A Galois Field multiplier circuit for multiplying two polynomials (multiplicands). The multiplier circuit can use any arbitrary primitive polynomial to preserve the Galois Field. The multiplier circuit includes at least one logic unit that receives as a first input one of the multiplicands and shift the multiplicand in question by 1 bit to the left. The logic unit receives as a second input a pre-determined primitive polynomial and multiplies the primitive polynomial by the highest bit of the multiplicand received at the other input of the logic unit. The bit-shifted multiplicand is XOR-ed with the primitive polynomial multiplied the highest bit of the multiplicand and the result of the XOR operation is provided to a second logic circuit that completes the multiplication of the two polynomials.