In this paper, the architecture for Fast Fourier Transform over Galois Field (2⁴) is described. The method used is cyclotomic decomposition. The Cyclotomic Fast Fourier Transforms (CFFTs) are preferred due to low multiplicative complexity. The approach used is the decomposition of the arbitrary polynomial into a sum of linearized polynomials. Also, Common Subexpression Elimination (CSE) algorithm is used to reduce the additive complexity of the architecture. By using CSE algorithm, the design with reduced operational complexity has been described.