The frequency transformation methods like fast fourier transform algorithms can be competently used in realization of discrete fourier transforms over Galois field, which have broad applications in network security and digital communication in error correcting codes. The cyclotomic fast fourier transform (CFFT) is a type of fast fourier transform algorithm over finite fields This method utilizes the benefit of cyclic decomposition. The cyclotomic breakdown of input data is used to reduce the number of operations which can be equally exploited to get a set by set treatment of the input sequence. Common subexpression elimination (CSE) is an useful optimization process to solve the multiple constant multiplication problems. In this paper, common subexpression elimination algorithm for cyclotomic fast fourier transform over fixed field 23 is designed. Using CSE algorithm, we reduce the additive complexities of cyclotomic fast fourier transform. The design of CSE is to spot regular patterns that are present in expressions more than once and replace them with a single variable. Using above method every regular pattern calculates only once, thus minimizing the area of CFFT architecture required in VLSI implementation.