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On isogeny calculation by solving p-adic differential equations

Elie Eid 1, 2 
2 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : In this thesis, we propose efficient algorithms for computing isogenies between elliptic curves and Jacobians of hyperelliptic curves via p-adic differential equations with a sharp analysis of the losses of precision. More precisely, in one hand, we are interested in computing elliptic curve isogenies defined over an extension of Q2. This work complements the work carried out over extensions of Qp for p odd. We give some applications, especially computing over finite fields of characteristic 2 isogenies of elliptic curves and irreducible polynomials, both in quasi-linear time in the degree. On the other hand, we present an algorithm for the explicit computation of rational representations between Jacobians of hyperelliptic curves defined over an extension of Qp. Consequently, after having possibly lifted the problem in the p-adics, we obtain efficient algorithms for computing isogenies between Jacobians of hyperelliptic curves defined over finite fields of odd characteristic. Another important application is the computation of Cantor’s l-division polynomials. The efficiency of these algorithms is based on an analysis of the solutions of p-adic differential equations.
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Submitted on : Tuesday, September 7, 2021 - 3:05:11 PM
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Elie Eid. On isogeny calculation by solving p-adic differential equations. Algebraic Geometry [math.AG]. Université Rennes 1, 2021. English. ⟨NNT : 2021REN1S012⟩. ⟨tel-03337021⟩

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