Fast Elliptic Curve Multiplications Resistant against Side Channel Attacks

Tetsuya IZU  Tsuyoshi TAKAGI  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E88-A   No.1   pp.161-171
Publication Date: 2005/01/01
Online ISSN: 
DOI: 10.1093/ietfec/e88-a.1.161
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)
Category: Tamper-Resistance
elliptic curve cryptosystems (ECC),  side channel attacks,  scalar multiplication,  SIMD operation,  

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This paper proposes fast elliptic curve multiplication algorithms resistant against side channel attacks, based on the Montgomery-type scalar multiplication. The proposed scalar multiplications can be applied to all curves over prime fields, e.g., any standardized curves over finite fields with characteristic larger than 3. The method utilizes the addition formulas xECDBL and xECADD assembled by only x-coordinates of points, and is applicable for any types of curves over finite fields. Then, we encapsulate two addition formulas into one formula xECADDDBL, which accomplishes a faster computation because several auxiliary variables of two formulas can be shared. We also develop a novel addition chain for the new formula, with which we can compute scalar multiplications. The improvement of our scalar multiplications over previous Coron's dummy operation method is about 18% for a 160-bit scalar multiplication. Our method requires no table-up of precomputed points and it is suitable for the implementation on memory constraint computing architectures, e.g., smart cards. Moreover, we optimize the proposed algorithms for parallelized implementations with SIMD operations. Compared with the similar scheme proposed by Fischer et al., our scheme is about 16% faster.