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Opérateurs Arithmétiques Parallèles pour la Cryptographie Asymétrique

Thomas Izard 1
1 ARITH - Arithmétique informatique
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : Protocols for asymmetric cryptography require arithmetic computations in several mathematical structures. In particular, many of them need computations in finite structures, imposing an efficient modular arithmetic. These modular calculations are composed of multiprecision operations between operands of sizes large enough to insure the required level of security (between several hundred and several thousand of bits). Finally, some protocols need arithmetic operations in the group of points of an elliptic curve, which are themselves composed of modular computations. The sizes of the keys used by the protocols make the arithmetic computations expansive in terms of execution time. Nowadays, current architectures have several computing units, which are distributed over the processors and GPU. These resources are now easily programmable using dedicated languages such as OpenMP or CUDA. This thesis focuses on the definition of parallel algorithms to take advantage of all the computing resources of multi-core shared-memory architectures. Parallelism at the lowest arithmetic level gives a moderate speedup since the sizes of the operands are not large enough so that the arithmetic intensity hides the latencies induced by the parallelism. We propose algorithms for parallelization at higher arithmetic: parallel algorithms for modular multiplication and scalar multiplication on elliptic curves. For modular multiplication, we study in particular several schedulings of modular computations. We also propose a two-level parallelization, at modular and multiprecision levels. This "coarse-grained" parallelism allows in practice a more substantial speedup. We also present a parallelization of modular and elliptic curves operations on GPU. Finally, we introduce a method to optimize scalar multiplication on elliptic curves for small scalars.
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  • HAL Id : tel-00685654, version 1

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Thomas Izard. Opérateurs Arithmétiques Parallèles pour la Cryptographie Asymétrique. Cryptographie et sécurité [cs.CR]. Université Montpellier II - Sciences et Techniques du Languedoc, 2011. Français. ⟨tel-00685654⟩

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