Modular Multiplicative Inverse

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.

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.

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.