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In this contribution, we introduce a low-complexity bit-parallel algorithm for computing square roots over binary extension fields. Our proposed method can be applied to any type of irreducible polynomials. We derive explicit formulas for the space and time complexities associated with the square root operator when working with binary extension fields generated using irreducible trinomials. We show that, for those finite fields, it is possible to compute the square root of an arbitrary field element with equal or better hardware efficiency than the one associated with the field squaring operation. Furthermore, a practical application of the square root operator in the domain of field exponentiation computation is presented.