International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: Low Complexity Bit-Parallel Square Root Computation over GF($2^m$) for all Trinomials

Francisco Rodríguez-Henríquez
Guillermo Morales-Luna
Julio López
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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.
  title={Low Complexity Bit-Parallel Square Root Computation over GF($2^m$) for all Trinomials},
  booktitle={IACR Eprint archive},
  keywords={foundations / number theory , implementation},
  note={ 13241 received 2 Apr 2006},
  author={Francisco Rodríguez-Henríquez and Guillermo Morales-Luna and Julio López},