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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.
Sample matrix inversion (or direct matrix inversion) is an algorithm that estimates weights of an array (adaptive filter) by replacing the correlation matrix with its estimate.
The modular inverse of aR mod N is REDC((aR mod N) −1 (R 3 mod N)). Modular exponentiation can be done using exponentiation by squaring by initializing the initial product to the Montgomery representation of 1, that is, to R mod N, and by replacing the multiply and square steps by Montgomery multiplies.
The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's method. In particular, if either exp {\displaystyle \exp } or log {\displaystyle \log } in the complex domain can be computed with some complexity, then that complexity is ...
A matrix (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix (in this case +) if and only if = =. We first verify that the right hand side ( Y {\displaystyle Y} ) satisfies X Y = I {\displaystyle XY=I} .
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.
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).
In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers : there is an integer matrix N that is its inverse (these are equivalent under Cramer's rule ).