Sorina Ionica

Another Approach to Pairing Computation in Edwards Coordinates

The recent introduction of Edwards curves has significantly reduced the cost of addition on elliptic curves. This paper presents new explicit formulae for pairing implementation in Edwards coordinates. We prove our method gives performances similar to those of Millers algorithm in Jacobian coordinates and is thus of cryptographic interest when one chooses Edwards curve implementations of protocols in elliptic curve cryptography. The method is faster than the recent proposal of Das and Sarkar for computing pairings on supersingular curves using Edwards coordinates.

Four-Dimensional GLV via the Weil Restriction

The Gallant-Lambert-Vanstone GLV algorithm uses efficiently computable endomorphisms to accelerate the computation of scalar multiplication of points on an abelian variety. Freeman and Satoh proposed for cryptographic use two families of genus 2 curves defined over

Pairing the volcano

\mathbb{F}_{p}

Pairing the Volcano

which have the property that the corresponding Jacobians are 2,2-isogenous over an extension field to a product of elliptic curves defined over

Efficient multiplication in finite field extensions of degree 5

\mathbb{F}_{p^2}

Pairing computation on elliptic curves with efficiently computable endomorphism and small embedding degree

. We exploit the relationship between the endomorphism rings of isogenous abelian varieties to exhibit efficiently computable endomorphisms on both the genus 2 Jacobian and the elliptic curve. This leads to a four-dimensional GLV method on Freeman and Satohs Jacobians and on two new families of elliptic curves defined over

Pairing-based algorithms for jacobians of genus 2 curves with maximal endomorphism ring

\mathbb{F}_{p^2}

A study of pairing computation for elliptic curves with embedding degree 15.

.

Isogeny graphs with maximal real multiplication

Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are l-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001). However, up to now, no method was known, to predict, before taking a step on the volcano, the direction of this step. Hence, in Kohels and Fouquet-Morain algorithms, many steps are taken before choosing the right direction. In particular, ascending or horizontal isogenies are usually found using a trial-and-error approach. In this paper, we propose an alternative method that efficiently finds all points P of order l such that the subgroup generated by P is the kernel of an horizontal or an ascending isogeny. In many cases, our method is faster than previous methods. This is an extended version of a paper published in the proceedings of ANTS 2010. In addition, we treat the case of 2-isogeny volcanoes and we derive from the group structure of the curve and the pairing a new invariant of the endomorphism class of an elliptic curve. Our benchmarks show that the resulting algorithm for endomorphism ring computation is faster than Kohels method for computing the l-adic valuation of the conductor of the endomorphism ring for small l.

Algorithmique des couplages et cryptographie

Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are l-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001). However, up to now, no method was known, to predict, before taking a step on the volcano, the direction of this step. Hence, in Kohel’s and Fouquet-Morain algorithms, we take many steps before choosing the right direction. In particular, ascending or horizontal isogenies are usually found using a trial-and-error approach. In this paper, we propose an alternative method that efficiently finds all points P of order l such that the subgroup generated by P is the kernel of an horizontal or an ascending isogeny. In many cases, our method is faster than previous methods.