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Character sums with exponential functions
Author(s) -
Friedlander John B.,
Hansen Jan,
Shparlinski Igor E.
Publication year - 2000
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300015734
Subject(s) - mathematics , modulo , integer (computer science) , multiplicative function , combinatorics , primitive root modulo n , character (mathematics) , prime (order theory) , exponential function , sequence (biology) , discrete mathematics , integer sequence , multiplicative inverse , multiplicative group , inverse , generating function , mathematical analysis , geometry , biology , computer science , genetics , programming language
Let ϑ be an integer of multiplicative order t ≥1 modulo a prime p . Sums of the formare introduced and estimated, with a sequence such that kz 1 , …, kz T is a permutation of z 1 , …, z T , both sequences taken modulo t , for sufficiently many distinct modulo t values of k . Such sequences include x n for x = 1,…, t with an integer n ≥1; x n for x = 1,…, t and gcd ( x , t ) = 1 with an integer n ≥1; e x for x = 1,…, T with an integer e , where T is the period of the sequence e x 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.