Foundations of Computational Mathematics

, Volume 14, Issue 2, pp 285–297 | Cite as

On the Distribution of Atkin and Elkies Primes

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Abstract

Given an elliptic curve \(E\) over a finite field \(\mathbb {F}_q\) of \(q\) elements, we say that an odd prime \(\ell \not \mid q\) is an Elkies prime for \(E\) if \(t_E^2 - 4q\) is a square modulo \(\ell \), where \(t_E = q+1 - \#E(\mathbb {F}_q)\) and \(\#E(\mathbb {F}_q)\) is the number of \(\mathbb {F}_q\)-rational points on \(E\); otherwise, \(\ell \) is called an Atkin prime. We show that there are asymptotically the same number of Atkin and Elkies primes \(\ell < L\) on average over all curves \(E\) over \(\mathbb {F}_q\), provided that \(L \ge (\log q)^\varepsilon \) for any fixed \(\varepsilon >0\) and a sufficiently large \(q\). We use this result to design and analyze a fast algorithm to generate random elliptic curves with \(\#E(\mathbb {F}_p)\) prime, where \(p\) varies uniformly over primes in a given interval \([x,2x]\).

Keywords

Elkies prime Elliptic curve Character sum 

Mathematical Subject Classification

11G07 11L40 11Y16 
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Notes

Acknowledgments

During the preparation I. E. Shparlinski was supported in part by ARC Grant DP130100237 and by NRF Grant CRP2-2007-03, Singapore. A. V. Sutherland received financial support from NSF Grant DMS-1115455.

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Copyright information

© SFoCM 2014

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of New South WalesSydneyAustralia
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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