1. Fast heuristic algorithms for computing relations in the class group of a quadratic order, with applications to isogeny evaluation
- Author
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Jean-François Biasse, Michael J. Jacobson, and Claus Fieker
- Subjects
Isogeny ,Discrete mathematics ,Ideal (set theory) ,Heuristic ,General Mathematics ,010102 general mathematics ,Ideal class group ,02 engineering and technology ,01 natural sciences ,Prime (order theory) ,Algebra ,Finite field ,Computational Theory and Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Order (group theory) ,020201 artificial intelligence & image processing ,Quadratic field ,0101 mathematics ,Algorithm ,Mathematics - Abstract
In this paper, we present novel algorithms for finding small relations and ideal factorizations in the ideal class group of an order in an imaginary quadratic field, where both the norms of the prime ideals and the size of the coefficients involved are bounded. We show how our methods can be used to improve the computation of large-degree isogenies and endomorphism rings of elliptic curves defined over finite fields. For these problems, we obtain improved heuristic complexity results in almost all cases and significantly improved performance in practice. The speed-up is especially high in situations where the ideal class group can be computed in advance.
- Published
- 2016