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...

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...

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

(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...

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

the remainder term (given below) which are valid under some additional regularity assumptions on f. These enhanced versions of Taylor's theorem typically...

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...

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...

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...

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

understand the construction of the profinite integers is by using the Chinese remainder theorem. Recall that for an integer n {\displaystyle n} with prime factorization...

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...

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...

attack cannot be applied regardless of how small d is. Using the Chinese remainder theorem: Suppose one chooses d such that both dp ≡ d (mod (p − 1)) and...

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...

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...

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...

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

CRT, a telnet client .crt, X.509 Certificate filename extension Chinese remainder theorem, in number theory Crater (constellation), in astronomy (abbreviated...

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...

the Chinese Remainder theorem to a minimal primary decomposition of the zero ideal. There is also the following result, due to Hungerford: Theorem (Hungerford):...

isomorphism. These statements are equivalent as a result of the Chinese remainder theorem, which implies that Z j k ≅ Z j ⊕ Z k {\displaystyle \mathbb {Z}...

⋅ 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...

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

contains discoveries which found their origins in China. Chinese remainder theorem: The Chinese remainder theorem, including simultaneous congruences in number...

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...

analogous to the Chinese remainder theorem. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials...

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...

exponent e, but different p, q and n, then one can recover m via the Chinese remainder theorem. Thus, if key encapsulations for several recipients need to be...

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...