž denotes the quotient of ๐‘Ž divided by ๐‘. The public key ๐‘๐‘˜ is than the ordered set of parameters ๐‘› and ๐‘” and the secret key ๐‘ ๐‘˜ is the ordered set of parameters ๐œ† and ๐œ‡. The encryption of a message ๐‘š is computed as ๐‘ = ๐‘”๐‘š ยท ๐‘Ÿ๐‘› mod ๐‘›2, (2.6) where ๐‘š must be in range โŸจ0, ๐‘›) and parameter ๐‘Ÿ is a generated number from interval (0, ๐‘›) at random. The decryption process is essentially exponentiation modulo ๐‘›2. The ciphertext is decrypted as ๐‘š = ๐ฟ(๐‘๐œ† mod ๐‘›2) ยท ๐œ‡ mod ๐‘›. (2.7)
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Paillier scheme.ย โ€Œ The Paillier cryptosystem [32] is a probabilistic asymmetric algorithm for public- key cryptography first proposed by โ–‡โ–‡โ–‡โ–‡โ–‡โ–‡ โ–‡โ–‡โ–‡โ–‡โ–‡โ–‡โ–‡โ–‡ in 1999 in his paper โ€œPublic- Key Cryptosystems Based on Composite Degree Residuosity Classesโ€. This paper offers a trapdoor mechanism from the family of trapdoors in the discrete logarithm technique based on composite residuosity classes. This scheme has an additive homomorphic property, i.e., allows adding two ci- phertexts without corrupting the result. The computation behaves just as if the corresponding plaintexts are added (without the need of decryption). However, a drawback of the Paillier cryptosystem is its efficiency which is a common issue for any scheme based on homomorphic encryption. Several optimi- sation propositions has already been made. For instance, โ–‡โ–‡โ–‡โ–‡โ–‡โ–‡โ–‡โ–‡ himself proposed a variant, namely Scheme 3, of the original proposal, namely Scheme 1, which uses abound ๐›ผ on exponentiation power size to speed up the computation of the encryp- tion in particular. Another improving method can be the use of pre-computation of some values, for instance, the exponentiation of either the message ๐‘”๐‘š or the noise ๐‘Ÿ๐‘› as suggested in the article [17]. The use-cases of the Paillier cryptosystem are, for instance, in electronic voting [2], machine learning on encrypted data [6], and can be also used in other algorithms that demand high confidentiality even during processing between mutually authen- ticated parties because of other features such as the self-binding ability (ability to change one ciphertext into another without changing the content of the decryption) [33].
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