Let \(a,b\in\mathbb{Z}\) be positive integers $a,b\in\mathbb Z$ with \(a\le b\) which are co-prime. The algorithm \(\operatorname{invmod}(a,b)\) calculates correctly the multiplicative inverse $a^{-1}$ in the ring of congruences $\mathbb Z_b,$ i.e. for which $$a\cdot a^{-1}\equiv 1\mod b.$$ In particular, if $b=p$ is a prime number, this is the unique inverse of $a$ modulo $b$ in the field of congruences $\mathbb Z_p.$

The algorithm requires \(\mathcal O(\log |b|)\) (worst case and average case) division operations, which corresponds to \(\mathcal O(\log^2 |b|)\) bit operations.

Algorithm: Calculation of Inverses Modulo a Number (Python)

def invmod(a, b): res = gcdext(a, b) if res[^0] != 1: raise NotCoPrimeException(a, b) else: if res[^1] < 0: res[^1] = res[^1] + (abs(res[^1]) // b + 1) * b return res[^1]


print(invmod(16, 21))

  1. will output
  2. will output

Proofs: 1

Propositions: 1

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  1. Hermann, D.: "Algorithmen Arbeitsbuch", Addison-Wesley Publishing Company, 1992
  2. Blömer, J.: "Lecture Notes Algorithmen in der Zahlentheorie", Goethe University Frankfurt, 1997