On Newton-Raphson iteration for multiplicative inverses modulo prime powers
Résumé
We study algorithms for the fast computation of modular inverses. We first give another proof of the formulas of Arazi and Qi for the modular inverse modulo $2^m$, derived from Newton-Raphson iteration over p-adic fields, namely Hensel's lifting. From the expression of Newton-Raphson's iteration we then derive 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 explicit formula can be $4$ times faster than Arazi and Qi's for small exponents. On the other hand, this algorithm becomes slower for arbitrary precision integers of more than $1700$ bits. Overall a hybrid combination of the two latter algorithms yield a constant factor improvement also for large exponents.
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