On the Distribution of Atkin and Elkies Primes
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10208_2013_Article_9181.pdf
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Author(s)
Shparlinski, Igor E.
Sutherland II, Andrew Victor
Date Issued
February 2014
Journal
Foundations of Computational Mathematics
Publisher
Springer US
Citation
Shparlinski, Igor E., and Andrew V. Sutherland. “On the Distribution of Atkin and Elkies Primes.” Foundations of Computational Mathematics 14.2 (2014): 285–297. © SFoCM 2014
Version
Author's final manuscript
Abstract
Given an elliptic curve E over a finite field F[subscript q] of q elements, we say that an odd prime ℓ∤q is an Elkies prime for E if t[superscript 2][subscript E]−4q is a square modulo ℓ, where t[subscript E]=q+1−#E(F[subscript q]) and #E(F[subscript q]) is the number of F[subscript q]-rational points on E; otherwise, ℓ is called an Atkin prime. We show that there are asymptotically the same number of Atkin and Elkies primes ℓ0 and a sufficiently large q. We use this result to design and analyze a fast algorithm to generate random elliptic curves with #E(F[subscript p]) prime, where p varies uniformly over primes in a given interval [x,2x].
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s10208-013-9181-9
