Advances in Alternative Non-adjacent Form Representations

Abstract

Abstract. From several decades, non-adjacent form (NAF) representations for integers have been extensively studied as an alternative to the usual binary number system where digits are in {0, 1}. In cryptography, the non-adjacent digit set (NADS) {−1, 0, 1} is used for optimization of arithmetic operations in elliptic curves. At SAC 2003, Muir and Stinson published new results on alternative digit sets: they proposed infinite families of integers x such that {0, 1, x} is a NADS as well as infinite families of integers x such that {0, 1, x} is not a NADS, so called a NON-NADS. Muir and Stinson also provided an algorithm that determines whether x leads to a NADS by checking if every integer n ∈ [0, −x 3] has a {0, 1, x}-NAF. In this paper, we extend these results by providing generators of NON-NADS infinite families. Furthermore, we reduce the search bound from. We introduce the notion of worst NON-NADS and give the complete characterization of such sets. Beyond the theoretical results, our contribution also aims at exploring some algorithmic aspects. We supply a much more efficient algorithm than those proposed by Muir and Stinson, which takes only 343 seconds to compute all x's from 0 to −10 7 such that {0, 1, x} is a NADS.

Reduced precomputed scalar multiplication cost for ECC

“…A Montgomery Ladder method keeps k DBLs & k ADDs, which is higher precomputed operations than the previous two but additional advantages is in favor of Side Channel Attack (SCA) [7]- [8]. A non-adjacent form (NAF) [9], [10] is another variation of algorithm representation in {−1, 0,1}, on average m 2 ⁄ bits of ADDS and m 2 ⁄ bits of DBLs, but in addition to this, it is resistant to the side-channel attacks [10]. The complexity of w-NAF [11] keeps m (w + 1) ⁄ in point ADDs only.…”