On the Cipolla–Lehmer type algorithms in finite fields

Abstract

In this paper, we present a refinement of the Cipolla–Lehmer type algorithm given by H. C. Williams in 1972, and later improved by K. S. Williams and K. Hardy in 1993. For a given r-th power residue c∈ 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 SageMath shows a substantial speed-up compared with the existing algorithms. It should be mentioned that our method also works for a composite r. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.