Deterministic polynomial factorization in large odd characteristic

Determine whether polynomial-time deterministic algorithms for factoring polynomials over finite fields of large odd characteristic exist, even assuming the Extended Riemann Hypothesis.

Background

The paper identifies the splitting step of finite-field polynomial factorization as the principal source of difficulty in large odd characteristic. Randomized methods are efficient, while known deterministic methods either have a cost polynomial in the field size or rely on conditional number-theoretic assumptions.

The unresolved issue is whether polynomial-time deterministic factorization is possible in this regime at all; the authors emphasize that the question remains open even under the Extended Riemann Hypothesis. The paper’s exact quantum procedure addresses the randomization directly but does not resolve the corresponding classical derandomization problem.

References

The hard regime is therefore large $q$ of odd characteristic. There, every known efficient classical splitting method is randomized. A random test element splits the current block with probability at least $1/2$, and repeating the trial drives the failure probability down exponentially, but no zero-failure-probability method is known, nor is an unconditional deterministic one. As Gao observes , it remains open whether polynomial-time deterministic factorization exists over such fields even assuming the Extended Riemann Hypothesis.

— Exact quantum splitting and the structure of finite algebras  (2608.30340 - Imran, 31 Aug 2026) in Section 1, Introduction, paragraph “What is known classically”