Fast algorithms for common-multiplicand multiplication and exponentiation by performing complements

The multiplications of common multiplicands and exponentiations of large integers with large modulus are the primary computation operations in several well-known public key cryptosystems. The Hamming weight of the multiplier or the exponent plays an important role for computation efficiency. By performing complements, the Hamming weight of an integer can be reduced. Based on this concept, we propose efficient algorithms for common-multiplicand multiplications (CMM) and exponentiations. In the average case, it takes k/2+2×log(k)+5 k-bit additions to compute the CMM. For exponentiation, the proposed method takes 5k/4+2 multiplications on average, but the pre-computation for a modular multiplicative inverse is required. Combining the original CMM, the number of multiplications can further be reduced to 9k/8+2.