ESTIMATION OF SOME EXPONENTIAL SUM BY MEANS OF q-DEGREE
Glasgow mathematical journal, Tome 52 (2010) no. 2, pp. 315-324

Voir la notice de l'article provenant de la source Cambridge University Press

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.
GILLOT, VALÉRIE; LANGEVIN, PHILIPPE. ESTIMATION OF SOME EXPONENTIAL SUM BY MEANS OF q-DEGREE. Glasgow mathematical journal, Tome 52 (2010) no. 2, pp. 315-324. doi: 10.1017/S0017089510000017
@article{10_1017_S0017089510000017,
     author = {GILLOT, VAL\'ERIE and LANGEVIN, PHILIPPE},
     title = {ESTIMATION {OF} {SOME} {EXPONENTIAL} {SUM} {BY} {MEANS} {OF} {q-DEGREE}},
     journal = {Glasgow mathematical journal},
     pages = {315--324},
     year = {2010},
     volume = {52},
     number = {2},
     doi = {10.1017/S0017089510000017},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089510000017/}
}
TY  - JOUR
AU  - GILLOT, VALÉRIE
AU  - LANGEVIN, PHILIPPE
TI  - ESTIMATION OF SOME EXPONENTIAL SUM BY MEANS OF q-DEGREE
JO  - Glasgow mathematical journal
PY  - 2010
SP  - 315
EP  - 324
VL  - 52
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089510000017/
DO  - 10.1017/S0017089510000017
ID  - 10_1017_S0017089510000017
ER  - 
%0 Journal Article
%A GILLOT, VALÉRIE
%A LANGEVIN, PHILIPPE
%T ESTIMATION OF SOME EXPONENTIAL SUM BY MEANS OF q-DEGREE
%J Glasgow mathematical journal
%D 2010
%P 315-324
%V 52
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089510000017/
%R 10.1017/S0017089510000017
%F 10_1017_S0017089510000017

[1] 1.Adolphson, A. and Sperber, S., Exponential sums and Newton polyhedra: Cohomology and estimates, Ann. of Maths. 130 (1989), 367–406. Google Scholar

[2] 2.Akulinicev, N. M., Estimates for rational trigonometric sums of a special type, Soviet Math. Dokl. 6 (1965), 480–482. Google Scholar

[3] 3.Chabaud, F. and Vaudenay, S., Links between differential and linear cryptanalysis, Eurocrypt 94 950 (1994), 356–365. Google Scholar

[4] 4.Deligne, P., La conjecture de Weil I, Publ. Math. IHES 43 (1974), 273–308. Google Scholar

[5] 5.Deligne, P., Cohomologie étale des schémas. Lecture notes in mathematics 569 (Springer Verlag, Berlin, 1977); Publ. Math. IHES, (1974), 273–308. Google Scholar

[6] 6.Gillot, V., Bounds for exponential sums over finite fields. Finite Fields Appl. 1 (1995), 421–436. Google Scholar

[7] 7.Helleseth, T. and Kholosha, A., Monomial and quadratic bent functions over finite fields of odd characteristic, to appear in IEEE. Google Scholar

[8] 8.Karatsuba, A. A., On estimates of complete trigonometric sums, Sov. Math. Dokl. 7 (1966), 133–139. Google Scholar

[9] 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] 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] 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. Google Scholar | DOI

[12] 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] 13.Langevinand, Ph.Véron, P., On the non-linearity of power functions. Des. Codes Cryptogr. 37 (1) (2005), 31–43. Google Scholar

[14] 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] 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] 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] 17.Sidel'Nikov, V. M., On the mutual correlation of sequences, Soviet Math. Dokl. 12 (1971), 197–201. Google Scholar

Cité par Sources :