Note on taking square-roots modulo N

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 .

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...

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...

ROOTS OF UNITY AND NULLITY MODULO n

For a fixed positive integer , we consider the function of n that counts the number of elements of order in Zn. We show that the average growth rate of this function is C (logn) d( )−1 for an explicitly given constant C , where d( ) is the number of divisors of . From this we conclude that the average growth rate of the number of primitive Dirichlet characters modulo n of order is (d( )− 1)C (l...

On group Elements Having Square Roots