Exponential sums and linear complexity of nonlinear pseudorandom number generators with polynomials of small $p$-weight degree

For a class of polynomials $f(X)$ of small $p$-weight degree over a finite field of characteristic $p$ we improve the general bounds on exponential sums and linear complexity of nonlinear pseudorandom number generators defined by $\mu_{n+1}= f(\mu_n), n=0,1,\ldots$ with some initial value $mu_0$. This extends the class of polynomials where a nontrivial exponential sum bound is known. From the bound on exponential sums we derive discrepancy bounds for nonlinear pseudorandom vectors.