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ESTIMATION OF SOME EXPONENTIAL SUM BY MEANS OF q-DEGREE

Published online by Cambridge University Press:  29 March 2010

VALÉRIE GILLOT  and
PHILIPPE LANGEVIN
VALÉRIE GILLOT
Affiliation:
Institut de Mathématiques de Toulon, Université du Sud Toulon-Var, France e-mail: [email protected], [email protected]
PHILIPPE LANGEVIN
Affiliation:
Institut de Mathématiques de Toulon, Université du Sud Toulon-Var, France e-mail: [email protected], [email protected]
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Abstract

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In this paper, we improve results of Gillot, Kumar and Moreno to estimate some exponential sums by means of q-degrees. The method consists in applying suitable elementary transformations to see an exponential sum over a finite field as an exponential sum over a product of subfields in order to apply Deligne bound. In particular, we obtain new results on the spectral amplitude of some monomials.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 2 , May 2010 , pp. 315 - 324
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Adolphson, A. and Sperber, S., Exponential sums and Newton polyhedra: Cohomology and estimates, Ann. of Maths. 130 (1989), 367–406.Google Scholar
2.Akulinicev, N. M., Estimates for rational trigonometric sums of a special type, Soviet Math. Dokl. 6 (1965), 480–482.Google Scholar
3.Chabaud, F. and Vaudenay, S., Links between differential and linear cryptanalysis, Eurocrypt 94 950 (1994), 356–365.Google Scholar
4.Deligne, P., La conjecture de Weil I, Publ. Math. IHES 43 (1974), 273–308.Google Scholar
5.Deligne, P., Cohomologie étale des schémas. Lecture notes in mathematics 569 (Springer Verlag, Berlin, 1977); Publ. Math. IHES, 43 (1974), 273–308.Google Scholar
6.Gillot, V., Bounds for exponential sums over finite fields. Finite Fields Appl. 1 (1995), 421–436.Google Scholar
7.Helleseth, T. and Kholosha, A., Monomial and quadratic bent functions over finite fields of odd characteristic, to appear in IEEE.Google Scholar
8.Karatsuba, A. A., On estimates of complete trigonometric sums, Sov. Math. Dokl. 7 (1966), 133–139.Google Scholar
9.Katz, N. M., Sommes exponentielles, Cours à Orsay, automne 1979, in Astérisque, vol. 79 (Société Mathématique de France, Paris, 1980), 209.Google Scholar
10.Kumar, P. V. and Moreno, O., Polyphase sequences with periodic correlation properties better than binary sequences, IEEE IT Trans. 37 (1991), 603–616.Google Scholar
11.Kumar, P. V., Scholtz, R. A. and Welch, L. R., Generalized bent functions and their properties, J. Comb. Theory (A) 40 (1985), 90–107.CrossRefGoogle Scholar
12.Lachaud, G., Exponential sums as discrete Fourier transform with invariant phase functions, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC) 673 (1993), 231–242.Google Scholar
13.Langevinand, Ph.Véron, P., On the non-linearity of power functions. Des. Codes Cryptogr. 37 (1) (2005), 31–43.Google Scholar
14.Lidl, R. and Niederreiter, H., Finite fields, vol. 20 of encyclopedia of mathematics and its applications (Addison-Wesley, Indianapolis, IN, 1983).Google Scholar
15.Paterson, K. G., Applications of exponential sums in communications theory. Cryptography and coding, in LNCS vol. 1746 (Walker, M., Editor), (Springer-Verlag, Berlin, 1999), 1–24.Google Scholar
16.Roquette, P., Exponential sums: The estimate of Hasse-Davenport-Weil. Available online: http://www.ma.utexas.edu/users/voloch/expsums.html.Google Scholar
17.Sidel'Nikov, V. M., On the mutual correlation of sequences, Soviet Math. Dokl. 12 (1971), 197–201.Google Scholar