"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 one can determine uniquely the remainder of the division of n by the product of these integers..."
A theorem that provides a solution to a system of linear congruences.
Modular arithmetic: This is the foundation of the Chinese remainder theorem. It involves arithmetic done on numbers that are restricted to a particular range or modulus.
Divisibility and GCD: The greatest common divisor (GCD) is an important concept in the Chinese remainder theorem, as it is used to determine whether or not a system of linear congruences has a solution.
Euclidean algorithm: This is one of the algorithms used to calculate the GCD of two numbers. It involves a series of successive divisions and is an efficient method for finding the GCD.
Least common multiple (LCM): The LCM is another important concept in the Chinese remainder theorem, as it is used to simplify calculations and determine if a solution exists.
Linear congruences: This is the type of equation that the Chinese remainder theorem solves. It involves finding the value of x that satisfies a system of equations of the form ax ≡ b (mod n).
Chinese remainder theorem: This theorem states that if a system of linear congruences is pairwise coprime, then a unique solution exists.
Bezout's identity: This identity states that for any two integers a and b, there exist integers x and y such that ax + by = gcd(a, b).
Rings and fields: The Chinese remainder theorem is based on concepts from algebra, specifically rings and fields. A ring is a set of elements with two operations (usually addition and multiplication), while a field is a ring in which every nonzero element has an inverse.
Modular inverses: An inverse of a number in modular arithmetic is another number that, when multiplied by the original number, results in a value of 1 (mod n).
Applications of the Chinese remainder theorem: The Chinese remainder theorem has several practical applications, such as in cryptography and computer science. It can be used to solve systems of modular equations, factor large numbers, and perform computations on encrypted data.
Standard Chinese Remainder Theorem: For a finite sequence of pairwise co-prime integers m1, m2, ..., mk, and given any sequence of integers a1, a2, ..., ak, there exists a unique solution x such that x is congruent to ai (mod mi) for all i.
Generalized Chinese Remainder Theorem: Extends the standard theorem to consider coprime ideals in a ring, so the given sets of congruences are over different ideals.
Interactive Chinese Remainder Theorem: An extension of the standard theorem that is used for secure multiparty computation. It allows parties to compute a public solution to their respective congruences without revealing their private values to each other.
Combined Chinese Remainder Theorem: A method used to simplify calculations by solving the congruences modulo the product of the moduli, apart from computing them separately.
Noncoprime Chinese Remainder Theorem: An extension of the standard theorem that can solve a system of congruences with noncoprime moduli by splitting the moduli into coprime parts using the Chinese Remainder Theorem.
Polynomial Chinese Remainder Theorem: A generalization of the standard theorem for polynomials with coefficients in a field or a ring.
Modular Chinese Remainder Theorem: A set of rules that allow solving arithmetic operations modulo two or more coprime integers.
Systolic Chinese Remainder Theorem: A variation of the CRT that uses systolic arrays to compute the solution more efficiently.
"...under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1)."
"...we can determine that the remainder of n divided by 105 (the product of 3, 5, and 7) is 23."
"...then without knowing the value of n, we can determine that the remainder of n divided by the product of these integers..."
"The earliest known statement of the theorem is by the Chinese mathematician Sunzi in the Sunzi Suanjing in the 3rd century CE."
"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 has been generalized to any ring, with a formulation involving two-sided ideals."
"...then one can determine uniquely the remainder of the division of n by the product of these integers..."
"...the divisors are pairwise coprime (no two divisors share a common factor other than 1)."
"...under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1)."
"...then without knowing the value of n, we can determine that the remainder of n divided by 105 (the product of 3, 5, and 7) is 23."
"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 earliest known statement of the theorem is by the Chinese mathematician Sunzi in the Sunzi Suanjing in the 3rd century CE."
"...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."
"...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 has been generalized to any ring, with a formulation involving two-sided ideals."
"Under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1)."
"...we can determine that the remainder of n divided by 105 (the product of 3, 5, and 7) is 23."