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International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 67 | Issue 6 | Year 2021 | Article Id. IJMTT-V67I6P508 | DOI : https://doi.org/10.14445/22315373/IJMTT-V67I6P508

Optimal Normal Bases Over Finite Fields


Duggirala Meher Krishna, Duggirala Ravi
Abstract

In this paper, a method for constructing a near optimal normal basis for algebraic extensions of a finite field is described. In each extension, except for the squares of the basis elements, the product of two distinct elements in the normal basis can be expressed as a linear combination of those two basis elements, with coefficients in a much smaller subfield.

Keywords
Finite fields; Algebraic field extensions; Normal basis; Optimal normal basis.
References

[1] L. Adleman, and H. Lenstra, Finding irreducible polynomials over finite fields. Proceedings of the 18th Annual Symposium on Theory of Computing (1986) 350--355
[2] S. H Gao,, and G. L. Mullen, Dickson Polynomials and Irreducible Polynomials over Finite Fields, J. Number Theory, 49 (1) (1994) 118--132
[3] R. Lidl, and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press (1986)
[4] R. C. Mullin, L. M. Onyszchuk, S. A/ Vanstone, and R. M. Wilson, Optimal Normal Bases in GF (pn), Discrete Applied Mathematics. 22 (1988/89) 149--161
[5] M. O. Rabin, Probabilistic Algorithms in Finite Fields, SIAM J. Computing, 9(2) (1980) 273--280
[6] V. Shoup, New Algorithms for Finding Irreducible Polynomials over Finite Fields, Mathematics of Computation. 54 (1990) 435-447. Extended abstract in Proceedings of the 29th Annual Symposium on Foundations of Computer Science. (1988) 283--290
[7] V. Shoup, Fast Construction of Irreducible Polynomials over Finite Fields. J. Symbolic Computation. 17 (1994) 371--391. Extended abstract in Proceedings of the 4th Annual Symposium on Discrete Algorithms (1993) 484--492


Citation :

Duggirala Meher Krishna, Duggirala Ravi, "Optimal Normal Bases Over Finite Fields," International Journal of Mathematics Trends and Technology (IJMTT), vol. 67, no. 6, pp. 58-61, 2021. Crossref, https://doi.org/10.14445/22315373/IJMTT-V67I6P508

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