On circuits for arithmetic in finite fields

In the work it is proved that for any ɛ> 0 for any m, n = ms and s ≥ sɛ , you can choose in the field GF(2n) basis, for which the circuit complexity of multiplication of less than n1+ɛ/2, and the difficulty of inverting is less than n1+ɛ. At n = 2 · 3k for some basis of the obtained for the multiplication of estimating complexity n(log3n)(log2 log3 n)/2+O(1) and in the order of magnitude of the same estimates are obtained for inverting.