3,755 research outputs found
Improvements in the computation of ideal class groups of imaginary quadratic number fields
We investigate improvements to the algorithm for the computation of ideal
class groups described by Jacobson in the imaginary quadratic case. These
improvements rely on the large prime strategy and a new method for performing
the linear algebra phase. We achieve a significant speed-up and are able to
compute ideal class groups with discriminants of 110 decimal digits in less
than a week.Comment: 14 pages, 5 figure
Practical improvements to class group and regulator computation of real quadratic fields
We present improvements to the index-calculus algorithm for the computation
of the ideal class group and regulator of a real quadratic field. Our
improvements consist of applying the double large prime strategy, an improved
structured Gaussian elimination strategy, and the use of Bernstein's batch
smoothness algorithm. We achieve a significant speed-up and are able to compute
the ideal class group structure and the regulator corresponding to a number
field with a 110-decimal digit discriminant
Security Estimates for Quadratic Field Based Cryptosystems
We describe implementations for solving the discrete logarithm problem in the
class group of an imaginary quadratic field and in the infrastructure of a real
quadratic field. The algorithms used incorporate improvements over
previously-used algorithms, and extensive numerical results are presented
demonstrating their efficiency. This data is used as the basis for
extrapolations, used to provide recommendations for parameter sizes providing
approximately the same level of security as block ciphers with
and -bit symmetric keys
Beyond Expansion II: Low-Lying Fundamental Geodesics
A closed geodesic on the modular surface is "low-lying" if it does not travel
"high" into the cusp. It is "fundamental" if it corresponds to an element in
the class group of a real quadratic field. We prove the existence of infinitely
many low-lying fundamental geodesics, answering a question of
Einsiedler-Lindenstrauss-Michel-Venkatesh.Comment: 39 pages, 1 figure. This paper is a complete re-write of the posting
arXiv:1310.7190, making the latter obsolete. The main tools are similar, but
the application is in some sense orthogonal to the initial goal. We hope to
return to the questions of the earlier paper at a later dat
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
Solving discrete logarithms on a 170-bit MNT curve by pairing reduction
Pairing based cryptography is in a dangerous position following the
breakthroughs on discrete logarithms computations in finite fields of small
characteristic. Remaining instances are built over finite fields of large
characteristic and their security relies on the fact that the embedding field
of the underlying curve is relatively large. How large is debatable. The aim of
our work is to sustain the claim that the combination of degree 3 embedding and
too small finite fields obviously does not provide enough security. As a
computational example, we solve the DLP on a 170-bit MNT curve, by exploiting
the pairing embedding to a 508-bit, degree-3 extension of the base field.Comment: to appear in the Lecture Notes in Computer Science (LNCS
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