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Fast Reconfigurable Elliptic Curve Cryptography Acceleration for GF(2m) on 32 bit Processors

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Abstract

This paper focuses on the design and implementation of a fast reconfigurable method for elliptic curve cryptography acceleration in GF(2m). The main contribution of this paper is comparing different reconfigurable modular multiplication methods and modular reduction methods for software implementation on Intel IA-32 processors, optimizing point arithmetic to reduce the number of expensive reduction operations through a novel reduction sharing technique, and measuring performance for scalar point multiplication in GF(2m) on Intel IA-32 processors. This paper determined that systematic reduction is best for fields defined with trinomials or pentanomials; however, for fields defined with reduction polynomials with large Hamming weight Barrett reduction is best. In GF(2571) for Intel P4 2.8 GHz processor, long multiplication with systematic reduction was 2.18 and 2.26 times faster than long multiplication with Barrett or Montgomery reduction. This paper determined that Montgomery Invariant scalar point multiplication with Systematic reduction in Projective coordinates was the fastest method for single scalar point multiplication for the NIST fields from GF(2163) to GF(2571). For single scalar point multiplication on a reconfigurable elliptic curve cryptography accelerator, we were able to achieve ∼6.1 times speedup using reconfigurable reduction methods with long multiplication, Montgomery’s MSB Invariant method in projective coordinates, and systematic reduction. Further extensions were made to implement fast reconfigurable elliptic curve cryptography for repeated scalar point multiplication on the same base point. We also show that for L > 20 the LSB invariant method combined with affine doubling precomputation outperforms the LSB invariant method combined with López-Dahab doubling precomputation for all reconfigurable reduction polynomial techniques in GF(2571) for Intel IA-32 processors. For L = 1000, the LSB invariant scalar point multiplication method was 13.78 to 34.32% faster than using the fastest Montgomery Invariant scalar point multiplication method on Intel IA-32 processors.

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  1. University of Minnesota, 200 Union St. SE., Minneapolis, MN, 55455, USA

    Aaron E. Cohen & Keshab K. Parhi

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  1. Aaron E. Cohen
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  2. Keshab K. Parhi
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Correspondence to Aaron E. Cohen.

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Cohen, A.E., Parhi, K.K. Fast Reconfigurable Elliptic Curve Cryptography Acceleration for GF(2m) on 32 bit Processors. J Sign Process Syst Sign Image Video Technol 60, 31–45 (2010). https://doi.org/10.1007/s11265-009-0398-4

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