Computing Modular Polynomials

Details Computing Modular Polynomials Computing Modular Polynomials

2004-08-03

arxiv.org/abs/math/0408051v2

Mathematics

Number Theory

8 pages, appendix added with correction to Elkies' running time, final version to appear in London Math Society Journal of Com

Scientific paper

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.

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