New Cube Root Algorithm Based on Third Order Linear Recurrence Relation in Finite Field

We present a new cube root algorithm in finite field Fq with q a power of prime, which extends Cipolla-Lehmer type algorithms and has lower complexity than Tonelli-Shanks type algorithms. Efficient computation of r-th root in Fq has many applications in computational number theory and many other related areas. There are two standard algorithms for computing rth root in finite field. One is Adleman-Manders-Miller algorithm which is a straightforward generalization of Tonelli-Shanks square root algorithm. Another algorithm is a also a natural generalization of Cipolla-Lehmer square root algorithm. Original Cipolla-Lehmer algorithm requires one to use extension field arithmetic in Fq2 , but one can use second order linear recurrence relation without any extension field arithmetic. Moreover a special type of Lucas sequence method of Müller gives a new square root algorithm which is consistently better than Tonelli-Shanks. However unlike the cases of Tonell-Shanks and Cipolla-Lehmer, extending the idea of Müller to cube root algorithm is not so obvious because, for given cubic residue c ∈ Fq, one needs to find a cubic polynomial f(x) with nice coefficients (i.e., with norm of f equal to one) and a suitable m such that Tr(α) = α + α + α 2 with f(α) = 0 is a cube root of c. In this paper, we show that the above question can be answered affirmatively. That is, for given cubic residue c ∈ Fq with q ≡ 1 (mod 9), we find an irreducible polynomial f(x) = x − ax + bx − 1 with root α ∈ Fq3 such that Tr(α q2+q−2 9 ) is a cube root of c. Consequently we find an efficient cube root algorithm which can be easily computed via simple third order linear recurrence sequence arising from f(x). Since it is easy to find closed formulas for cube root when q ≡ 4, 7 (mod 9) or when q ≡ 2 (mod 3), our cube root algorithm is applicable for any prime power q. Complexity estimation shows that our algorithm is consistently better than previously proposed Tonelli-Shanks and CipollaLehmer type algorithms.