A Simple Left-to-Right Algorithm for the Computation of the Arithmetic Weight of Integers

We present a simple algorithm for computing the arithmetic weight of an integer with respect to a given radix r ≥ 2. The arithmetic weight of n is the minimum number of nonzero digits in any signed radix-r representation of n. This algorithm leads to a new family of minimal weight signed radix-r representations which can be constructed using a left-to-right on-line algorithm. These representations are different from the ones previously discovered by Joye and Yen [3]. The idea behind our algorithm is that of choosing closest elements which was introduced in [5]. Our results have applications in coding theory and in the efficient implementation of public-key cryptography. Index Terms computer arithmetic, signed radix-r representations, redundant representations, minimal weight representations, left-to-right recoding, elliptic curve cryptography.