A number-theoretic conjecture implying faster algorithms for polynomial factorization and integer factorization
Chris Umans, Siki Wang
TL;DR
The paper reframes polynomial factorization over finite fields as a problem tied to a number-theoretic conjecture about structured difference-sets $S-T$ and their divisors. By replacing the standard $X^{q^i}-X$ GCD checks with $X^{q^A}-X$ for carefully chosen $A$ drawn from $S-T$, the authors propose a framework that could dramatically reduce the exponent in polynomial factoring from $3/2$ toward $4/3$ (and potentially to $1/2$ for special cases), contingent on the conjecture holding with favorable parameters. They also extend these ideas to integer factoring, yielding potential deterministic improvements to the current best exponents under the same conjectural assumptions. A variant based on arithmetic progressions strengthens the connection and highlights both the promise and the challenges of grounding these algorithmic gains in solid number-theoretic foundations. The work opens multiple avenues for future research, including derandomization, AP-variant analyses, and practical implementations of nearly-linear time factoring via structured difference-sets.
Abstract
The fastest known algorithm for factoring a degree $n$ univariate polynomial over a finite field $\mathbb{F}_q$ runs in time $O(n^{3/2 + o(1)}\text{polylog } q)$, and there is a reason to believe that the $3/2$ exponent represents a ''barrier'' inherent in algorithms that employ a so-called baby-steps-giant-steps strategy. In this paper, we propose a new strategy with the potential to overcome the $3/2$ barrier. In doing so we are led to a number-theoretic conjecture, one form of which is that there are sets $S, T$ of cardinality $n^β$, consisting of positive integers of magnitude at most $\exp(n^α)$, such that every integer $i \in [n]$ divides $s-t$ for some $s \in S, t \in T$. Achieving $α+ β\le 1 + o(1)$ is trivial; we show that achieving $α, β< 1/2$ (together with an assumption that $S, T$ are structured) implies an improvement to the exponent 3/2 for univariate polynomial factorization. Achieving $α= β= 1/3$ is best-possible and would imply an exponent 4/3 algorithm for univariate polynomial factorization. Interestingly, a second consequence would be a reduction of the current-best exponent for deterministic (exponential) algorithms for factoring integers, from $1/5$ to $1/6$.