The computational refined Humbert invariant problem is equivalent to the computational isogeny problem

Eda Kırımlı; Chloe Martindale

Paper 2025/1295

The computational refined Humbert invariant problem is equivalent to the computational isogeny problem

Eda Kırımlı, University of Bristol, Université de Neuchâtel, University of Birmingham

Chloe Martindale, University of Bristol

Abstract

In this paper, we discuss refined Humbert invariants, introduced by Kani in 1994. We show that in the contexts of SQISign and CSIDH, under GRH, the computational isogeny problem is polynomially equivalent to the computational refined Humbert invariant problem. This includes an algorithm to compute the basis of a maximal order of a quaternion algebra given its norm form. Finally, we also review what is known in the literature about computing refined Humbert invariants.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Preprint.
Keywords: Isogenies Abelian surfaces Refined Humbert invariants
Contact author(s): eda kirimli @ bristol ac uk
chloe martindale @ bristol ac uk
History: 2025-07-16: approved; 2025-07-15: received; See all versions
Short URL: https://ia.cr/2025/1295
License: CC BY

BibTeX

@misc{cryptoeprint:2025/1295,
      author = {Eda Kırımlı and Chloe Martindale},
      title = {The computational refined Humbert invariant problem is equivalent to the computational isogeny problem},
      howpublished = {Cryptology {ePrint} Archive, Paper 2025/1295},
      year = {2025},
      url = {https://eprint.iacr.org/2025/1295}
}