Using partial smoothness of 𝑝-1 for factoring polynomials modulo 𝑝

Abstract

Let an arbitrarily small positive constant δ \delta less than 1 1 and a polynomial f f with integer coefficients be fixed. We prove unconditionally that f f modulo p p can be completely factored in deterministic polynomial time if p − 1 p-1 has a ( ln ⁡ p ) O ( 1 ) (\ln p)^{O(1)} -smooth divisor exceeding p δ p^\delta . We also address the issue of factoring f f modulo p p over finite extensions of the prime field F p \mathbb {F}_p and show that p − 1 p-1 can be replaced by p k − 1 p^k-1 ( k ∈ N k\in \mathbb {N} ) for explicit classes of primes p p .