Division polynomials on the Hessian model of elliptic curves
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Division polynomials on the Hessian model of elliptic curves Perez Broon Fouazou Lontouo1 · Emmanuel Fouotsa2 · Daniel Tieudjo3 Received: 8 April 2020 / Revised: 1 August 2020 / Accepted: 28 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper we derive formulas for the scalar multiplication by n map, denoted [n], on the Hessian model of elliptic curve. This enables to characterize n-torsion points on this curve. The computation involves three families of polynomials Pn , Qn and Vn and we show some properties on the coefficients and degrees of these polynomials. We also show some functional equations satisfied by these polynomials. As application we provide a type of mean-value theorem for the Hessian elliptic curve. Keywords Elliptic curves · Division polynomial · Hessian curves · Mean-value theorem Mathematics Subject Classification 14H52 · 1990S
1 Introduction Division polynomials over elliptic curves provide formulas for the scalar multiplication by [n] map on elliptic curves endowed with the additive group law. This then enables to characterize n-torsion points on this curve. Several work have been done on the computation of these polynomials: we can cite [3, 12, 17] (the aim being to compute the division polynomial without doing polynomial multiplication). The paper [26, 31] factorize * Perez Broon Fouazou Lontouo [email protected] Emmanuel Fouotsa [email protected] Daniel Tieudjo [email protected] 1
Department of Mathematics and Computer Science, Faculty of Sciences, The University of Maroua, P.O. Box 814, Maroua, Cameroon
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Department of Mathematics, Higher Teacher Training College, The University of Bamenda, P.O. Box 37, Bambili, Cameroon
3
ENSAI, The University of Ngaoundere, P.O. Box 454, Ngaoundere, Cameroon
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the division polynomials. Several works on elliptic curves use division polynomials: the most popular one is the Schoof algorithm [27] as a polynomial time algorithm for computing the cardinality of an elliptic curve over finite fields. We can also cite [18] which gives an algorithm to compute the l-Sylow subgroup of an elliptic curve and its generators. The work in [30] provides a third proof to Nagel’s theorem [22] using division polynomials. The works [1–3] consider division polynomials and the intersection of projective torsion points of elliptic curves. The research work in [6] uses p-division polynomial for testing if a j-invariant is of a super singular elliptic curve or not. Efficient scalar multiplication over elliptic curves can be done using division polynomials [5, 11]. Several other works characterize torsion points of elliptic curves and corresponding generated fields using division polynomials [4, 10, 13, 16, 23, 29]. Formulas for division polynomials have been derived and used mostly on the Weierstrass model of elliptic curve. Considering their many applications, the computation of division polynomials have been extended to other model of elliptic curves such
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