Deterministic polynomial factorisation modulo many primes
By: Daniel Altman
Potential Business Impact:
Breaks down math problems into smaller pieces.
Designing a deterministic polynomial time algorithm for factoring univariate polynomials over finite fields remains a notorious open problem. In this paper, we present an unconditional deterministic algorithm that takes as input an irreducible polynomial $f \in \mathbb{Z}[x]$, and computes the factorisation of its reductions modulo $p$ for all primes $p$ up to a prescribed bound $N$. The \emph{average running time per prime} is polynomial in the size of the input and the degree of the splitting field of $f$ over $\mathbb{Q}$. In particular, if $f$ is Galois, we succeed in factoring in (amortised) deterministic polynomial time.
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