A New Method of Generating Optimal Addition Chain Based on Graph

In many number theoretic cryptographic algorithms, encryption and decryption is of the form xn mod p, where n and p are integers. Exponentiation normally takes more time than any arithmetic operations. It may be performed by repeated multiplication which will reduce the computational time. To reduce the time further fewer multiplications are performed in computing the same exponentiation operation using addition chain. The problem of determining correct sequence of multiplications requires in performing modular exponentiation can be elegantly formulated using the concept of addition chains. There are several methods available in literature in generating the optimal addition chain. But novel graph based methods have been proposed in this paper to generate the optimal addition chain where the vertices of the graph represent the numbers used in the addition chain and edges represent the move from one number to another number in the addition chain. Method 1 termed as GBAPAC which generates all possible optimum addition chains for the given integer n by considering the edge weight of all possible numbers generated from every number in addition chain. Method 2 termed as GBMAC which generates the minimum number of optimum addition chains by considering mutually exclusive edges starting from every number. Further, the optimal addition chain generated for an integer using the proposed methods are verified with the conjectures which already existed in the literature with respect to addition chains.