Abstract
There are at least two points of view when representing elements of \(\mathbb F_{2^n}\), the field of 2n elements. We could represent the (nonzero) elements as powers of a generating element, the exponent ranging from 0 to 2n–2. On the other hand, we could represent the elements as strings of n bits. In the former representation, multiplication becomes a very easy task, whereas in the latter one, addition is obvious. In this note, we focus on representing \(\mathbb F_{2^n}\) as strings of n bits in such a way that the natural basis (1,0,...,0), (0,1,...,0), ..., (0,0,...,1) becomes self-dual. We also outline an idea which leads to a very simple algorithm for finding a self-dual basis. Finally we study multiplication tables for the natural basis and present necessary and sufficient conditions for a multiplication table to give \(\mathbb F_2^n\) a field structure in such a way that the natural basis is self-dual.
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References
Brandman, Y., Orlitsky, A., Hennessy, J.: A Spectral Lower Bound Technique for the Size of Decision Trees and Two-Level and/or Circuits. IEEE Transactions on Computers 39(2), 282–287 (1990)
Hirvensalo, M.: Quantum Computing. Springer, Heidelberg (2001)
Hirvensalo, M.: Studies on Boolean Functions Related to Quantum Computing, Ph.D Thesis, University of Turku (2003)
Lempel, A.: Matrix factorization over GF(2) and trace-orthogonal bases of GF(2n)*. SIAM Journal on Computing 4(2), 175–186 (1975)
MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. North-Holland, Amsterdam (1981)
Mullin, R.C., Onyszchuk, I.M., Vanstone, S.A., Wilson, R.M.: Optimal normal bases in GF(pn)*. Discrete Applied Mathematics 22, 149–161 (1989)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)
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© 2004 Springer-Verlag Berlin Heidelberg
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Hirvensalo, M., Lahtonen, J. (2004). On Self-Dual Bases of the Extensions of the Binary Field. In: Karhumäki, J., Maurer, H., Păun, G., Rozenberg, G. (eds) Theory Is Forever. Lecture Notes in Computer Science, vol 3113. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27812-2_10
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DOI: https://doi.org/10.1007/978-3-540-27812-2_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22393-1
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