A Fast Modular Multiexponentiation Algorithm Revisited

Recently, a fast modular multiexponentiation algorithm for computing A^XB^Y (mod N) was proposed. The authors claimed that on average their algorithm only requires to perform 1.306 k modular multiplications (MMs), where k is the bit length of the exponents. This claimed performance is significantly better than all other comparable algorithms, where the best known result by other algorithms achieves 1.503 k MMs only. However, neither the complexity analysis of the algorithm given by its original proposers nor that by other authors are correct. In this paper, we give a formal complexity analysis and show that the claimed performance is not true. We find that the actual computational complexity of the algorithm should be 1.556 k. This means that the best modular multiexponentiations algorithm based on canonical-sighed-digit technique is still not able to overcome the 1.5 k barrier.