VLSI architectures for computing exponentiations, multiplicative inverses, and divisions in GF(2m)

A modified parallel-in-parallel-out linear-systolic power-sum circuit designed to perform AB²+C computations in the finite field GF(2^m) is presented, where A, B, and C are arbitrary elements of GF(2^m). On the basis of the linear-systolic power-sum circuits, a VLSI architecture for exponentiation in GF(2^m ) is developed. Furthermore, two modified architectures that can be used to compute inverses and divisions over GF(2^m) are proposed. All the architectures are constructed from m-1 linear-systolic power-sum circuits. It should be noted that the presented exponentiator, inverter, and divider are the only such circuits having a throughput of 100%. The latency of the presented pipeline exponentiator, inverter, and divider is m(m-1) clock cycles. The cycle time (i.e., clock period) of the presented architectures is only two logic gate delays plus a short routing delay. For moderate values of m, say m⩽10, the circuit complexity of the presented circuits is realizable using presently available VLSI technology. The computation time of near two gate delays and 100% throughput enables the greatest computation speed in finite field arithmetic