We propose efficient algorithms and formulas that improve the performance of side channel protected elliptic curve computations with special focus on scalar multiplication exploiting the Gallant-Lambert-Vanstone (CRYPTO 2001) and Galbraith-Lin-Scott (EUROCRYPT 2009) methods. Firstly, by adapting Feng et al.'s recoding to the GLV setting, we derive new regular algorithms for variable-base scalar multiplication that offer protection against simple side-channel and timing attacks. Secondly, we propose an efficient, sidechannel protected algorithm for fixed-base scalar multiplication which combines Feng et al.'s recoding with Lim-Lee's comb method. Thirdly, we propose an efficient technique that interleaves ARM and NEON-based multiprecision operations over an extension field to improve performance of GLS curves on modern ARM processors. Finally, we showcase the efficiency of the proposed techniques by implementing a state-of-the-art GLV-GLS curve in twisted Edwards form defined over F p 2 , which supports a four dimensional decomposition of the scalar and is fully protected against timing attacks. Analysis and performance results are reported for modern x64 and ARM processors. For instance, we compute a variable-base scalar multiplication in 89,000 and 244,000 cycles on an Intel Ivy Bridge and an ARM Cortex-A15 processor (respect.); using a precomputed table of 6KB, we compute a fixed-base scalar multiplication in 49,000 and 116,000 cycles (respect.); and using a precomputed table of 3KB, we compute a double scalar multiplication in 115,000 and 285,000 cycles (respect.). The proposed techniques represent an important improvement of the state-ofthe-art performance of elliptic curve computations, and allow us to set new speed records in several modern processors. The techniques also reduce the cost of adding protection against timing attacks in the computation of GLV-based variable-base scalar multiplication to below 10%. This work is the extended version of a publication that appeared at CT-RSA 2014 [12].
The availability of a new carry-less multiplication instruction in the latest Intel desktop processors significantly accelerates multiplication in binary fields and hence presents the opportunity for reevaluating algorithms for binary field arithmetic and scalar multiplication over elliptic curves. We describe how to best employ this instruction in field multiplication and the effect on performance of doubling and halving operations. Alternate strategies for implementing inversion and half-trace are examined to restore most of their competitiveness relative to the new multiplier. These improvements in field arithmetic are complemented by a study on serial and parallel approaches for Koblitz and random curves, where parallelization strategies are implemented J. and compared. The contributions are illustrated with experimental results improving the state-of-the-art performance of halving and doubling-based scalar multiplication on NIST curves at the 112-and 192-bit security levels and a new speed record for side-channel-resistant scalar multiplication in a random curve at the 128-bit security level. The algorithms presented in this work were implemented on Westmere and Sandy Bridge processors, the latest generation Intel microarchitectures.
Elliptic curve cryptosystems are considered an efficient alternative to conventional systems such as DSA and RSA. Recently, Montgomery and Edwards elliptic curves have been used to implement cryptosystems. In particular, the elliptic curves Curve25519 and Curve448 were used for instantiating Diffie-Hellman protocols named X25519 and X448. Mapping these curves to twisted Edwards curves allowed deriving two new signature instances, called Ed25519 and Ed448, of the Edwards Digital Signature Algorithm. In this work, we focus on the secure and efficient software implementation of these algorithms using SIMD parallel processing. We present software techniques that target the Intel AVX2 vector instruction set for accelerating prime field arithmetic and elliptic curve operations. Our contributions result in a high-performance software library for AVX2-ready processors. For example, our library computes digital signatures 19% (for Ed25519) and 29% (for Ed448) faster than previous optimized implementations. Also, our library improves by 10% and 20% the execution time of X25519 and X448, respectively.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.