Efficient Computation of Modular Multiplicative Inverse w.r.t 2i

Computing Modular multiplicative inverse is an important step in many arithmetic algorithms used in Cryptography. For example, Montgomery modular multiplication [2] is required to find out multiplicative inverse. In this paper, we propose an improved modular multiplicative inverse algorithm w.r.t modulo some power of 2 which is very fast and can be easily implemented on a FPGA, for it uses mainly the addition and shift operations. We also provide a formal proof for the correctness of the algorithm. One beauty of our algorithm is while computing modular inverse of an integer x w.r.t $2^{i}(i \geq 3)$, it also computes modular inverses of the integer x for all values of j, where $2 \leq j \leq i-1$.