Elliptic curves and braid groups in cryptography

Abstract

In this thesis, we study recent results of two kinds of cryptographic objects: elliptic curve and braid group cryptosystem and our contributions on it. For elliptic curve cryptosystem, we focus on two topics: elliptic curve point counting and pairing based cryptosystems. After Satoh proposed a p-adic method for counting points on elliptic curves over finite fields, several useful techniques have evolved to improve the computational efficiency of the basic Satoh algorithm. The evolution of these techniques has proved remarkably successful and reduced the computational efficiency by asymptotically optimal. We briefly review p-adic methods and present an improved algorithm. It is mainly based on the Satoh-Skjernaa-Taguchi (SST) algorithm and the modified SST algorithm, and uses a Gaussian normal basis (GNB) of small type. We show that a Gaussian normal basis can be lifted form $\mathbb{F}_q$ to $\mathbb{Z}_q$ in a natural way. From the specific properties of GNBs, efficient multiplication and the Frobenius substitution are available. Thus a fast norm computation algorithm is derived. As a result, we reduced the time complexity of both algorithms from $O(N^{2μ+0.5})$ to $O(N^{2μ +{1\choosμ +1}})$ and the space complexity still fits in $O(N^2)$ for either a small characteristic. So, applying our contribution to other recent improvements allows to compute the number of points of an elliptic curve defined over large finite fields. Pairing based cryptosystems are currently one of the most active areas of research in elliptic curve cryptography. Especially, the identity based encryption scheme of Boneh and Franklin has spurred a tremendous amount of new cryptographic research. We describe a number of simple yet amazing applications of pairings and propose a certificate-based signature scheme that can share parameters and certificate revocation strategy with the encryption scheme proposed by Gentry. We first suggest a formal security model of a certificate-based sign...