Parallel algorithms for addition in non-standard number systems

In 1961 Avizienis proposed a parallel algorithm for addition in base 10 with digit set A = {-6, -5, ..., 5, 6}. Such an algorithm performs addition in constant time, independently of the length of the representation of the summands. In computer arithmetic parallel addition is used for speeding up multiplication and division algorithms. In this work we consider number systems where the base is a complex number β such that |β| > 1. We show that we can find a set of signed-digits on which addition is realizable by a parallel algorithm if and only if β is an algebraic number with no conjugate of modulus 1. We then address the question of the size of the digit set that permits parallel addition. We also investigate block parallel addition.