Related papers: Improved Weil and Tate pairings for elliptic and h…

We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by 3-isogeny, to that coming from a full…

Fisher and Newton have given an explicit description of the Tate local pairing associated with the 3-torsion of an elliptic curve. The present paper summarizes the work from the author's master's thesis and gives an explicit formula for any…

For an elliptic curve $E$ over any field $K$, the Weil pairing $e_n$ is a bilinear map on $n$-torsion. For $K$ of characteristic $p>0$, the map $e_n$ is degenerate if and only if $n$ is divisible by $p$. In this paper, we consider $E$ over…

In this paper we give an algorithm of how to determine a Weierstrass equation with minimal discriminant for superelliptic curves generalizing work of Tate for elliptic curves and Liu for genus 2 curves.

We present normal forms for elliptic curves over a field of characteristic $2$ analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient…

The decision-Diffie-Hellman problem (DDH) is a central computational problem in cryptography. It is known that the Weil and Tate pairings can be used to solve many DDH problems on elliptic curves. Distortion maps are an important tool for…

Some numerical algorithms for elliptic eigenvalue problems are proposed, analyzed, and numerically tested. The methods combine advantages of the two-grid algorithm, two-space method, the shifted inverse power method, and the polynomial…

We present an efficient algorithm to compute the Hasse-Witt matrix of a hyperelliptic curve C/Q modulo all primes of good reduction up to a given bound N, based on the average polynomial-time algorithm recently introduced by Harvey. An…

We study a popular algorithm for fitting polynomial curves to scattered data based on the least squares with gradient weights. We show that sometimes this algorithm admits a substantial reduction of complexity, and, furthermore, find…

We apply some of the latest techniques from machine-learning to the arithmetic of hyperelliptic curves. More precisely we show that, with impressive accuracy and confidence (between 99 and 100 percent precision), and in very short time…

We describe an algorithm for determining a minimal Weierstrass equation for hyperelliptic curves over principal ideal domains. When the curve has a rational Weierstrass point $w_0$, we also give a similar algorithm for determining the…

For positive rank $r$ elliptic curves $E(\mathbb{Q})$, we employ ideal class pairings $$ E(\mathbb{Q})\times E_{-D}(\mathbb{Q}) \rightarrow \mathrm{CL}(-D), $$ for quadratic twists $E_{-D}(\mathbb{Q})$ with a suitable ``small $y$-height''…

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These…

We compute twists of the modular curve $X(13)$ that parametrise the elliptic curves 13-congruent to a given elliptic curve. Searching for rational points on these twists enables us to find non-trivial pairs of 13-congruent elliptic curves…

Much attention has been given to the efficient computation of pairings on elliptic curves with even embedding degree since the advent of pairing-based cryptography. The few existing works in the case of odd embedding degrees require some…

Short Weierstrass's elliptic curves with underlying hard Elliptic Curve Discrete Logarithm Problems was widely used in Cryptographic applications. This paper introduces a new security notation 'trusted security' for computation methods of…

We state the graph-theoretic computational problem of finding tight paths in a directed, edge-weighted graph, as well as its simplification of finding tight pairs. These problems are motivated by the need of algorithms that find so-called…

Efficient computations of pairings with Miller Algorithm have recently received a great attention due to the many applications in cryptography. In this work, we give formulae for the optimal Ate pairing in terms of elliptic nets associated…

In this paper, we study some cohomology groups and quadratic twists of elliptic curves, and apply Tate local duality and the results of Kramer-Tunnell on local norm cokernel to give a refined version of Yu's formula in the case of elliptic…

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree $\ell$ ($\ell$ different from the…