A New Representation of Elements of Finite Fields GF(2
Germain Drolet · 1998
n+ 1. The new representation simultaneously satisfies the properties of various traditional representations, which leads, in some cases, to architectures of parallel-in-parallel-out arithmetic circuits (adder, multiplier, exponentiator/inverter, squarer, divider) with average to small complexity. We show that the implementation of all the arithmetic circuits designed for the new representation on an integrated circuit sometimes has smaller complexity than the implementation of all the arithmetic circuits designed for other representations. In addition, we derive a serial multiplier for the field F 2 m which comprises the least number of gates of all the serial multipliers known to the author, when m + 1 is a prime such that 2 is primitive in the field Z m+1 . Index Terms—Galois field arithmetic, normal basis, dual basis, canonical basis, VLSI implementation. ——————————F——————————