A low-complexity power-sum circuit for GF(2m) and its applications

This brief presents a new hardware-efficient bit-parallel circuit for computing C+AB² in finite fields GF(2^m) over the canonical basis. It consists of two parts: a normal poser-sum part and modular-reduction part, where each part is realized in a binary XOR tree structure. The proposed power-sum circuit works for the general-form generating polynomial and requires 3 m²-2 m AND gates and 3 m²-4 m+2 XOR gates to reach low time complexity of O(log₂ m). As compared to the conventional cellular-array structures for C+AB² in GF(2^m), the proposed one involves less hardware complexity and achieves a significant reduction in time complexity. The hardware requirement can further be reduced when a special-form generating polynomial is adopted. The corresponding reduced structures based on three special-form generating polynomials, including the trinomial x^m+x+1, the all-one polynomial, and the equally spaced polynomial, are given to demonstrate this property. A versatile structure, which can be programmed to compute inverses/divisions and exponentiations in GF(2^m), is also constructed based on the proposed power-sum circuit