Square Roots Modulo p
نویسنده
چکیده
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 field extensions due to Cipolla, and give an explicit condition on a prime number to decide which algorithm is faster. Finally, we show that there exists an infinite sequence of prime numbers for which the algorithm of Tonelli and Shanks is asymptotically worse.
منابع مشابه
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 .
متن کاملInformation Protection Based on Extraction of Square Roots of Gaussian Integers
A cryptosystem, based on computation of square roots of complex integers modulo composite n, is described in this paper. This paper provides an algorithm extracting a square root of Gaussian integer. Various properties of square roots and a method for finding Gaussian generators are demonstrated. The generators can be instrumental in constructing other cryptosystems. It is shown how to signific...
متن کاملAn Analogue of Artin’s Primitive Root Conjecture
Let S = {a1, a2, . . . , an} be a set of nonzero integers such that for any nonempty subset T of S, the product of all the elements in T is not a perfect square. Then the density of the set of primes p for which the ai’s are quadratic non-residues modulo p, but not primitive roots modulo p, is at least 1 2n(q 1)qm , where m is a non-negative integer with m n and q is the least odd prime which...
متن کاملConstructing elliptic curves with a given number of points over a finite field
In using elliptic curves for cryptography, one often needs to construct elliptic curves with a given or known number of points over a given finite field. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is the calculation of the Hilbert class polynomial HD(X) modulo some inte...
متن کاملON THE SECOND MOMENT ESTIMATE INVOLVING THE λ-PRIMITIVE ROOTS MODULO n
Artin’s Conjecture on Primitive Roots states that a non-square non unit integer a is a primitive root modulo p for positive proportion of p. This conjecture remains open, but on average, there are many results due to P. J. Stephens (see [14], also [15]). There is a natural generalization of the conjecture for composite moduli. We can consider a as the primitive root modulo (Z/nZ)∗ if a is an el...
متن کامل