Incomplete character sums and polynomial interpolation of the discrete logarithm

In the first part of the paper, certain incomplete character sums over a finite field F-p(r) are considered which in the case of finite prime fields F-p are of the form Sigma(n=A)(A+N-1) chi(g(n))psi(f(n)), where A and N are integers with 1 < N < p, g and f are polynomials over F-p, and chi denotes a multiplicative and psi an additive character of F-p. Excluding trivial cases. it is shown that the above sums are at most of the order of magnitude N(1/2)p(r/4), Recently. Shparlinski showed that a polynomial f over the integers which coincides with the discrete logarithm of the finite prime field Fp for N consecutive elements of Fp must have a degree at least of the order of magnitude Np-1/2. In this paper this result is extended to arbitrary F-p(r). The proof is based on the above new bound for incomplete hybrid character sums. (C) 2002 Elsevier Science (USA).