Computing Modular Polynomials

Denis Charles; Kristin Lauter

Paper 2004/189

Computing Modular Polynomials

Denis Charles and Kristin Lauter

Abstract

We present a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography. Our algorithm has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. We avoid computing the exponentially large integral coefficients by working directly modulo a prime and computing isogenies between elliptic curves via Velu's formulas.

Note: Small improvements have been made, running times without fast multiplication have been added, and an appendix correcting the run-time analysis of Elkies' method has been added.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Published elsewhere. to appear in London Math Society Journal of Computation and Mathematics
Keywords: elliptic curve cryptosystems number theory
Contact author(s): klauter @ microsoft com
History: 2005-06-15: revised; 2004-08-07: received; See all versions
Short URL: https://ia.cr/2004/189
License: CC BY

BibTeX

@misc{cryptoeprint:2004/189,
      author = {Denis Charles and Kristin Lauter},
      title = {Computing Modular Polynomials},
      howpublished = {Cryptology {ePrint} Archive, Paper 2004/189},
      year = {2004},
      url = {https://eprint.iacr.org/2004/189}
}