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Secret sharing consists of recovering a secret S from a set of shares, each containing partial information about the secret. The Chinese remainder theorem (CRT) states that for a given system of simultaneous congruence equations, the solution is unique in some Z/nZ, with n > 0 under some appropriate conditions on the congruences.
The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers. The Chinese remainder theorem (expressed in terms of congruences) is true over every principal ideal domain.
It is used in cryptography, integer factorization, and primality testing. It is an abelian, ... More specifically, the Chinese remainder theorem [7] ...
The Chinese remainder theorem can also be used in secret sharing, for it provides us with a method to uniquely determine a number S modulo k many pairwise coprime integers ,,...,, given that < =. There are two secret sharing schemes that make use of the Chinese remainder theorem, Mignotte's and Asmuth-Bloom's Schemes.
Most of the implementations of RSA will accept exponents generated using either method (if they use the private exponent d at all, rather than using the optimized decryption method based on the Chinese remainder theorem described below), but some standards such as FIPS 186-4 (Section B.3.1) may require that d < λ(n).
The public key in the RSA system is a tuple of integers (,), where N is the product of two primes p and q.The secret key is given by an integer d satisfying (() ()); equivalently, the secret key may be given by () and () if the Chinese remainder theorem is used to improve the speed of decryption, see CRT-RSA.
A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if M is the product of the moduli, there is, in an interval of length M, exactly one integer having any given set of modular values.
Euler's Phi Function and the Chinese Remainder Theorem — proof that φ(n) is multiplicative Archived 2021-02-28 at the Wayback Machine; Euler's totient function calculator in JavaScript — up to 20 digits; Dineva, Rosica, The Euler Totient, the Möbius, and the Divisor Functions Archived 2021-01-16 at the Wayback Machine