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 ${\BBF_{{q^2}}}$ , 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 ${\BBF_{{q^2}}}$ . The first algorithm is devoted to the case when $q \equiv 1\, {\rm mod}\, 4$ , whereas the second one handles the case when $q \equiv 3\, {\rm mod}\,4$ . 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.