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Let an arbitrarily small positive constant δ less than 1 and a poly-nomial f with integer coefficients be fixed. We prove unconditionally that J modulo p can be completely factored in deterministic polynomial time if p— 1 has a (lnp) O(1) -smooth divisor exceeding p δ . We also address the issue of factoring f modulo p over finite extensions of the prime field F p and show that p - 1 can be replaced by p k ― 1 (k ∈ N) for explicit classes of primes p.

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