  The generation of pseudorandom elements over finite fields is fundamental to
the time, space and randomness complexity of randomized algorithms and data
structures. We consider the problem of generating $k$-independent random values
over a finite field $\mathbb{F}$ in a word RAM model equipped with constant
time addition and multiplication in $\mathbb{F}$, and present the first
nontrivial construction of a generator that outputs each value in constant
time, not dependent on $k$. Our generator has period length
$|\mathbb{F}|\,\mbox{poly} \log k$ and uses $k\,\mbox{poly}(\log k) \log
|\mathbb{F}|$ bits of space, which is optimal up to a $\mbox{poly} \log k$
factor. We are able to bypass Siegel's lower bound on the time-space tradeoff
for $k$-independent functions by a restriction to sequential evaluation.
