Tool to compute the modular inverse of a number. The modular multiplicative inverse of an integer N modulo m is an integer n such as the inverse of N modulo m equals n.

Modular Multiplicative Inverse - dCode

Tag(s) : Arithmetics

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The value of the modular inverse of $ a $ by the modulo $ n $ is the value $ a^{-1} $ such that $ a \cdot a^{-1} \equiv 1 \pmod n $

It is common to note this modular inverse $ u $ and to use these equations $$ u \equiv a^{-1} \pmod n \\ a u \equiv 1 \pmod n $$

If a modular inverse exists then it is unique.

To calculate the value of the modulo inverse, use the extended euclidean algorithm which finds solutions to the Bezout identity $ au + bv = \text{G.C.D.}(a, b) $. Here, the gcd value is known, it is 1: $ \text{G.C.D.}(a, b) = 1 $, thus, only the value of $ u $ is needed.

__Example:__ $ 3^{-1} \equiv 4 \mod 11 $ because $ 4 \times 3 = 12 $ and $ 12 \equiv 1 \mod 11 $

dCode uses the Extended Euclidean algorithm for its inverse modulo N calculator and arbitrary precision functions to get results with big integers.

Use the Bezout identity, also available on dCode.

The keyword `invmod` is the abbreviation of `inverse modular`.

A multiplicative inverse is the other name of a modular inverse.

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Cite as source (bibliography):

*Modular Multiplicative Inverse* on dCode.fr [online website], retrieved on 2023-09-27,

invmod,inverse,modular,modulo,mod,euclide,gcd,bezout,calculator,multiplicative

https://www.dcode.fr/modular-inverse

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