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<p><b>Abstract</b>—Let <b>F</b><sub>2</sub> denote the binary field and <tmath>${\schmi{\bf F}}_{2^m}$</tmath> an algebraic extension of degree <it>m</it> > 1 over <b>F</b><sub>2</sub>. Traditionally, elements of <tmath>${\schmi{\bf F}}_{2^m}$</tmath> are either represented as powers of a primitive element of <tmath>${\schmi{\bf F}}_{2^m}$</tmath> together with 0, or by an expansion in a basis of the vector space <tmath>${\schmi{\bf F}}_{2^m}$</tmath> over <b>F</b><sub>2</sub>. We propose a new representation based on an isomorphism from <tmath>${\schmi{\bf F}}_{2^m}$</tmath> into the residue polynomial ring modulo <it>X</it><super><it>n</it></super> + 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 <tmath>${\schmi{\bf F}}_{2^m}$</tmath> which comprises the least number of gates of all the serial multipliers known to the author, when <it>m</it> + 1 is a prime such that 2 is primitive in the field <b>Z</b><sub><it>m</it>+1</sub>.</p>
Galois field arithmetic, normal basis, dual basis, canonical basis, VLSI implementation.
Germain Drolet, "A New Representation of Elements of Finite Fields GF(2m) Yielding Small Complexity Arithmetic Circuits", IEEE Transactions on Computers, vol. 47, no. , pp. 938-946, September 1998, doi:10.1109/12.713313
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