High-Performance Two-Dimensional Finite Field Multiplication and Exponentiation for Cryptographic Applications

Finite field arithmetic operations have been traditionally used in different applications ranging from error control coding to cryptographic computations. Among these computations are normal basis multiplications and exponentiations which are utilized in efficient applications due to their advantageous characteristics and the fact that squaring (and subsequent powering by two) of elements can be obtained with no hardware complexity. In this paper, we present 2-D decomposition systolic-oriented algorithms to develop systolic structures for digit-level Gaussian normal basis multiplication and exponentiation over GF(2^m). The proposed high-performance architectures are suitable for a number of applications, e.g., architectures for elliptic curve Diffie-Hellman key agreement scheme in cryptography. The results of the benchmark of efficiency, performance, and implementation metrics of such architectures through a 65-nm application-specific integrated circuit platform confirm high-performance structures for the multiplication and exponentiation architectures presented in this paper are suitable for high-speed architectures, including cryptographic applications.