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Computing (l,l)-isogenies in polynomial time on Jacobians of genus 2 curves

Romain Cosset 1 Damien Robert 2, 3
1 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
2 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : In this paper, we compute l-isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l, when l is an odd prime which is coprime to the characteristic. We use level n symmetric theta structure where n = 2 or n = 4. In a second part of this paper we explain how to convert between Mumford coordinates of Jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4. Combined with the preceding algorithm, this gives a method to compute (l,l)-isogenies in polynomial time on Jacobians of genus 2 curves.
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Romain Cosset, Damien Robert. Computing (l,l)-isogenies in polynomial time on Jacobians of genus 2 curves. Mathematics of Computation, American Mathematical Society, 2015, 84 (294), pp.1953-1975 ⟨10.1090/S0025-5718-2014-02899-8⟩. ⟨hal-00578991⟩

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