Related papers: Faster computation of the Tate pairing
An algorithm is given to compute a normal form for hyperelliptic curves. The elliptic case has been treated in a previous paper. In this paper the hyperelliptic case is treated.
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the…
Koblitz curves are a special set of elliptic curves and have improved performance in computing scalar multiplication in elliptic curve cryptography due to the Frobenius endomorphism. Double-base number system approach for Frobenius…
In this paper, we present efficient algorithms for computing the number of points and the order of the Jacobian group of a superelliptic curve over finite fields of prime order p. Our method employs the Hasse-Weil bounds in conjunction with…
We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose…
We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then…
We propose a new method for accelerating the computation of a concurrency relation, that is all pairs of places in a Petri net that can be marked together. Our approach relies on a state space abstraction, that involves a mix between…
We present a simple and efficient algorithm to compute the sum of the algebraic conjugates of a point on an elliptic curve.
The study of alternative models for elliptic curves has found recent interest from cryptographic applications, once it was recognized that such models provide more efficiently computable algorithms for the group law than the standard…
We assemble and reorganize the recent work in the area of hyperelliptic pairings: We survey the research on constructing hyperelliptic curves suitable for pairing-based cryptography. We also showcase the hyperelliptic pairings proposed to…
We study the ranking of individuals, teams, or objects, based on pairwise comparisons between them, using the Bradley-Terry model. Estimates of rankings within this model are commonly made using a simple iterative algorithm first introduced…
The problem of constructing elliptic curves suitable for pairing applications has received a lot of attention. To solve this, we propose a variant algorithm of a known method by Brezing and Weng. We produce new families of parameters using…
We revisit the group structure on elliptic curves and give a simple and elementary proof of the associativity of the addition. We do this by providing an explicit formula for the sum of three points, only using the explicit definition of…
This study reports on an implementation of cryptographic pairings in a general purpose computer algebra system. For security levels equivalent to the different AES flavours, we exhibit suitable curves in parametric families and show that…
We introduce the twisted $\boldsymbol{\mu}_4$-normal form for elliptic curves, deriving in particular addition algorithms with complexity $9\mathbf{M} + 2\mathbf{S}$ and doubling algorithms with complexity $2\mathbf{M} + 5\mathbf{S} +…
We propose two optimal representations for the elements of trace zero subgroups of twisted Edwards curves. For both representations, we provide efficient compression and decompression algorithms. The efficiency of the algorithm is compared…
We give an elementary and self-contained introduction to pairings on elliptic curves over finite fields. For the first time in the literature, the three different definitions of the Weil pairing are stated correctly and proved to be…
In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption…
This article concerns the computational complexity of a fundamental problem in number theory: counting points on curves and surfaces over finite fields. There is no subexponential-time algorithm known and it is unclear if it can be…
A new procedure is presented for computing the matrix cosine and sine simultaneously by means of Taylor polynomial approximations. These are factorized so as to reduce the number of matrix products involved. Two versions are developed to be…