Square root computation over even extension fields

Abstract

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 \F_{q^{2}}, with $q=p^n,$ $p$ an odd prime and $n\geq 1$. Both algorithms have an associate computational cost roughly equivalent to one exponentiation in \F_{q^{2}}. The first algorithm is devoted to the case when $q\equiv 1 \bmod 4$, whereas the second one handles the case when $q\equiv 3 \bmod 4$. 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