Fast Elliptic Curve Multiplications with SIMD Operations

Tetsuya IZU; Tsuyoshi TAKAGI

Fast Elliptic Curve Multiplications with SIMD Operations

Tetsuya IZU, Tsuyoshi TAKAGI

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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.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E87-A No.1 pp.85-93

Publication Date: 2004/01/01

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Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)

Category: Asymmetric Cipher

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Tetsuya IZU, Tsuyoshi TAKAGI, "Fast Elliptic Curve Multiplications with SIMD Operations" in IEICE TRANSACTIONS on Fundamentals, vol. E87-A, no. 1, pp. 85-93, January 2004, doi: .
Abstract: 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.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e87-a_1_85/_p

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@ARTICLE{e87-a_1_85,
author={Tetsuya IZU, Tsuyoshi TAKAGI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Fast Elliptic Curve Multiplications with SIMD Operations},
year={2004},
volume={E87-A},
number={1},
pages={85-93},
abstract={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.},
keywords={},
doi={},
ISSN={},
month={January},}

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TY - JOUR
TI - Fast Elliptic Curve Multiplications with SIMD Operations
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 85
EP - 93
AU - Tetsuya IZU
AU - Tsuyoshi TAKAGI
PY - 2004
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E87-A
IS - 1
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - January 2004
AB - 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.
ER -