Deterministic polynomial-time factorization of polynomials over prime fields

Theorem 1.1: a uniform deterministic algorithm computes the complete factorization (irreducible factors with multiplicities) of any nonzero dense $f\in\mathbb F_p[x]$ in $O(((n+1)\lceil\log_2 p\rceil)^{10^{12}})$ bit operations, with no randomness, oracle or GRH. Theorem 1.2 (proved here, unconditional) does this given small auxiliary primes $\ell\equiv1\pmod{12q}$ with $p$ not a $q$th power mod $\ell$; the odd-degree splitter uses Jacobians of cyclic curves $Y^q=F(X)$ and norm equations in cyclic algebras. Small auxiliary primes are deduced from a uniform zero-free strip $\mathrm{Re}\,s>1-10^{-6}$ for all finite-order Hecke L-functions over cyclotomic fields containing $\mu_{12}$, imported from the companion on primitive roots and not proved here. The exponent is not practical.

Theorem 1.1: a uniform deterministic algorithm computes the complete factorization (irreducible factors with multiplicities) of any nonzero dense $f\in\mathbb F_p[x]$ in $O(((n+1)\lceil\log_2 p\rceil)^{10^{12}})$ bit operations, with no randomness, oracle or GRH. Theorem 1.2 (proved here, unconditional) does this given small auxiliary primes $\ell\equiv1\pmod{12q}$ with $p$ not a $q$th power mod $\ell$; the odd-degree splitter uses Jacobians of cyclic curves $Y^q=F(X)$ and norm equations in cyclic algebras. Small auxiliary primes are deduced from a uniform zero-free strip $\mathrm{Re}\,s>1-10^{-6}$ for all finite-order Hecke L-functions over cyclotomic fields containing $\mu_{12}$, imported from the companion on primitive roots and not proved here. The exponent is not practical.