Displaying similar documents to “An implementation of the number field sieve.”

A computer algorithm for finding new euclidean number fields

Roland Quême (1998)

Journal de théorie des nombres de Bordeaux

Similarity:

This article describes a computer algorithm which exhibits a sufficient condition for a number field to be euclidean for the norm. In the survey [3] p 405, Franz Lemmermeyer pointed out that 743 number fields where known (march 1994) to be euclidean (the first one, ℚ , discovered by Euclid, three centuries B.C.!). In the first months of 1997, we found more than 1200 new euclidean number fields of degree 4, 5 and 6 with a computer algorithm involving classical lattice properties of the...

On the class numbers of real cyclotomic fields of conductor pq

Eleni Agathocleous (2014)

Acta Arithmetica

Similarity:

The class numbers h⁺ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows, and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed, which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields...

What is the inverse of repeated square and multiply algorithm?

H. Gopalkrishna Gadiyar, K. M. Sangeeta Maini, R. Padma, Mario Romsy (2009)

Colloquium Mathematicae

Similarity:

It is well known that the repeated square and multiply algorithm is an efficient way of modular exponentiation. The obvious question to ask is if this algorithm has an inverse which would calculate the discrete logarithm and what is its time compexity. The technical hitch is in fixing the right sign of the square root and this is the heart of the discrete logarithm problem over finite fields of characteristic not equal to 2. In this paper a couple of probabilistic algorithms to compute...

Extending automorphisms to the rational fractions field.

Fernando Fernández Rodríguez, Agustín Llerena Achutegui (1991)

Extracta Mathematicae

Similarity:

We say that a field K has the Extension Property if every automorphism of K(X) extends to an automorphism of K. J.M. Gamboa and T. Recio [2] have introduced this concept, naive in appearance, because of its crucial role in the study of homogeneity conditions in spaces of orderings of functions fields. Gamboa [1] has studied several classes of fields with this property: Algebraic extensions of the field Q of rational numbers; euclidean, algebraically closed and pythagorean fields; fields...

Integer factorization and discrete logarithm problems

Pierrick Gaudry (2014)

Les cours du CIRM

Similarity:

These are notes for a lecture given at CIRM in 2014, for the “Journées Nationales du Calcul Formel”. We explain the basic algorithms based on combining congruences for solving the integer factorization and the discrete logarithm problems. We highlight two particular situations where the interaction with symbolic computation is visible: the use of Gröbner basis in Joux’s algorithm for discrete logarithm in finite field of small characteristic, and the exact sparse linear algebra tools...