Applications of signed digit representations of an integer include computer arithmetic, cryptography, and digital signal processing. An integer of length n bits can have several binary signed digit (BSD) representations and their number depends on its value and varies with its length. In this paper, we present an algorithm that calculates the exact number of BSD representations of an integer of a certain length. We formulate the integer that has the maximum number of BSD representations among all integers of the same length. We also present an algorithm to generate a random BSD representation for an integer starting from the most significant end and its modified version which generates all possible BSD representations. We show how the number of BSD representations of k increases as we prepend 0s to its binary representation.
The Montgomery ladder method of computing elliptic curve scalar multiplication is esteemed as an efficient algorithm, inherently resistant to simple side-channel attacks as well as to various fault attacks. In FDTC 08, Fouque et al. present an attack on the Montgomery ladder in the presence of a point validation countermeasure, when the y-coordinate is not used.In this paper, we present an efficient countermeasure that renders the algorithm resistant to this attack as well as to other known fault attacks.
Abstract. Differential power analysis (DPA) attacks can be of major concern when applied to cryptosystems that are embedded into small devices such as smart cards. To immunize elliptic curve cryptosystems (ECCs) against DPA attacks, recently several countermeasures have been proposed. A class of countermeasures is based on randomizing the paths taken by the scalar multiplication algorithm throughout its execution which also implies generating a random binary signed-digit (BSD) representation of the scalar. This scalar is an integer and is the secret key of the cryptosystem. In this paper, we investigate issues related to the BSD representation of an integer such as the average and the exact number of these representations, and integers with maximum number of BSD representations within a specific range. This knowledge helps a cryptographer to choose a key that provides better resistance against DPA attacks. Here, we also present an algorithm that generates a random BSD representation of an integer starting from the most significant signed bit. We also present another algorithm that generates all existing BSD representations of an integer to investigate the relation between increasing the number of bits in which an integer is represented and the increase in the number of its BSD representations.
Elliptic curve cryptosystems have become increasingly popular due to their efficiency and the small size of the keys they use. Particularly, the anomalous curves introduced by Koblitz allow a complex representation of the keys, denoted τ NAF, that make the computations over these curves more efficient. In this report, we propose an efficient method for randomizing a τ NAF to produce different equivalent representations of the same key to the same complex base τ . We prove that the average Hamming density of the resulting representations is 0.5. We identify the pattern of the τ NAFs yielding the maximum number of representations and the formula governing this number. We also present deterministic methods to compute the average and the exact number of possible representations of a τ NAF.
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