Division polynomials for arbitrary isogenies

Katherine E. Stange

Paper 2025/521

Division polynomials for arbitrary isogenies

Katherine E. Stange

, University of Colorado Boulder

Abstract

Following work of Mazur-Tate and Satoh, we extend the definition of division polynomials to arbitrary isogenies of elliptic curves, including those whose kernels do not sum to the identity. In analogy to the classical case of division polynomials for multiplication-by-n, we demonstrate recurrence relations, identities relating to classical elliptic functions, the chain rule describing relationships between division polynomials on source and target curve, and generalizations to higher dimension (i.e., elliptic nets).

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Preprint.
Keywords: isogeny-based cryptography division polynomials kernel polynomials isogenies elliptic curves
Contact author(s): kstange @ math colorado edu
History: 2025-03-21: approved; 2025-03-19: received; See all versions
Short URL: https://ia.cr/2025/521
License: CC BY-NC-ND

BibTeX

@misc{cryptoeprint:2025/521,
      author = {Katherine E. Stange},
      title = {Division polynomials for arbitrary isogenies},
      howpublished = {Cryptology {ePrint} Archive, Paper 2025/521},
      year = {2025},
      url = {https://eprint.iacr.org/2025/521}
}