21,674 research outputs found
Detecting Simultaneous Integer Relations for Several Real Vectors
An algorithm which either finds an nonzero integer vector for
given real -dimensional vectors such
that or proves that no such integer vector with
norm less than a given bound exists is presented in this paper. The cost of the
algorithm is at most exact arithmetic
operations in dimension and the least Euclidean norm of such
integer vectors. It matches the best complexity upper bound known for this
problem. Experimental data show that the algorithm is better than an already
existing algorithm in the literature. In application, the algorithm is used to
get a complete method for finding the minimal polynomial of an unknown complex
algebraic number from its approximation, which runs even faster than the
corresponding \emph{Maple} built-in function.Comment: 10 page
Algorithms in algebraic number theory
In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope to show that the study of algorithms not only increases our
understanding of algebraic number fields but also stimulates our curiosity
about them. The discussion is concentrated of three topics: the determination
of Galois groups, the determination of the ring of integers of an algebraic
number field, and the computation of the group of units and the class group of
that ring of integers.Comment: 34 page
Parallel integer relation detection: techniques and applications
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New Shortest Lattice Vector Problems of Polynomial Complexity
The Shortest Lattice Vector (SLV) problem is in general hard to solve, except
for special cases (such as root lattices and lattices for which an obtuse
superbase is known). In this paper, we present a new class of SLV problems that
can be solved efficiently. Specifically, if for an -dimensional lattice, a
Gram matrix is known that can be written as the difference of a diagonal matrix
and a positive semidefinite matrix of rank (for some constant ), we show
that the SLV problem can be reduced to a -dimensional optimization problem
with countably many candidate points. Moreover, we show that the number of
candidate points is bounded by a polynomial function of the ratio of the
smallest diagonal element and the smallest eigenvalue of the Gram matrix.
Hence, as long as this ratio is upper bounded by a polynomial function of ,
the corresponding SLV problem can be solved in polynomial complexity. Our
investigations are motivated by the emergence of such lattices in the field of
Network Information Theory. Further applications may exist in other areas.Comment: 13 page
A kilobit hidden SNFS discrete logarithm computation
We perform a special number field sieve discrete logarithm computation in a
1024-bit prime field. To our knowledge, this is the first kilobit-sized
discrete logarithm computation ever reported for prime fields. This computation
took a little over two months of calendar time on an academic cluster using the
open-source CADO-NFS software. Our chosen prime looks random, and
has a 160-bit prime factor, in line with recommended parameters for the Digital
Signature Algorithm. However, our p has been trapdoored in such a way that the
special number field sieve can be used to compute discrete logarithms in
, yet detecting that p has this trapdoor seems out of reach.
Twenty-five years ago, there was considerable controversy around the
possibility of back-doored parameters for DSA. Our computations show that
trapdoored primes are entirely feasible with current computing technology. We
also describe special number field sieve discrete log computations carried out
for multiple weak primes found in use in the wild. As can be expected from a
trapdoor mechanism which we say is hard to detect, our research did not reveal
any trapdoored prime in wide use. The only way for a user to defend against a
hypothetical trapdoor of this kind is to require verifiably random primes
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