An Improvement of the Cipolla-Lehmer Type Algorithms

Let Fq be a finite field with q elements with prime power q and let r > 1 be an integer with q ≡ 1 (mod 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 ∈ Fq where r is an odd prime, the algorithm of H. C. Williams determines a solution of X = c in O(r log q) multiplications in Fq, and the algorithm of K. S. Williams and K. Hardy finds a solution in O(r+r log q) multiplications in Fq. Our refinement finds a solution in O(r 3 + r log q) multiplications in Fq. 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.