**On the evaluation of modular polynomials**

*Andrew V. Sutherland*

**Abstract: **We present two algorithms that, given a prime ell and an elliptic curve E/Fq, directly compute the polynomial $\Phi_\ell(j(E),Y)\in\Fq[Y] whose roots are the j-invariants of the elliptic curves that are ell-isogenous to E. We do not assume that the modular polynomial Phi_ell(X,Y) is given. The algorithms may be adapted to handle other types of modular polynomials, and we consider applications to point counting and the computation of endomorphism rings. We demonstrate the practical efficiency of the algorithms by setting a new point-counting record, modulo a prime q with more than 5,000 decimal digits, and by evaluating a modular polynomial of level ell=100,019.

**Category / Keywords: **public-key cryptography / elliptic curves, isogenies

**Publication Info: **ANTS X

**Date: **received 30 Mar 2013, last revised 7 May 2013

**Contact author: **drew at math mit edu

**Available format(s): **PDF | BibTeX Citation

**Note: **Corrected a typo in one of the complexity bounds in the introduction (it now matches the theorem in the body of the paper).

**Version: **20130507:162554 (All versions of this report)

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