High-speed multioperand addition and subtraction using a p-ary representation with necessary and minimum redundancy

In the usual P-ary number digit for the addition and subtraction of two numbers (operands), 0, 1, 2, …, P - 1 are employed. In the minimum redundant P-ary representation, 1 is further used, and the addition or subtraction can be performed in a constant time independent of the number of operand digits (Pϵ{2, 3, 4, …}). This paper extends this property. First, a high-speed addition-subtraction algorithm is discussed. It can execute the addition and subtraction of L(ϵ {2, 3, …, P}) operands in a constant time independent of n and L. The following property is shown as the result. Using the redundant P-ary representation (called K-redundant P-ary representation), where K,…, 2, 1, 0, 1, 2, …, P-l can be used as the digit, the minimum necessary value for K is L - 1, and the desired algorithm can be found if L-1K(ϵP—1). The presented algorithm can easily be extended to an algorithm which can execute the addition and subtraction of L operands in a constant time independent of n, where L is an arbitrary number such that P+IL. It is shown that if f(x) = [logpx]+1 ([y] is the minimum integer not less than y), the computation time T is O(loga), where α is the minimum number of operations for the function f( ) satisfying f(f(f(F))))P; α increases stepwise with L. When P is increased, T decreases, and the range of L, for which T is constant, increases exponentially. Using the proposed algorithm, one can expect that a multi-input high-speed adder-subtractor can be constructed. Various kinds of high-speed data processings can be realized, including the multiplication, where the shifted partial products are added.