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Modular Multiplicative Inverse

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.

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Modular Multiplicative Inverse -

Tag(s) : Arithmetics

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# Modular Multiplicative Inverse

## 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 $a a ^{-1} \equiv 1 \pmod n$

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.

### How to calculate a modular inverse?

To calculate the value of the modulo inverse, use the 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 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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