Character sums with exponential functions

Let uθ be an integer of multiplicative order t ≥ 1 modulo a prime p. Sums of the form Sf (p, t, a) = Σ^T_{s =1} exp (2πiaθ^zs/p) are introduced and estimated, with f = (Z₁ , . . . , z_T) a sequence such that kz₁ , . . . , kz_T is a permutation of z₁ , . . . , z_T, 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; 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.