Iterative procedures for taking roots based on square roots

Note on Taking Square-Roots Modulo

In this contribution it is shown how Gauss’ famous cyclotomic sum formula can be used for extracting square-roots modulo .

On taking square roots without quadratic nonresidues over finite fields

We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in Õ(log q) bit operations over finite fields with q elements. As an application, we construct a deterministic primality-proving al...

On group Elements Having Square Roots

Square Roots Modulo p

The algorithm of Tonelli and Shanks for computing square roots modulo a prime number is the most used, and probably the fastest among the known algorithms when averaged over all prime numbers. However, for some particular prime numbers, there are other algorithms which are considerably faster. In this paper we compare the algorithm of Tonelli and Shanks with an algorithm based in quadratic fiel...

Ancient Indian Square Roots

Our modern system of positional decimal notation with zero, together with efficient algorithms for computation, which were discovered in India some time prior to 500 CE, certainly must rank among the most significant achievements of all time. And it was not easy. As Pierre-Simon Laplace (1923) explained, “the difficulty of inventing it will be better appreciated if we consider that it escaped t...