What is the Chinese remainder theorem?

What is the Chinese remainder theorem?

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, under the condition that the divisors are pairwise coprime (no two …

How do you find the inverse of the Chinese remainder theorem?

We can find the multiplicative inverse of b mod a if a and b are relatively prime (if not, the multiplicative inverse does not exist). Since gcd(a, b) = 1, the extended Euclidean algorithm gives us s and t such that sa + tb = 1. Taking mod a on both sides, we get that t = b−1 mod a. Chinese remainder theorem.

Why is it called the Chinese remainder theorem?

There was a much greater emphasis on algorithms used to solve more or less practical problems. The Chinese remainder theorem in its original form was an algorithm devised to solve a problem often known as “an unknown quantity of things” (物不知數), so that was the name used to refer to the original method.

What is Chinese remainder theorem in cryptography and network security?

One of the most useful results of number theory is the Chinese remainder theorem (CRT). In essence, the CRT says it is possible to reconstruct integers in a certain range from their residues modulo a set of pairwise relatively prime moduli.

How is Chinese remainder theorem used in cryptography?

Proposed technique. Our secret image sharing scheme is based on Chinese Remainder Theorem (CRT), where compression and encryption is achieved by solving r different congruent equations, used for encrypting r pixels at a time in secret image. Here, r is also equal to the compression factor of the image.

Is Sun Tzu a mathematician?

Sun Tzu or Sun Zi was a Chinese mathematician of the third century CE. His interests were in astronomy. He tried to develop a calendar and for this he investigated Diophantine equations.

Who proved the Chinese remainder theorem?

Sun Zi
Chinese remainder theorem, ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its origin in the work of the 3rd-century-ad Chinese mathematician Sun Zi, although the complete theorem was first given in 1247 by Qin Jiushao.

How do you use the Chinese remainder theorem to solve equations?

Example: Solve the simultaneous congruences x ≡ 6 (mod 11), x ≡ 13 (mod 16), x ≡ 9 (mod 21), x ≡ 19 (mod 25). Solution: Since 11, 16, 21, and 25 are pairwise relatively prime, the Chinese Remainder Theorem tells us that there is a unique solution modulo m, where m = 11⋅16⋅21⋅25 = 92400.

Who first proved the Chinese remainder theorem?

Chinese remainder theorem, ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its origin in the work of the 3rd-century-ad Chinese mathematician Sun Zi, although the complete theorem was first given in 1247 by Qin Jiushao.

What do you mean by Remainder Theorem?

Definition of remainder theorem : a theorem in algebra: if f(x) is a polynomial in x then the remainder on dividing f(x) by x − a is f(a)

How do you use the Chinese Remainder Theorem?

For any system of equations like this, the Chinese Remainder Theorem tells us there is always a unique solution up to a certain modulus, and describes how to find the solution efficiently. Theorem: Let p, q be coprime. Then the system of equations x = a (mod p)

What is the Chinese remainder problem?

The Chinese Remainder Problem appeared around the first century AD in Sun Zie’s book. Its uses ranged from the computation of calendars and counting soldiers to building the wall and base of a house. Later on, it became known as the Chinese Remainder Theorem involving integers and remainders under division. Over a period of

Is the Chinese Remainder Theorem for two ideals isomorphic?

The Chinese Remainder Theorem for Two Ideals If R is a commutative ring and I and J are proper ideals with I + J = R, then R/ (IAJ) is isomorphic to R/I ffi R/ J.

What is Chinese remainder algorithm for integers?

2.4 Chinese Remainder Algorithm for Integers The Chinese Remainder Algorithm was generated based on Sun Zi’s method to solve the original problem. By the 13th century, Quin Jiushao gave a more general method which did not restrict the modulimito pairwise relatively prime numbers. His method,