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A new representation of elements of finite fields GF(2m) yielding small complexity arithmetic circuits

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1 Author(s)
Drolet, G. ; Dept. of Electr. & Comput. Eng., R. Mil. Coll. of Canada, Kingston, Ont., Canada

Let F2 denote the binary field and F2m, an algebraic extension of degree m>1 over F2. Traditionally, elements of F2m are either represented as powers of a primitive element of F2m together with 0, or by an expansion in a basis of the vector space F2m over F2. We propose a new representation based on an isomorphism from F2m into the residue polynomial ring module Xn+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 F2m 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 Zm+1

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Computers, IEEE Transactions on  (Volume:47 ,  Issue: 9 )