On Newton-Raphson iteration for multiplicative inverses modulo prime powers

Jean-Guillaume Dumas (1)

1 LJK - Laboratoire Jean Kuntzmann

We study algorithms for the fast computation of modular inverses. We first give another proof of the formulas of [1] for the modular inverse modulo 2 , derived from Newton-Raphson iteration over p-adic fields, namely Hensel's lifting. From the expression of Newton-Raphson's iteration we then derive a recurrence relation and an actually explicit formula for the modular inverse, generalizing to any prime power modulus. On the one hand, we show then that despite a worse complexity the two newly obtained algorithms can be 4 times faster for small exponents. On the other hand, these two algorithms become slower for arbitrary precision integers of more than 1500 bits.

Domaines: Calcul formel [cs.SC]; Logiciel mathématique [cs.MS]

Jean-Guillaume Dumas. On Newton-Raphson iteration for multiplicative inverses modulo prime powers. 2012. ⟨hal-00736701v1⟩