Square Root Computation over Even Extension Fields

This paper presents a comprehensive study of the computation of square roots over finite extension fields. We propose two novel algorithms for computing square roots over even field extensions of the form BBFq², with q = pⁿ, p an odd prime and n ≥ 1. Both algorithms have an associate computational cost roughly equivalent to one exponentiation in BBFq². The first algorithm is devoted to the case when q ≡ 1 mod 4, whereas the second one handles the case when q ≡ 3 mod 4. Numerical comparisons show that the two algorithms presented in this paper are competitive and in some cases more efficient than the square root methods previously known.