Fast Elliptic Curve Multiplications with SIMD Operations

Tetsuya IZU  Tsuyoshi TAKAGI  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E87-A    No.1    pp.85-93
Publication Date: 2004/01/01
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)
Category: Asymmetric Cipher
Elliptic Curve Cryptosystems (ECC),  scalar multiplication,  window method,  SIMD operations,  side channel attacks,  

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The Single Instruction, Multiple Data (SIMD) architecture enables computation in parallel on a single processor. The SIMD operations are implemented on some processors such as Pentium 3/4, Athlon, SPARC, or even on smart cards. This paper proposes efficient algorithms for assembling an elliptic curve addition (ECADD), doubling (ECDBL), and k-iterated ECDBL (k-ECDBL) with SIMD operations. We optimize the number of auxiliary variables and the order of basic field operations used for these addition formulas. If an addition chain has k-bit zero run, we can replace k-time ECDBLs to the proposed faster k-ECDBL and the total efficiency of the scalar multiplication can be improved. Using the singed binary chain, we can compute a scalar multiplication about 10% faster than the previously fastest algorithm proposed by Aoki et al. Combined with the sliding window method or the width-w NAF window method, we also achieve about 10% faster parallelized scalar multiplication algorithms with SIMD operations. For the implementation on smart cards, we establish two fast parallelized scalar multiplication algorithms with SIMD resistant against side channel attacks.