On binary signed digit representations of integers

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.