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Modular Multiplicative Inverse

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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.
Summary

Modular Inverse Calculator

What is the modular Inverse? (Definition)

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

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.

How to calculate a modular inverse?

To calculate the value of the modulo inverse, use the gcd">extended euclidean algorithm which find 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 gcd">Extended Euclidean algorithm for its inverse modulo N calculator and arbitrary precision functions to get results with big integers.

How to calculate v?

Use the Bezout identity, also available on dCode.

What does invmod mean?

The keyword invmod is the abbreviation of inverse modular.

What is a multiplicative inverse?

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

Source code

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