Using the smoothness of p-1 for computing roots modulo p

TL;DR

This paper presents a deterministic polynomial-time method for factoring polynomials and computing roots modulo a prime p, based on the smoothness properties of p-1, without relying on the Extended Riemann Hypothesis.

Contribution

It introduces a new approach that leverages the smoothness of p-1 to enable efficient deterministic algorithms for polynomial factorization and root finding modulo p.

Findings

Complete polynomial factorization modulo p in deterministic polynomial time.

Deterministic computation of roots modulo p under smoothness conditions.

No reliance on the Extended Riemann Hypothesis for these results.

Abstract

We prove, without recourse to the Extended Riemann Hypothesis, that the projection modulo $p$ of any prefixed polynomial with integer coefficients can be completely factored in deterministic polynomial time if $p-1$ has a $(\ln p)^{O(1)}$-smooth divisor exceeding $(p-1)^{{1/2}+\delta}$ for some arbitrary small $\delta$. We also address the issue of computing roots modulo $p$ in deterministic time.