Existence of Rational Primitive Normal Pairs with Prescribed Norm and Trace

Authors: Soniya Takshak, R. K. Sharma

Abstract:

Let q be a prime power and n be a positive integer, Fq stands for the finite field of q elements, and Fqn denotes the extension of Fq of degree n. Also, F∗q represents the multiplicative group of non-zero elements of Fq, and the generators of F∗q are called primitive elements. A normal element of a finite field Fqn is an element α such that the set of α and its all conjugates in Fqn forms a basis for Fqn over Fq. Primitive normal elements have several applications in coding theory and cryptography. So, establishing the existence of primitive normal elements under certain conditions is theoretically essential and a genuine issue. In this article, we provide a sufficient condition for the existence of a primitive normal element α in Fqn of a prescribed primitive norm and non-zero trace over Fq such that f(α) is also primitive, where f(x) is a rational function of degree sum m over Fqn. Particularly, for the rational functions of degree sum 4 over Fqn, where Fq is the field of characteristic 11 and n is greater than or equal to 7, we demonstrated that there are only 3 exceptional pairs (q, n) for which such kind of primitive normal elements may not exist. In general, we show that such elements always exist except for finitely many choices of (q, n). We used additive and multiplicative character sums as important tools to arrive at our conclusion.

Keywords: Finite Field, Primitive Element, Normal Element, norm, trace, character.

References:

[1] A. K. Sharma, M. Rani, and S. K. Tiwari, Primitive normal pairs with prescribed norm and trace, Finite Fields Their Appl., vol. 78, pp. 101976, 2022.
[2] Anju, and R. K. Sharma, Existence of some special primitive normal elements over finite fields, Finite Fields Appl., vol. 46, pp. 280-303, 2017.
[3] S. D. Cohen, and S. Huczynska, The primitive normal basis theorem-without a computer, J. Lond. Math. Soc., vol. 67, no. 1, pp. 41-56, 2003.
[4] S. D. Cohen, and S. Huczynska, The strong primitive normal basis theorem, Acta Arith., vol. 143, no. 4, pp. 299-332, 2010.
[5] T. Cochrane and C. Pinner, Using Stepanov’s method for exponential sums involving rational functions, Journal of Number Theory, vol. 116, no. 2, pp. 270-292, 2006.
[6] L. Fu, and D. Q. Wan, A class of incomplete character sums, Quart. J. Math., vol. 65, pp. 1195-1211, 2014.
[7] H. W. Lenstra, Jr. and R. J. Schoof, Primitive normal bases for finite fields, Math. Comp., vol. 48, pp. 217-231, 1987.
[8] A. Gupta, R. K. Sharma, and S. D. Cohen, Some Special Primitive Elements with Prescribed Trace over Finite Fields, Finite Fields Appl., vol. 54, pp. 1-18, 2018.