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

Abstract

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 ∈ [47, 541].

Very efficient point multiplication on Koblitz curves

“…Binary curves have attracted many researchers to reduce point multiplication. These methods include parallelization, by using multiple parallel field multipliers in the finite field computations [8,9,10,11], and by interleaving [12,13]. Recently, several methods to perform parallel computations for point addition on Koblitz curves have been proposed in [8,9,11,13,14,15].…”

Formulas for p th root computations in finite fields of characteristic p

“…In Table 1, we give the Hamming wts of our heptanomials. In comparison with the equally spaced pentanomials in [8], these heptanomials have similar or better weight values. There are 21 new extensions for m ≤ 1024 where equally spaced heptanomials exists but no polynomials with less number of non-zero terms exist.…”

“…Cube roots can be used in Tate pairing algorithms in F 3 m in order to enhance its performance. As a consequence, work has been done to analyse the behaviour of polynomials with few non-zero coefficients (trinomials, tetranomials and pentanomials) over F 3 , as well as binomials over F p , p odd prime, where p ≥ 5 [7][8][9][10][11].…”

“…The Hamming wt for pth root friendly k-nomials can be computed directly from (4) using the polynomial expansion of f (x) ji . Therefore, from the multinomial theorem, an upper bound for wt(x i/p ) can be expressed as To show the potentiality of the previous Hamming wt results, we slightly improve the weights of x 1/3 and x 2/3 for the tetranomials in Section 4.2 of [8]. Let Q(x) = x m + ax k + bx l + c be an irreducible polynomial over F 3 where m ; k ; l ; r( mod 3) with r ∈ {1, 2}.…”