EUDML

For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p, w} (u)$ are defined by $q_{p, w} (u) \equiv (u^{w} - u^{w p}) / p m o d p$ with $0 \leq q_{p, w} (u) \leq p - 1$ , u ≥ 0, which are generalizations of Fermat quotients $q_{p, p - 1} (u)$ . First, we estimate the number of elements $1 \leq u < N \leq p$ for which $f (u) \equiv q_{p, w} (u) m o d p$ for a given polynomial f(x) over the finite field $_{p}$ . In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each element of...

Approximation of the discrete logarithm in finite fields of even characteristic by real polynomials

Nina Brandstätter; Arne Winterhof — 2006

Archivum Mathematicum

We obtain lower bounds on degree and additive complexity of real polynomials approximating the discrete logarithm in finite fields of even characteristic. These bounds complement earlier results for finite fields of odd characteristic.

Exact solutions to Waring's problem for finite fields

Arne Winterhof; Christiaan van de Woestijne — 2010

Acta Arithmetica

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On Waring's problem in finite fields

A note on Waring's problem in finite fields

Incomplete exponential sums over finite fields and their applications to new inversive pseudorandom number generators

Multiplicative character sums for nonlinear recurring sequences

Incomplete character sums over finite fields and their application to the interpolation of the discrete logarithm by Boolean functions

Polynomial quotients: Interpolation, value sets and Waring's problem

Approximation of the discrete logarithm in finite fields of even characteristic by real polynomials

Exact solutions to Waring's problem for finite fields