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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



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.

Answers to Questions

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

dCode retains ownership of the source code of the script Modular Multiplicative Inverse online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Modular Multiplicative Inverse script for offline use on PC, iPhone or Android, ask for price quote on contact page !

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