An Improvement of the Cipolla-Lehmer Type Algorithms

Let F_q be a finite field with q elements with prime power q and let r>1 be an integer with $q\equiv 1 \pmod{r}$. In this paper, we present a refinement of the Cipolla-Lehmer type algorithm given by H. C. Williams, and subsequently improved by K. S. Williams and K. Hardy. For a given r-th power residue c in F_q where r is an odd prime, the algorithm of H. C. Williams determines a solution of X^r=c in $O(r^3\log q)$ multiplications in F_q, and the algorithm of K. S. Williams and K. Hardy finds a solution in $O(r^4+r^2\log q)$ multiplications in F_q. Our refinement finds a solution in $O(r^3+r^2\log q)$ multiplications in F_q. Therefore our new method is better than the previously proposed algorithms independent of the size of r, and the implementation result via SAGE shows a substantial speed-up compared with the existing algorithms.