Character sums with exponential functions

John B. Friedlander*, Jan Hansen, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

30 Citations (Scopus)

Abstract

Let uθ be an integer of multiplicative order t ≥ 1 modulo a prime p. Sums of the form Sf (p, t, a) = ΣTs =1 exp (2πiaθzs/p) are introduced and estimated, with f = (Z1 , . . . , zT) a sequence such that kz1 , . . . , kzT is a permutation of z1 , . . . , zT, both sequences taken modulo t, for sufficiently many distinct modulo t values of k. Such sequences include xn for x = 1 , . . . , t with an integer n ≥ 1; xn for x = 1 , . . . , t and gcd (x, t) = 1 with an integer n ≥ 1; ex for x 1 , . . . , T with an integer e, where T is the period of the sequence ex modulo t. Some of the results can be extended to composite moduli and to sums of multiplicative characters as well. Character sums with the above sequences have some cryptographic motivation and applications and have been considered in several papers by J. B. Friedlander, D. Lieman and I. E. Shparlinski. In particular several previous bounds are generalized and improved.

Original languageEnglish
Pages (from-to)75-85
Number of pages11
JournalMathematika
Volume47
Issue number1-2
Publication statusPublished - Jun 2000

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