A Fast Galois-Field Transform Algorithm Using Normal Bases

Abstract

The Good-Thomas prime-factor FFT algorithm is applied over the Galois field GF( pm ). The intermediate results are represented by using normal basis representation. A significant computational reduction is obtained by applying a conjugacy relation to a cyclotomic coset of the intermediate variables, and by using the cyclic-shift property of powers of the variables within the normal basis representation. Once the VLSI architecture for the butterfly module of a cyclotomic coset based on the algorithm is developed, these module arrays are used to form the stages of the fast transform. For the case of = 2, m = 8, perfonnance analysis shows a six-fold reduction in computational complexity, which is achieved in terms of an added hardware budget. MacWilliams and Sloan suggest [2] that there always exists a normal basis in the finite field GF( p' ) for all positive integers m. In other words, one can find a field element a such that N, = [ a , ap, dZ,. . .,&*I ) is a basis set of GF( p' ). Thus, the field element Vj E GF( p ) can be uniquely expressed in terms of the basis N,, as