Low-Power Radix-8 Booth Encoded Modulo 2n − 1 Multiplier with Adaptive Delay

A special moduli set Residue Number System (RNS) of high dynamic range (DR) can speed up the execution of very-large word-length repetitive multiplications found in applications like public key cryptography. The modulo 2n − 1 multiplier is usually the noncritical datapath among all modulo multipliers in such high-DR RNS multiplier. This timing slack can be exploited to reduce the system area and power consumption without compromising the system performance. With this precept, a family of radix-8 Booth encoded modulo 2n − 1 multipliers, with delay adaptable to the RNS multiplier delay, is proposed. The modulo 2n − 1 multiplier delay is made scalable by controlling the word-length of the ripple carry adder, k employed for radix-8 hard multiple generation. Formal criteria for the selection of the adder word-length are established by analysing the effect of varying k on the timing of multiplier components. It is proven that for a given n, there exist a number of feasible values of k such that the total bias incurred from the partially-redundant partial products can be counteracted by only a single constant binary string. This compensation constant for different valid combinations of n and k can be pre-computed at design time using number theoretic properties of modulo 2n−1 arithmetic and hardwired as a partial product to be accumulated in the carry save adder tree. The proposed radix-8 Booth encoded modulo 2n −1 multiplier saves substantial area and power consumption over the radix-4 Booth encoded multiplier in medium to large word-length RNS multiplication.