Efficient algorithms and architectures for field multiplication using Gaussian normal bases

Recently, implementations of normal basis multiplication over the extended binary field GF(2^m) have received considerable attention. A class of low complexity normal bases called Gaussian normal bases has been included in a number of standards, such as IEEE and NIST for an elliptic curve digital signature algorithm. The multiplication algorithms presented there are slow in software since they rely on bit-wise inner product operations. In this paper, we present two vector-level software algorithms which essentially eliminate such bit-wise operations for Gaussian normal bases. Our analysis and timing results show that the software implementation of the proposed algorithm is faster than previously reported normal basis multiplication algorithms. The proposed algorithm is also more memory efficient compared with its look-up table-based counterpart. Moreover, two new digit-level multiplier architectures are proposed and it is shown that they outperform the existing normal basis multiplier structures. As compared with similar digit-level normal basis multipliers, the proposed multiplier with serial output requires the fewest number of XOR gates and the one with parallel output is the fastest multiplier.