In Cryptography, computing a^xb^y mod n is the most important and the most time-consuming calculation The problem can be solved by classical binary method. Later research is based on this basis to increase computational efficiency. Furthermore, Binary signed-digit representation recoding algorithm, the Sparse Form, the DJM recoding method, and the Joint Sparse Form can be used to decrease the number of multiplication by aligning more non-zero bits. Another method is to pre-compute and store the part of the results to decrease the number of computations by shifting bits. Joint Sparse Form recording method is not a table driven algorithm in converting source codes into joint sparse form. In this paper, we first proposed a table driven algorithm for joint sparse form to simply recording concept. This algorithm can be constructed a finite state machine to denote the recording procedure. According to this finite state machine, we show that the average joint Hamming weight among joint sparse form is 0.5n when n approaches infinity. Finally, we show that the average joint Hamming weights of SS1 method and DS1 method among joint sparse form are 0.469n and 0.438n by using a similar method, respectively.

Computing A^X mod n or A^XB^Y mod n for large X, Y, and n is very important in many ElGamal-like public key cryptosystems. In this paper, we proposed using block method in sparse form to improve the performance of modular exponentiation and analyzing the computational cost by state transition diagram. We also extended the concept of Block Method and make it more general. This method is suitable for some devices with limited storage space, such as smart card.