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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Elliptic curves over finite fields and the computation of square roots mod $p$


Author: René Schoof
Journal: Math. Comp. 44 (1985), 483-494
MSC: Primary 11Y16; Secondary 11G20, 14G15
DOI: https://doi.org/10.1090/S0025-5718-1985-0777280-6
MathSciNet review: 777280
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Abstract: In this paper we present a deterministic algorithm to compute the number of ${{\mathbf {F}}_q}$-points of an elliptic curve that is defined over a finite field ${{\mathbf {F}}_q}$ and which is given by a Weierstrass equation. The algorithm takes $O({\log ^9}q)$ elementary operations. As an application we give an algorithm to compute square roots $\bmod p$. For fixed $x \in {\mathbf {Z}}$, it takes $O({\log ^9}p)$ elementary operations to compute $\sqrt x \bmod p$.


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Keywords: Elliptic curves, finite fields, factorization, polynomials, computational number theory
Article copyright: © Copyright 1985 American Mathematical Society