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Paper 2013/181

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.

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

Metadata
Available format(s)
PDF
Category
Public-key cryptography
Publication info
Published elsewhere. ANTS X
Keywords
elliptic curvesisogenies
Contact author(s)
drew @ math mit edu
History
2013-05-07: revised
2013-04-01: received
See all versions
Short URL
https://ia.cr/2013/181
License
Creative Commons Attribution
CC BY
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