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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae , published in 1801.
In modular arithmetic, the integers coprime (relatively prime) to n from the set {,, …,} of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n.
To use REDC to compute the product of 7 and 15 modulo 17, first convert to Montgomery form and multiply as integers to get 12 as above. Then apply REDC with R = 100, N = 17, N′ = 47, and T = 12. The first step sets m to 12 ⋅ 47 mod 100 = 64. The second step sets t to (12 + 64 ⋅ 17) / 100.
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation.. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.
In number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that (). [1]In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n.
t 2 = 6 is the modular multiplicative inverse of 5 × 11 (mod 7) and t 3 = 6 is the modular multiplicative inverse of 5 × 7 (mod 11). Thus, X = 3 × (7 × 11) × 4 + 6 × (5 × 11) × 4 + 6 × (5 × 7) × 6 = 3504. and in its unique reduced form X ≡ 3504 ≡ 39 (mod 385) since 385 is the LCM of 5,7 and 11. Also, the modular multiplicative ...
In modular arithmetic, Barrett reduction is an algorithm designed to optimize the calculation of [1] without needing a fast division algorithm. It replaces divisions with multiplications, and can be used when n {\displaystyle n} is constant and a < n 2 {\displaystyle a<n^{2}} .
Modulus (modular arithmetic), base of modular arithmetic; Modulus, the absolute value of a real or complex number ( | a |) Moduli space, in mathematics a geometric space whose points represent algebro-geometric objects; Conformal modulus, a measure of the size of a curve family; Modulus of continuity, a function gauging the uniform continuity ...