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An Improved Subexponential Bound for Deterministic Integer Factorization

Primarily AI-generated textHuman understanding: all partsmath.NT — Number Theorycs.CC — Computational Complexity

Contributed by Joseph Shunia ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We prove that the complete prime factorization of an nn-bit integer can be computed deterministically in time 2O(n/log⁡n)2^{O(n/\sqrt{\log n})} and polynomial space. This establishes the bound conjectured by Rudich and Rousseau, sharpening their qualitative No(1)N^{o(1)} result. The proof combines OpenAI's quantitative Boolean tensor-contraction theorem with the linear-size multiplication encoding of Akshima et al. The encoding permits exact counting of divisors with prescribed binary prefixes without loss in the exponent. The least prime factor is then recovered by successive prefix queries, and iteration yields the complete factorization within the same asymptotic bound.

Provenance statement

OpenAI's ChatGPT assisted the author in developing, drafting, and checking the argument. The author supplied the source manuscripts and directed the investigation.

Tools used

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References

  1. Akshima, R. Bacho, Y.-H. Huang, E. Kiltz, A. Nishaburi, S. Nooripoor, N. Tittelbach, and E. Wiedijk, A Subexponential Time Algorithm for Elliptic Curve Discrete Logarithm , preprint, version 0.2, October 29, 2026. https://github.com/bochum-crypto/subexponential-ecdl.link
  2. OpenAI, The Exponential Time Hypothesis is False , preprint, October 17, 2026; revised October 25, 2026. https://github.com/openai/math/blob/main/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf.link
  3. I. Rudich and L.-M. Rousseau, Deterministic Factoring in Time No(1)N^{o(1)} , Hexagon preprint 2610.00600, version 1, October 2026. https://hexagonmath.org/2610.00600v1.hexagon

Version history

  1. v1 · 4 pagesSubmitted by Joseph ShuniaInitial depositCurrentOct 31, 2026