Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic

It was proved that the complexity of square root computation in the Galois field GF(3s), s = 2kr, is equal to O(M(2k)M(r)k + M(r) log2r) + 2kkr1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3s) is equal to O(M(2k)M(r)) and O(M(2k)M(r)) + r1+o(1), respectively. If the basis in the field GF(3r) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2kkr1+o(1) and r1+o(1) can be omitted. For M(n) one may take n log2nψ(n) where ψ(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form Or (M (s) log 2s) = s (log 2s)2ψ(s).