On the power of straight- line computations in finite fields

It is shown that a lower hound of
n^{3}
or more on the straight-line complexity of a function
f
over GF
(2^{n})
is also a lower bound on the network complexity of
f
and, hence, on the product of run time and program size of Turing machines. It is further shown that most functions over a finite field are hard to compute and that for most hard functions there exists no approximation via an easy algorithm.