1,372 research outputs found
Computing discrete logarithms in subfields of residue class rings
Recent breakthrough methods \cite{gggz,joux,bgjt} on computing discrete
logarithms in small characteristic finite fields share an interesting feature
in common with the earlier medium prime function field sieve method \cite{jl}.
To solve discrete logarithms in a finite extension of a finite field \F, a
polynomial h(x) \in \F[x] of a special form is constructed with an
irreducible factor g(x) \in \F[x] of the desired degree. The special form of
is then exploited in generating multiplicative relations that hold in
the residue class ring \F[x]/h(x)\F[x] hence also in the target residue class
field \F[x]/g(x)\F[x]. An interesting question in this context and addressed
in this paper is: when and how does a set of relations on the residue class
ring determine the discrete logarithms in the finite fields contained in it? We
give necessary and sufficient conditions for a set of relations on the residue
class ring to determine discrete logarithms in the finite fields contained in
it. We also present efficient algorithms to derive discrete logarithms from the
relations when the conditions are met. The derived necessary conditions allow
us to clearly identify structural obstructions intrinsic to the special
polynomial in each of the aforementioned methods, and propose
modifications to the selection of so as to avoid obstructions.Comment: arXiv admin note: substantial text overlap with arXiv:1312.167
Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language
The traditional split-up into a low level language and a high level language
in the design of computer algebra systems may become obsolete with the advent
of more versatile computer languages. We describe GiNaC, a special-purpose
system that deliberately denies the need for such a distinction. It is entirely
written in C++ and the user can interact with it directly in that language. It
was designed to provide efficient handling of multivariate polynomials,
algebras and special functions that are needed for loop calculations in
theoretical quantum field theory. It also bears some potential to become a more
general purpose symbolic package
Time- and Space-Efficient Evaluation of Some Hypergeometric Constants
The currently best known algorithms for the numerical evaluation of
hypergeometric constants such as to decimal digits have time
complexity and space complexity of or .
Following work from Cheng, Gergel, Kim and Zima, we present a new algorithm
with the same asymptotic complexity, but more efficient in practice. Our
implementation of this algorithm improves slightly over existing programs for
the computation of , and we announce a new record of 2 billion digits for
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