Abstract
The complexity of the multiplication operation in finite fields is of interest for both theoretical and practical reasons. For example, an optimal normal basis for F2N has complexity 2N−1. A construction described in J. H. Silverman, (“Cryptographic Hardware and Embedded Systems,” Lecture Notes in Computer Science, Vol. 1717, pp. 122–134, Springer–Verlag, Berlin, 1999.) allows multiplication of complexity N+1 to be performed in F2N by working in a larger ring R of dimension N+1 over F2. In this paper we give a complete classification of all such rings and show that this construction is the only one which also has a certain useful permutability property.
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