The Width-w NAF Method Provides Small Memory and Fast Elliptic Scalar Multiplications Secure against Side Channel Attacks

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. Möller proposed an SPA-resistant window method based on 2-ary window method, which replaces w-consecutive zeros to 1 plus w-consecutive 1̄ and it requires 2 points of table (or 2w−1 +1 points if the signed 2-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 < 2. 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 Möller’s scheme for w = 2, 3, 4, 5, which are relevant choices in the sense of efficiency for 160-bit ECC.