Towards minimal addition chains using ant colony optimisation

Modular exponentiation is one of the most important operations in almost all modern cryptosystems. It is performed using a series of modular multiplications. This operation is time consuming for large operands as is always the case in cryptography. Hence fast public-key cryptography software or hardware requires optimisation of the time consumed by a single modular multiplication and/or the reduction of the total number of modular multiplications required. This paper introduces a novel idea based on the principles of ant colony optimisation for finding a minimal addition chain that allows one to reduce the number of modular multiplications so that modular exponentiation can be implemented efficiently. The best addition chain reached by the ant system is compared to the one used in the m-ary and sliding window methods as well as with the best addition chain evolved by genetic algorithms. We demonstrate that the ant system significantly outperforms all these methods for any exponent size.