A probabilistic polynomial-time algorithm for computing the square root of a number x \in {\bf Z}/P{\bf Z} , where P = 2^{S}Q + 1(Q odd, s > 0) is a prime number, is described. In contrast to the Adleman, Manders, and Miller algorithm, this algorithm gets faster as s grows. As with the Berlekamp-Rabin algorithm, the expected running time of the algorithm is independent of x . However, the algorithm presented here is considerably faster for values of s greater than 2 .

A Simple an Fast Probabilistic Algorithm for Computing Square Roots Modulo a Prime Number.

We consider the computation of r-th roots in finite field-s. For the computation of square roots, there are two typical probabilistic methods: the Tonelli-Shanks method and the Cipolla-Lehmer method. The former method can be extended to the case of r-th roots, which is called the Adleman-Manders-Miller(AMM) method. The latter method had been generalized to the case of r-th roots with r prime. In this paper, we extend the Cipolla-Lehmer to the case of r-th root with r prime power and give the expected running time of our algorithm.

Generalized Cipolla-Lehmer Root Computation in Finite Fields

We propose a novel algorithm for finding square roots modulo p. Although there exists a direct formula to calculate square root of an element modulo prime (3 mod 4), but calculating square root modulo prime (1 mod 4) is non trivial. Tonelli-Shanks algorithm remains the most widely used and probably the fastest when averaged over all primes [19]. This paper proposes a new algorithm for finding square roots modulo all odd primes, which shows improvement over existing method in practical terms although asymptotically gives the same run time as Tonelli-Shanks. Apart from practically efficient computation time, the proposed method does not necessarily require availability of non-residue and can work with `relative non-residue' also. Such `relative non-residues' are much easier to find ( probability 2/3) compared to non-residues ( probability 1/2).

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An algorithm for finding square root modulo p

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An algorithm for finding square root modulo p