Low Register-Complexity Systolic Digit-Serial Multiplier Over $GF(2^m)$ Based on Trinomials

Digit-serial systolic multipliers over $GF(2^m)$ based on the National Institute of Standards and Technology (NIST) recommended trinomials play a critical role in the real-time operations of cryptosystems. Systolic multipliers over $GF(2^m)$ involve a large number of registers of size $O(m^2)$ which results in significant increase in area complexity. In this paper, we propose a novel low register-complexity digit-serial trinomial-based finite field multiplier. The proposed architecture is derived through two novel coherent interdependent stages: (i) derivation of an efficient hardware-oriented algorithm based on a novel input-operand feeding scheme and (ii) appropriate design of novel low register-complexity systolic structure based on the proposed algorithm. The extension of the proposed design to Karatsuba algorithm (KA)-based structure is also presented. The proposed design is synthesized for FPGA implementation and it is shown that it (the design based on regular multiplication process) could achieve more than 12.1 percent saving in area-delay product and nearly 2.8 percent saving in power-delay product. To the best of the authors’ knowledge, the register-complexity of proposed structure is so far the least among the competing designs for trinomial based systolic multipliers (for the same type of multiplication algorithm).