On the number of binary signed digit representations of a given weight

summary:Binary signed digit representations (BSDR's) of integers have been studied since the 1950's. Their study was originally motivated by multiplication and division algorithms for integers and later by arithmetics on elliptic curves. Our paper is motivated by differential cryptanalysis of hash functions. We give an upper bound for the number of BSDR's of a given weight. Our result improves the upper bound on the number of BSDR's with minimal weight stated by Grabner and Heuberger in On the number of optimal base $2$ representations, Des. Codes Cryptogr. 40 (2006), 25--39, and introduce a new recursive upper bound for the number of BSDR's of any given weight