Smooth evaluations of cyclotomic quadratics

Bruno Sterner

Paper 2026/2116

Smooth evaluations of cyclotomic quadratics

Bruno Sterner

, University of Waterloo

Abstract

In this note we find large primes $p$ such that $\Phi_k(p)$ is $B$-smooth for the cyclotomic polynomials $\Phi_k$, with $k = 3,4$ and $6$, and a sufficiently small smoothness bound $B$. These primes hence provide the first instantiation of integer factoring challenges that, a priori would be difficult to factor but, are actually easy to factor with traditional algorithms not typically used in practice.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Preprint.
Keywords: Smooth integers cyclotomic polynomials integer factorisation
Contact author(s): bsterner @ uwaterloo ca
History: 2026-09-22: approved; 2026-09-19: received; See all versions
Short URL: https://ia.cr/2026/2116
License: CC0

BibTeX

@misc{cryptoeprint:2026/2116,
      author = {Bruno Sterner},
      title = {Smooth evaluations of cyclotomic quadratics},
      howpublished = {Cryptology {ePrint} Archive, Paper 2026/2116},
      year = {2026},
      url = {https://eprint.iacr.org/2026/2116}
}