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1999•LMS Journal of Computation and MathematicsOpen access

Constructing Isogenies between Elliptic Curves Over Finite Fields

Steven D. Galbraith⋆

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Abstract

Abstract Let E1 and E2 be ordinary elliptic curves over a finite field Fp such that #E1(Fp) = #E2(Fp). Tate's isogeny theorem states that there is an isogeny from E1 to E2 which is defined over Fp. The goal of this paper is to describe a probabilistic algorithm for constructing such an isogeny. The algorithm proposed in this paper has exponential complexity in the worst case. Nevertheless, it is efficient in certain situations (that is, when the class number of the endomorphism ring is small). The significance of these results to elliptic curve cryptography is discussed.

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Abstract Let E1 and E2 be ordinary elliptic curves over a finite field Fp such that #E1(Fp) = #E2(Fp). Tate's isogeny theorem states that there is an isogeny from E1 to E2 which is defined over Fp. The goal of this paper is to describe a probabilistic algorithm for constructing such an isogeny. The algorithm proposed in this paper has exponential complexity in the worst case. Nevertheless, it is efficient in certain situations (that is, when the class number of the endomorphism ring is small). The significance of these results to elliptic curve cryptography is discussed.

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Available abstract

Abstract Let E1 and E2 be ordinary elliptic curves over a finite field Fp such that #E1(Fp) = #E2(Fp). Tate's isogeny theorem states that there is an isogeny from E1 to E2 which is defined over Fp. The goal of this paper is to describe a probabilistic algorithm for constructing such an isogeny. The algorithm proposed in this paper has exponential complexity in the worst case. Nevertheless, it is efficient in certain situations (that is, when the class number of the endomorphism ring is small). The significance of these results to elliptic curve cryptography is discussed.

Key concepts: Isogeny, Mathematics, Endomorphism ring, Schoof's algorithm, Elliptic curve, Supersingular elliptic curve, Endomorphism, Finite field

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