Pseudorandomness and Dynamics of Fermat Quotients

We obtain some theoretic and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0, ..., p-1\}$. We also consider pseudorandom properties of Fermat quotients such as joint distribution and linear complexity

On the Hidden Shifted Power Problem

We consider the problem of recovering a hidden element $s$ of a finite field \F_q of $q$ elements from queries to an oracle that for a given x\in \F_q returns $(x+s)^e$ for a given divisor $e\mid q-1$. We use some techniques from additive combinatorics and analytic number theory that lead to more efficient algorithms than the naive interpolation algorithm, for example, they use substantially fewer queries to the oracle.Comment: Moubariz Garaev (who has now become a co-author) has introduced some new ideas that have led to stronger results. Several imprecision of the previous version have been corrected to