The Goldwasser–Micali (GM) cryptosystem is an asymmetric key encryption algorithm developed by Shafi Goldwasser and Silvio Micali in 1982. GM has the distinction of being the first probabilistic public-key encryption scheme which is provably secure under standard cryptographic assumptions. However, it is not an efficient cryptosystem, as ciphertexts may be several hundred times larger than the initial plaintext. To prove the security properties of the cryptosystem, Goldwasser and Micali proposed the widely-used definition of semantic security.
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... There is a simple reduction from breaking this cryptosystem to the problem of determining whether a random value modulo N with Jacobi symbol +1 is a ... If an algorithm A breaks the cryptosystem, then to determine if a given value x is a quadratic residue modulo N, we test A to see if it can break the cryptosystem using (x,N) as a public key ... The GM cryptosystem has homomorphic properties, in the sense that if c0, c1 are the encryptions of bits m0, m1, then c0c1 mod N will be an encryption of ...