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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



Extended GCD Calculator (with steps)

Bezout Identity Calculator (with steps)

Batch InvMod Calculator

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Answers to Questions (FAQ)

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 \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.

How to calculate a modular inverse?

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.

How to calculate v in au+bv?

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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Modular Multiplicative Inverse on dCode.fr [online website], retrieved on 2024-12-23, https://www.dcode.fr/modular-inverse

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