The Single Instruction, Multiple Data (SIMD) architecture enables to compute in parallel on a single processor. The SIMD operations are implemented on some processors such as Pentium 3/4, Athlon, SPARC, and 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. Using the singed binary chain, we can compute a scalar multiplication about 10 faster than the previously fastest algorithm 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 propose two fast parallelized scalar multiplication algorithms with SIMD resistant against side channel attacks.
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