Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms

Cédric Pilatte

Paper 2024/629

Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms

Cédric Pilatte

, University of Oxford

Abstract

In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in $(Z / N Z)^{\times}$ that can be written as short products of very small prime numbers. We prove a version of this conjecture using tools from analytic number theory such as zero-density estimates. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm and of subsequent variants.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Preprint.
Keywords: factoring quantum number theory lattices Generalized Riemann Hypothesis
Contact author(s): cedric pilatte @ maths ox ac uk
History: 2024-04-26: approved; 2024-04-24: received; See all versions
Short URL: https://ia.cr/2024/629
License: CC BY-NC-SA

BibTeX

@misc{cryptoeprint:2024/629,
      author = {Cédric Pilatte},
      title = {Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms},
      howpublished = {Cryptology {ePrint} Archive, Paper 2024/629},
      year = {2024},
      url = {https://eprint.iacr.org/2024/629}
}