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

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