High-radix Modular Exponentiation for hardware implementation of Public-Key Cryptography

Abstract

Public Key Cryptography (PKC) is an asymmetric security system where encryption is done publicly by disseminating public keys and decryption is done by using private keys, known only to the owner. The process of sending these keys involve encrypting a message in using various techniques like Rivest-Shamir-Adleman and Deffie Hellman. Majority of the cryptosystems use Modular Multiplication and Modular Exponentiation to bring about the encryption. Efficient hardware implementation of this is a necessity however is difficult as implementation of modular operations in hardware consumes large areas and power. Montgomery method does faster Modular Multiplications by converting numbers to special representations (Residual Number System) where multiplications and divisions are done by shift operations. This paper presents the modular exponential techniques in higher radix namely, radix-4 and radix-8. These algorithms are compatible to the hardware implementations. The modular multiplications involved in modular exponentiation are evaluated with the help of Montgomery multiplication method. We have introduced modifications in the existing Montgomery method and named as Adapted Montgomery Multiplication (AMM) Technique. This efficiently will reduce the number of clock cycles for implementation. The algorithms implemented are analyzed in comparison to other existing algorithms in terms of number of multiplications.