Improved Randomized Algorithm for Square Root Extraction

Square root extraction plays a key role in cryptosystems based on elliptic curves. Motwani and Raghavan had proposed an algorithm for square root extraction over finite field F-p, where p is an odd prime. It is a randomized algorithm with expected running time O(len(p)(4)). Its complexity relies mainly on the loops calling Euclid's algorithm for polynomials over F-p. In this paper, we propose an improvement of it. The new algorithm calls a subroutine for computing a Legendre symbol. Since the running time of computing a Legendre symbol is much less than that of Euclid's algorithm for polynomials over F-p, the new algorithm is more efficient. It only takes time O(len(p)(3)). We also compare the new algorithm with those algorithms for square root extraction over finite fields.