A Genetic Algorithm for the Problem of Minimal Brauer Chains for Large Exponents

Exponentiation is an important and complex task used in cryptosystems such RSA. The reduction of the number of multiplications needed during the exponentiation can significantly improve the execution time of cryptosystems. The problem of determining the minimal sequence of multiplications required for performing a modular exponentiation can be formulated using the concept of Brauer Chains. This paper, shows a new approach to face the problem of getting Brauer Chains of minimal length by using a Genetic Algorithm (GA). The implementation details of the GA includes a representation based on the Factorial Number System (FNS), a mixture of Neighborhood Functions (NF), a mixture of Distribution Functions (DF) and a fine-tuning process to set the parameter values. We compare the proposed GA approach with another relevant solutions presented in the literature by using three benchmarks considered difficult to show that it is a viable alternative to solve the problem of getting shortest Brauer Chains.