In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then...

Remainder theorem may refer to: Polynomial remainder theorem Chinese remainder theorem This disambiguation page lists mathematics articles associated with...

equations. Although a special case of the Chinese remainder theorem had already been described in the Chinese book Sunzi Suanjing, the general solution...

well beyond mere proofs of existence. By an ingenious use of the Chinese remainder theorem, we can constructively define such a recursive function β {\displaystyle...

x_{2}=x_{1}+kv,\quad y_{2}=y_{1}-ku,} which completes the proof. The Chinese remainder theorem describes an important class of linear Diophantine systems of...

resulting scheme is equivalent to Shamir's polynomial system. The Chinese remainder theorem can also be used in secret sharing, for it provides us with a...

(mod λ(pq)). This is part of the Chinese remainder theorem, although it is not the significant part of that theorem. Although the original paper of Rivest...

\;\mathbb {Z} /R\mathbb {Z} ,} an isomorphism that follows from the Chinese Remainder Theorem (CRT). For an integer T {\displaystyle T} with 0 ≤ T < N R {\displaystyle...

completion of the integers Z {\displaystyle \mathbb {Z} } . By the Chinese remainder theorem, Z ^ {\displaystyle {\widehat {\mathbb {Z} }}} can also be understood...

interpolating polynomial must satisfy. For another method, see Chinese remainder theorem § Hermite interpolation. For yet another method, see, which uses...

coefficients −9 and 2. Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity...

of the form x ≡ k (mod a) and x ≡ m (mod b), has a solution (Chinese remainder theorem); in fact the solutions are described by a single congruence relation...

First compute Mp ≡ Cdp (mod p) and Mq ≡ Cdq (mod q). Use the Chinese remainder theorem to compute the unique value of 0 ≤ M < N that satisfies M ≡ Mp...

⋅ q = 1 {\displaystyle y_{p}\cdot p+y_{q}\cdot q=1} . Use the Chinese remainder theorem to find the four square roots of c {\displaystyle c} modulo n...

shares, each containing partial information about the secret. The Chinese remainder theorem (CRT) states that for a given system of simultaneous congruence...

polynomials with integer coefficients through modular computation and Chinese remainder theorem (see below). Let P = p 0 + p 1 X + ⋯ + p m X m , Q = q 0 + q 1...

In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups G over global fields k. Eichler...

one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime...

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...

isomorphic to the direct product of these rings. For example, the Chinese remainder theorem may be stated as: if m and n are coprime integers, the quotient...

logarithm modulo each prime power in the group order) and the Chinese remainder theorem (to combine these to a logarithm in the full group). (Again, we...

prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime...

used to generate sets of equations, that can be used in CRT (Chinese remainder theorem): g ( M p , n − 1 ) 2 ≡ − 1 ( mod M p , n ) {\displaystyle g^{\frac...

polynomial remainder theorem: If a polynomial f(x) is divided by x − k, the remainder is the constant r = f(k). Chinese remainder theorem Divisibility...

φ(m), etc. Then there is a bijection between A × B and C by the Chinese remainder theorem. If p is prime and k ≥ 1, then φ ( p k ) = p k − p k − 1 = p k...

important theorems relating to modular arithmetic: Carmichael's theorem Chinese remainder theorem Euler's theorem Fermat's little theorem (a special...

mod q − 1 ) {\displaystyle d_{q}\equiv d{\pmod {q-1}}} if the Chinese remainder theorem is used to improve the speed of decryption, see CRT-RSA. Encryption...

for each i, m can be recovered by a direct application of the Chinese remainder theorem. The semantic security of the Naccache–Stern cryptosystem rests...

modular arithmetic) is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form Z / k Z {\displaystyle \mathbb...

Edmund F. (January 2004), "Chinese numerals", MacTutor History of Mathematics Archive, University of St Andrews "Chinese numerals". Maths History. Retrieved...