Low-Complexity Bit-Parallel Square Root Computation over GF(2^{m}) for All Trinomials

Abstract

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 for any type of irreducible polynomials. We derive explicit formulae for the space and time complexities associated to 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 to the field squaring operation. Furthermore, a practical application of the square root operator in the domain of field exponentiation computation is presented. It is shown that by using as building blocks squarers, multipliers and square root blocks, a parallel version of the classical square-and-multiply exponentiation algorithm can be obtained. A hardware implementation of that parallel version may provide a speedup of up to 50% percent when compared with the traditional version.

Automatic synthesis of quantum circuits for point addition on ordinary binary elliptic curves

“…be of degree ≤ n/2 . From the explicit formulae for a classical implementation by Rodríguez-Henríquez et al [21] we obtain the following.…”

“…Let A := a 0 + a 1 x + · · · + a n−1 x n−1 be a representative of an F 2 n -element a. In [21] explicit expressions for computing the representations of a 2 and √ a from a 0 , . .…”

Low Complexity Cubing and Cube Root Computation over $\F_{3^m}$ in Polynomial Basis

“…However, in the case that the irreducible polynomial P (x) is a trinomial, P (x) = x m + x n + 1 with m an odd prime number, then the square root of an arbitrary element A ∈ F 2 m can be obtained at a very low price: the computation of some few additions and shift operations [15]. 5 Furthermore, Rodríguez-Henríquez et al showed in [22] that for all practical cases, the cost of computing in hardware the square root over binary fields generated with irreducible trinomials, is not more expensive than the computational effort required for computing field squarings.…”

On Repeated Squarings in Binary Fields

“…Square root can be implemented with fewer resources than squaring if the irreducible is a trinomial [20]. Hence, it might be possible to reduce the complexity of a repeated squaring (especially, varying exponent with the second solution) by, first, "shooting over" the required exponent e with an e opt chain and, then, reversing back with (repeated) square roots, rather than using fewer e opt 's and then reaching e with a few repeated squarings.…”