On the computation of r-th roots in finite fields

Abstract

Let q be a power of a prime such that $q\equiv 1\left(\mathrm{mod}r\right)$. Let c be an r-th power residue over $F_q$. In this paper, we present a new r-th root formula which generalizes G.H. Cho et al.'s cube root algorithm, and which provides a refinement of Williams' Cipolla-Lehmer based procedure. Our algorithm which is based on the recurrence relations arising from irreducible polynomial $h\left(x\right)=x^r+{(-1)}^{r+1}(b+{(-1)}^{r}r)(x-1)$ with $b=c+{(-1)}^{r+1}r$ requires only $O(r^2\log q+r^4)$ multiplications for $r>1$. The multiplications for computation of the main exponentiation of our algorithm are half of that of the Williams' Cipolla-Lehmer type algorithms.