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founded on problems that are believed to be intractable. The higher residuosity problem (also called the nth-residuosity problem) is one such problem. This
Higher_residuosity_problem
Problem in computational number theory
The quadratic residuosity problem (QRP) in computational number theory is to decide, given integers a {\displaystyle a} and N {\displaystyle N} , whether
Quadratic_residuosity_problem
Hypothesis in computational complexity theory
(decisional composite residuosity problem) Benaloh cryptosystem (higher residuosity problem) Naccache–Stern cryptosystem (higher residuosity problem) For a composite
Computational hardness assumption
Computational_hardness_assumption
Decidability assumption
\,} Quadratic residuosity problem Higher residuosity problem P. Paillier, Public-Key Cryptosystems Based on Composite Degree Residuosity Classes, Eurocrypt
Decisional composite residuosity assumption
Decisional_composite_residuosity_assumption
Public-key security system
homomorphic public-key cryptosystem whose security rests on the higher residuosity problem. The Naccache–Stern cryptosystem was discovered by David Naccache
Naccache–Stern_cryptosystem
Integer that is a perfect square modulo some integer
must be coprime to the modulus. Gauss used R and N to denote residuosity and non-residuosity, respectively; for example, 2 R 7 and 5 N 7, or 1 R 8 and 3
Quadratic_residue
{r}})} time and space. The security of this scheme rests on the Higher residuosity problem, specifically, given z,r and n where the factorization of n is
Benaloh_cryptosystem
Algorithm for public key cryptography
cryptography. The problem of computing n-th residue classes is believed to be computationally difficult. The decisional composite residuosity assumption is
Paillier_cryptosystem
order p. This is very similar to the quadratic residuosity problem and the higher residuosity problem. Okamoto, Tatsuaki; Uchiyama, Shigenori (1998).
Okamoto–Uchiyama_cryptosystem
Cryptography method
reduced to solving some hard mathematical problem (e.g., Decisional Diffie-Hellman or the Quadratic Residuosity Problem). Other, semantically insecure algorithms
Semantic_security
Type of functions designed for being unsolvable by root-finding algorithms
based on the difficulty of the quadratic residuosity problem. Since the only known way to solve that problem is to factor the modulus, it is generally
Cryptographically secure pseudorandom number generator
Cryptographically_secure_pseudorandom_number_generator
Cryptographic problem
cryptographic protocols used by Schindelhauer are based on quadratic residuosity, and the general scheme is similar in spirit to the above protocol. The
Mental_poker
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