Hüseyin Hisil

Twisted Edwards Curves Revisited

This paper introduces fast algorithms for performing group operations on twisted Edwards curves, pushing the recent speed limits of Elliptic Curve Cryptography (ECC) forward in a wide range of applications. Notably, the new addition algorithm uses

New formulae for efficient elliptic curve arithmetic

8\mathrm{\textbf{M}}

Faster Pairings on Special Weierstrass Curves

for suitably selected curve constants. In comparison, the fastest point addition algorithms for (twisted) Edwards curves stated in the literature use

Jacobi Quartic Curves Revisited

9\mathrm{\textbf{M}} + 1\mathrm{\textbf{S}}

High-Performance scalar multiplication using 8-dimensional GLV/GLS decomposition

. It is also shown that the new addition algorithm can be implemented with four processors dropping the effective cost to

Faster Compact Diffie-Hellman: Endomorphisms on the x-line

2\mathrm{\textbf{M}}

Jacobian Coordinates on Genus 2 Curves

. This implies an effective speed increase by the full factor of 4 over the sequential case. Our results allow faster implementation of elliptic curve scalar multiplication. In addition, the new point addition algorithm can be used to provide a natural protection from side channel attacks based on simple power analysis (SPA).

A Simple and Compact Algorithm for SIDH with Arbitrary Degree Isogenies

This paper is on efficient implementation techniques of Elliptic Curve Cryptography. In particular, we improve timings for Jacobiquartic (3M+4S) and Hessian (7M+1S or 3M+6S) doubling operations. We provide a faster mixed-addition (7M+3S+1d) on modified Jacobiquartic coordinates. We introduce tripling formulae for Jacobi-quartic (4M+11S+2d), Jacobi-intersection (4M+10S+5d or 7M+7S+3d), Edwards (9M+4S) and Hessian (8M+6S+1d) forms. We show that Hessian tripling costs 6M+4C+1d for Hessian curves defined over a field of characteristic 3. We discuss an alternative way of choosing the base point in successive squaring based scalar multiplication algorithms. Using this technique, we improve the latest mixed-addition formulae for Jacobi-intersection (10M+2S+1d), Hessian (5M+6S) and Edwards (9M+1S+ 1d+4a) forms. We discuss the significance of these optimizations for elliptic curve cryptography.

An exploration of affine group laws for elliptic curves

This paper presents efficient formulas for computing cryptographic pairings on the curve y 2 = c x 3 + 1 over fields of large characteristic. We provide examples of pairing-friendly elliptic curves of this form which are of interest for efficient pairing implementations.

How to (Pre-)Compute a Ladder

This paper provides new results about efficient arithmetic on Jacobi quartic form elliptic curves, y 2 = d x 4 + 2 a x 2 + 1. With recent bandwidth-efficient proposals, the arithmetic on Jacobi quartic curves became solidly faster than that of Weierstrass curves. These proposals use up to 7 coordinates to represent a single point. However, fast scalar multiplication algorithms based on windowing techniques, precompute and store several points which require more space than what it takes with 3 coordinates. Also note that some of these proposals require d = 1 for full speed. Unfortunately, elliptic curves having 2-times-a-prime number of points, cannot be written in Jacobi quartic form if d = 1. Even worse the contemporary formulae may fail to output correct coordinates for some inputs. This paper provides improved speeds using fewer coordinates without causing the above mentioned problems. For instance, our proposed point doubling algorithm takes only 2 multiplications, 5 squarings, and no multiplication with curve constants when d is arbitrary and a = ±1/2.