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Integer factorization via tensor-network Schnorr's sieving

Marco Tesoro1,2,3, Ilaria Siloi1,2,3, Daniel Jaschke1,2,3,4, Giuseppe Magnifico1,2,3,5, and Simone Montangero1,2,3

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Phys. Rev. A 113, 032418 – Published 11 March, 2026

DOI: https://doi.org/10.1103/d9dl-ctt4

Abstract

Classical public-key cryptography standards rely on the Rivest-Shamir-Adleman (RSA) encryption protocol. The security of this protocol is based on the exponential computational complexity of the most efficient classical algorithms for factoring large semiprime numbers into their two prime components. Here, we address RSA factorization building on Schnorr's mathematical framework where factorization translates into a combinatorial optimization problem. We solve the optimization task via tensor network methods, a quantum-inspired classical numerical technique. This tensor network Schnorr's sieving algorithm displays numerical evidence of polynomial scaling of resources with the bit-length of the semiprime. We factorize RSA numbers up to 100 bits and assess how computational resources scale through numerical simulations up to 130 bits, encoding the optimization problem in quantum systems with up to 256 qubits. Only the high-order polynomial scaling of the required resources limits the factorization of larger numbers. Although these results do not currently undermine the security of the present communication infrastructure, they strongly highlight the urgency of implementing postquantum cryptography or quantum key distribution.

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Outline

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Phys. Rev. A 113, 032418– Published 11 March, 2026
  • Received 17 December 2024
  • Accepted 22 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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