The Goldwasser-Micali encryption scheme (see public key cryptography) is the first encryption scheme that achieved semantic security against a passive adversary under the assumption that solving the quadratic residuosity problem is hard. The scheme encrypts 1 bit of information, and the resulting ciphertext is typically 1024 bits long.
In the Goldwasser-Micali encryption scheme, a public key is a number n, that is a product of two primes numbers, say p and q. Let Y be a quadratic nonresidue modulo n (see quadratic residue and modular arithmetic), whose Jacobi Symbol is 1. The decryption key is formed by the prime factors of n.
The Goldwasser-Micali encryption scheme encrypts a bit b as follows. One picks an integer \( r\, (1<r<n-1) \) and outputs \( c=Y^{\,b} r^{2} \) mod n as ciphertext. That is, c is quadratic residue if and only if \( b=0 \) . Therefore a person knowing the prime factors of n can compute the quadratic residuosity of the ciphertext c, thus obtaining the value of b.
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References
Goldwasser, S. and S. Micali (1984). “Probabilistic encryption.” Journal of Computer and System Sciences, 28, 270–299.
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© 2005 International Federation for Information Processing
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Sako, K. (2005). GOLDWASSER-MICALI ENCRYPTION SCHEME. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_177
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DOI: https://doi.org/10.1007/0-387-23483-7_177
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