Incomplete character sums and a special class of permutations

Cohen, S. D.; Niederreiter, H.; Shparlinski, I. E.; Zieve, M.

Cohen, S. D. ; Niederreiter, H. ; Shparlinski, I. E. ; Zieve, M.

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 53-63.

Résumé
Abstract

Nous donnons une méthode pour majorer des sommes incomplètes des valeurs d’un caractère d’un groupe abélien fini, en des éléments générés par une récurrence d’ $.$ Cette méthode est particulièrement explicite lorsque la récurrence implique un type spécial de permutations, appelées $ℛ$ -orthomorphismes. Nous donnons quelques exemples de ces $ℛ$ -orthomorphismes.

We present a method of bounding incomplete character sums for finite abelian groups with arguments produced by a first-order recursion. This method is particularly effective if the recursion involves a special type of permutation called an $ℛ$ -orthomorphism. Examples of $ℛ$ -orthomorphisms are given.

MR Zbl

@article{JTNB_2001__13_1_53_0,
     author = {Cohen, S. D. and Niederreiter, H. and Shparlinski, I. E. and Zieve, M.},
     title = {Incomplete character sums and a special class of permutations},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {53--63},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     mrnumber = {1839900},
     zbl = {1065.11097},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2001__13_1_53_0/}
}

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JO  - Journal de théorie des nombres de Bordeaux
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Cohen, S. D.; Niederreiter, H.; Shparlinski, I. E.; Zieve, M. Incomplete character sums and a special class of permutations. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 53-63. http://www.numdam.org/item/JTNB_2001__13_1_53_0/

Bibliographie
Cité par

[1] T. Cochrane, On a trigonometric inequality of Vinogradov. J. Number Theory 27 (1987), 9-16. | MR | Zbl

[2] J. Dénes, P.J. Owens, Some new Latin power sets not based on groups. J. Combinatorial Theory Ser. A 85 (1999), 69-82. | MR | Zbl

[3] J. Gutierrez, H. Niederreiter, I.E. Shparlinski, On the multidimensional distribution of inversive congruential pseudorandom numbers in parts of the period. Monatsh. Math. 129 (2000), 31-36. | MR | Zbl

[4] N.M. Korobov, On the distribution of digits in periodic fractions. Math. USSR Sbornik 18 (1972), 659-676. | MR | Zbl

[5] R. Lidl, H. Niederreiter, Finite Fields. Cambridge Univ. Press, Cambridge, 1997. | MR | Zbl

[6] H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Amer. Math. Soc. 84 (1978), 957-1041. | MR | Zbl

[7] H. Niederreiter, I.E. Shparlinski, On the distribution of inversive congruential pseudorandom numbers in parts of the period. Math. Comp., to appear. | MR | Zbl

[8] H. Niederreiter, I.E. Shparlinski, On the distribution and lattice structure of nonlinear congruential pseudorandom numbers. Finite Fields Appl. 5 (1999), 246-253. | MR | Zbl

[9] H. Niederreiter, I.E. Shparlinski, Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus. Acta Arith. 92 (2000), 89-98. | MR | Zbl

[10] H. Niederreiter, I.E. Shparlinski, On the distribution of pseudorandom numbers and vectors generated by inversive methods. Applicable Algebra Engrg. Comm. Comput. 10 (2000),189-202. | MR | Zbl

[11] C.P. Schnorr, S. Vaudenay, Black box cryptanalysis of hash networks based on multipermutations. Advances in Cryptology - EUROCRYPT '94 (A. De Santis, ed.), Lecture Notes in Computer Science, Vol. 950, pp. 47-57, Springer, Berlin, 1995. | MR | Zbl

[12] I.E. Shparlinski, Finite Fields: Theory and Computation. Kluwer Academic Publ., Dordrecht, 1999. | MR | Zbl

[13] J.H. Van Lint, R.M. Wilson, A Course in Combinatorics. Cambridge Univ. Press, Cambridge, 1992. | MR | Zbl