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A modular multiplicative inverse of a modulo m can be found by using the extended Euclidean algorithm. The Euclidean algorithm determines the greatest common divisor (gcd) of two integers, say a and m. If a has a multiplicative inverse modulo m, this gcd must be 1. The last of several equations produced by the algorithm may be solved for this gcd.
In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that ax ≡ 1 (mod n). This multiplicative inverse exists if and only if a and n are coprime. For example, the inverse of 3 modulo 11 is 4 because 4 ⋅ 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it.
Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ 1 (mod n). It exists precisely when a is coprime to n , because in that case gcd( a , n ) = 1 and by Bézout's lemma there are integers x and y satisfying ax + ny = 1 .
This integer a −1 is called a modular multiplicative inverse of a modulo m. If a ≡ b (mod m) and a −1 exists, then a −1 ≡ b −1 (mod m) (compatibility with multiplicative inverse, and, if a = b, uniqueness modulo m). If ax ≡ b (mod m) and a is coprime to m, then the solution to this linear congruence is given by x ≡ a −1 b (mod m).
When R is a power of a small positive integer b, N′ can be computed by Hensel's lemma: The inverse of N modulo b is computed by a naïve algorithm (for instance, if b = 2 then the inverse is 1), and Hensel's lemma is used repeatedly to find the inverse modulo higher and higher powers of b, stopping when the inverse modulo R is known; N′ is ...
Modular exponentiation can be performed with a negative exponent e by finding the modular multiplicative inverse d of b modulo m using the extended Euclidean algorithm. That is: c = b e mod m = d −e mod m, where e < 0 and b ⋅ d ≡ 1 (mod m). Modular exponentiation is efficient to compute, even for very large integers.
Italy's competition watchdog AGCM closed an investigation into Booking.com as it found commitments offered by the online travel platform sufficient to address the competition concerns it had ...
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z n; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or ...