Arithmetic of pairings on algebraic curves for cryptography. (Étude de l'arithmétique des couplages sur les courbes algébriques pour la cryptographie)

Since 2000 pairings became a very useful tool to design new protocols in cryptography. Short signaturesand identity-based encryption became also practical thanks to these pairings.This thesis contains two parts. One part is about optimized pairing implementation on different ellip-tic curves according to the targeted protocol. Pairings are implemented on supersingular elliptic curvesin large characteristic and on Barreto-Naehrig curves. The pairing library developed at Thales is usedin a broadcast encryption scheme prototype. The prototype implements pairings over Barreto-Naehrigcurves. Pairings over supersingular curves are much slower and have larger parameters. However thesecurves are interesting when implementing protocols which use composite-order elliptic curves (the grouporder is an RSA modulus). We implement two protocols that use pairings on composite-order groupsand compare the benchmarks and the parameter size with their counterpart in a prime-order setting. Thecomposite-order case is 30 up to 250 times much slower according to the considered step in the protocols:the efficiency difference in between the two cases is very important.A second part in this thesis is about two families of genus 2 curves. Their Jacobians are isogenousto the product of two elliptic curves over a small extension field. The properties of elliptic curves canbe translated to the Jacobians thanks to this isogeny. Point counting is as easy as for elliptic curves inthis case. We also construct two endomorphisms both on the Jacobians and the elliptic curves. These en-domorphisms can be used for scalar multiplication improved with a four-dimensional Gallant-Lambert-Vanstone method.keywords: elliptic curves, genus 2 curves, endomorphisms, pairings, implementation, composite-order groups.tel-00921940,version1-22Dec2013