Multiplication and exponentiation of big integers with hybrid Montgomery and distributed Karatsuba algorithm

This paper presents a Montgomery algorithm used for modular reduction with distributed Karatsuba algorithm to achieve the goal. The main objective of this proposal is to reduce the running time and increase the performance of modular reduction. Distributed Karatsuba is proposed to reduce the time of multiplication of two very large digits. Modular exponentiation is performed by repeated modular multiplication. In this algorithm, the multiplication operations must be performed sequentially and the order of operation must be implemented in sequential order. In this work, the Montgomery´s algorithm is enhanced with distributed Karatsuba algorithm. The distributed Karatsuba algorithm is applied to perform multiplication operations that use very large numbers (more than 256 digits) in the Montgomery algorithm. Both hardware and software are important parameters in the successful implementation of the purposed idea.