Hardware Addition Over Finite Fields Based On Booth–Karatsuba Algorithm

Abstract

Two algorithms, both based around multiplication, one defined by Andrew Donald Booth in 1950 and the other defined by Anatoly Alexeevitch Karatsuba in 1960 can be applied to other types of operations. We know from recent results that to perform some algebraic transformations it is more efficient to calculate an inverse effect of a smaller magnitude together with an original action with a higher rank, than the initial operation, reaching the final element with less transitions. In this paper, we present an addition algorithm on $\mathbb {Z}/m\mathbb {Z}$ of big-integer numbers based on these concepts, using an alternative to traditional hardware implementation for binary addition based on FULL-ADDER cells, allowing the reduction of space complexity compared with other techniques, such as carry-lookahead, letting us calculate a modular addition in an optimal order complexity of $\mathcal {O}(n)$ without adding more complexity due to reduction operations.