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Inspired by the Montgomery and generalized polynomial basis (GPB) squaring operation, we introduce and study the notion of  asymptotic square root  over binary extension fields    $GF(2^{m})$   . This new notion is a natural generalization of Montgomery-like square root, a special form of square root that was recently introduced by Li  et al . Given an arbitrary element    $Ain GF(2^{m})$    and a fixed element    $omega $   , the asymptotic square root is defined as    $A^{1/2}cdot omega $   . We show that, by choosing a proper parameter    $omega $    regarding different kind of irreducible polynomials that define    $GF(2^{m})$   , such square root operation can achieve better space and time complexity. Meanwhile, we have proved that the complexity of asymptotic is linear with the generating polynomials. Specifically, for the field    $GF(2^{m})$    is defined by two type of specific irreducible pentanomials, i.e., Type C.1 and Type C.2 pentanomials, we derived explicit formulae for space and time complexities associated with the asymptotic square root operator. On top of that, a practical application of the asymptotic square root in exponentiation computation is also presented.

Fast Asymptotic Square Root for Two Types of Special Pentanomials