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|Title:||Incomplete character sums and polynomial interpolation of the discrete logarithm|
|Authors:||Niederreiter, H. |
|Source:||Niederreiter, H., Winterhof, A. (2002). Incomplete character sums and polynomial interpolation of the discrete logarithm. Finite Fields and their Applications 8 (2) : 184-192. ScholarBank@NUS Repository. https://doi.org/10.1006/ffta.2001.0334|
|Abstract:||In the first part of the paper, certain incomplete character sums over a finite field Fp r are considered which in the case of finite prime fields Fp are of the form ∑n=A A+N-1 x(g(n)ψ(f(n)), where A and N are integers with 1 ≤ N < p, g and f are polynomials over Fp, and χ denotes a multiplicative and ψ an additive character of Fp. Excluding trivial cases, it is shown that the above sums are at most of the order of magnitude N1/2pr/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 Fp r. The proof is based on the above new bound for incomplete hybrid character sums. © 2002 Elsevier Science (USA).|
|Source Title:||Finite Fields and their Applications|
|Appears in Collections:||Staff Publications|
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