Let n be the product of two distinct odd prime numbers p, q, and let x be an integer such that the Jacobi symbol \(({x}/{n}) = +1\). The Quadratic Residuosity Problem (QRP) is to determine, given x and n, whether x is a quadratic residue modulo n (see modular arithmetic). (All quadratic residues have Jacobi symbol \(+1\), but not necessarily the reverse.) This problem is easy to solve given the factors p and q, but is believed to be difficult given only x and n. However, it is not known whether the problem is equivalent to factoring the modulus n.
The QRP is the basis for several cryptosystems, including the Goldwasser–Micali encryption scheme and Cocks’ identity-based encryption scheme [1] (see identity-based cryptosystems).
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References
Cocks, Clifford (2001). “An identity based encryption scheme based on quadratic residues.” Cryptography and Coding, Lecture Notes in Computer Science, vol. 2260, ed. B. Honary. Springer, Berlin, 360–363.
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Kaliski, B. (2005). Quadratic Residuosity Problem. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_336
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