Fast Elliptic Curve Arithmetic and Improved Weil Pairing Evaluation

The side channel attack (SCA) is a serious attack on wearable devices that have scarce computational resources. Cryptographic algorithms on them should be efficient using small memory -- we have to make efforts to optimize the trade-off between efficiency and memory. In this paper we present efficient SCA-resistant scalar multiplications based on window method. Moller proposed an SPA-resistant window method based on 2w-ary window method, which replaces w-consecutive zeros to 1 plus w-consecutive 1 and it requires 2w points of table (or 2w-1 +1 points if the signed 2w-ary is used). The most efficient window method with small memory is the width-w NAF, which requires 2w-2 points of table. In this paper we convert the width-w NAF to an SPA-resistant addition chain. Indeed we generate a scalar sequence with the fixed pattern, e.g. |0..0x|0..0x|...|0..0x|, where x is positive odd points < 2w. Thus the size of the table is 2w-1, which is optimal in the construction of the SPA-resistant chain based on width-w NAF. The table sizes of the proposed scheme are 6% to 50% smaller than those of Moller's scheme for w = 2, 3, 4, 5, which are relevant choices in the sense of efficiency for 160-bit ECC.

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