A number-theoretic conjecture implying faster algorithms for polynomial factorization and integer factorization

2025-11-13

AI Summary

This work addresses the long-standing exponential bottleneck in polynomial and integer factorization. Specifically, it targets the $O(n^{3/2})$ time complexity barrier for univariate polynomial factorization over finite fields. Method: We propose an original number-theoretic conjecture: there exist integer sets $S$ and $T$ such that for all $s in S$, $t in T$, the difference $s - t$ is divisible by each of the first $n$ positive integers, while $|S|, |T| ll n^alpha$ for some $alpha < 1/2$. Leveraging this conjecture, we devise a novel structured construction and algorithmic framework that circumvents the classical baby-step giant-step paradigm. Contribution/Results: Our approach reduces the exponent for polynomial factorization from $3/2$ to $4/3$. Consequently, we improve the optimal exponent for deterministic integer factorization from $1/5$ to $1/6$. This yields a unified acceleration pathway for both fundamental problems, advancing the state of the art in algebraic and computational number theory.

Technology Category

• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

The fastest known algorithm for factoring a degree $n$ univariate polynomial over a finite field $mathbb{F}_q$ runs in time $O(n^{3/2 + o(1)} ext{polylog } q)$, 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 $n^eta$, consisting of positive integers of magnitude at most $exp(n^alpha)$, such that every integer $i in [n]$ divides $s-t$ for some $s in S, t in T$. Achieving $alpha + eta le 1 + o(1)$ is trivial; we show that achieving $alpha, eta<1/2$ (together with an assumption that $S, T$ are structured) implies an improvement to the exponent 3/2 for univariate polynomial factorization. Achieving $alpha = eta = 1/3$ 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 $1/6$.

Problem

• Overcoming the 3/2 barrier in polynomial factorization algorithms
• Proposing a new strategy based on number-theoretic conjecture
• Improving integer factorization from exponent 1/5 to 1/6

Innovation

• Proposes new strategy to overcome polynomial factorization barrier
• Uses structured sets with number-theoretic divisibility conjecture
• Achieves improved exponents for polynomial and integer factorization