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On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average

Part of: Computational number theory Arithmetic algebraic geometry Exponential sums and character sums

Published online by Cambridge University Press:  01 April 2015

Igor E. Shparlinski  and
Andrew V. Sutherland
Igor E. Shparlinski
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney NSW 2052, Australia email igor.shparlinski@unsw.edu.au
Andrew V. Sutherland
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA email drew@math.mit.edu

Abstract

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For an elliptic curve $E/\mathbb{Q}$ without complex multiplication we study the distribution of Atkin and Elkies primes $\ell$, on average, over all good reductions of $E$ modulo primes $p$. We show that, under the generalized Riemann hypothesis, for almost all primes $p$ there are enough small Elkies primes $\ell$ to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in $(\log p)^{4+o(1)}$ expected time.

MSC classification

Primary: 11Y16: Algorithms; complexity
Secondary: 11G05: Elliptic curves over global fields 11G07: Elliptic curves over local fields 11L40: Estimates on character sums
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 308 - 322
Copyright
© The Author(s) 2015 

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