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

We present low complexity formulae for the computation of cubing and cube root over \F_{3^m} constructed using special classes of irreducible trinomials, tetranomials and pentanomials. We show that for all those special classes of polynomials, field cubing and field cube root operation have the same computational complexity when implemented in hardware or software platforms. As one of the main applications of these two field arithmetic operations lies in pairing-based cryptography, we also give in this paper a selection of irreducible polynomials that lead to low cost field cubing and field cube root computations for supersingular elliptic curves defined over \F_{3^m}, where $m$ is a prime number in the pairing-based cryptographic range of interest, namely, $m\in [47, 541]$