39 research outputs found
Resolution of Linear Algebra for the Discrete Logarithm Problem Using GPU and Multi-core Architectures
In cryptanalysis, solving the discrete logarithm problem (DLP) is key to
assessing the security of many public-key cryptosystems. The index-calculus
methods, that attack the DLP in multiplicative subgroups of finite fields,
require solving large sparse systems of linear equations modulo large primes.
This article deals with how we can run this computation on GPU- and
multi-core-based clusters, featuring InfiniBand networking. More specifically,
we present the sparse linear algebra algorithms that are proposed in the
literature, in particular the block Wiedemann algorithm. We discuss the
parallelization of the central matrix--vector product operation from both
algorithmic and practical points of view, and illustrate how our approach has
contributed to the recent record-sized DLP computation in GF().Comment: Euro-Par 2014 Parallel Processing, Aug 2014, Porto, Portugal.
\<http://europar2014.dcc.fc.up.pt/\>
Hadronic freeze-out following a first order hadronization phase transition in ultrarelativistic heavy-ion collisions
We analyze the hadronic freeze-out in ultra-relativistic heavy ion collisions
at RHIC in a transport approach which combines hydrodynamics for the early,
dense, deconfined stage of the reaction with a microscopic non-equilibrium
model for the later hadronic stage at which the hydrodynamic equilibrium
assumptions are not valid. With this ansatz we are able to self-consistently
calculate the freeze-out of the system and determine space-time hypersurfaces
for individual hadron species. The space-time domains of the freeze-out for
several hadron species are found to be actually four-dimensional, and differ
drastically for the individual hadrons species. Freeze-out radii distributions
are similar in width for most hadron species, even though the Omega-baryon is
found to be emitted rather close to the phase boundary and shows the smallest
freeze-out radii and times among all baryon species. The total lifetime of the
system does not change by more than 10% when going from SPS to RHIC energies.Comment: 11 pages, 4 eps-figures included, revised versio
Space-time evolution and HBT analysis of relativistic heavy ion collisions in a chiral SU(3) x SU(3) model
The space-time dynamics and pion-HBT radii in central heavy ion-collisions at
CERN-SPS and BNL-RHIC are investigated within a hydrodynamic simulation. The
dependence of the dynamics and the HBT-parameters on the EoS is studied with
different parametrisations of a chiral SU(3) sigma-omega model. The
selfconsistent collective expansion includes the effects of effective hadron
masses, generated by the nonstrange and strange scalar condensates. Different
chiral EoS show different types of phase transitions and even a crossover. The
influence of the order of the phase transition and of the difference in the
latent heat on the space-time dynamics and pion-HBT radii is studied. A small
latent heat, i.e. a weak first-order chiral phase transition, or even a smooth
crossover leads to distinctly different HBT predictions than a strong first
order phase transition. A quantitative description of the data, both at SPS
energies as well as at RHIC energies, appears difficult to achieve within the
ideal hydrodynamical approach using the SU(3) chiral EoS. A strong first-order
quasi-adiabatic chiral phase transition seems to be disfavored by the pion-HBT
data from CERN-SPS and BNL-RHIC
Comparison of space-time evolutions of hot/dense matter in =17 and 130 GeV relativistic heavy ion collisions based on a hydrodynamical model
Based on a hydrodynamical model, we compare 130 GeV/ Au+Au collisions at
RHIC and 17 GeV/ Pb+Pb collisions at SPS. The model well reproduces the
single-particle distributions of both RHIC and SPS.
The numerical solution indicates that huge amount of collision energy in RHIC
is mainly used to produce a large extent of hot fluid rather than to make a
high temperature matter; longitudinal extent of the hot fluid in RHIC is much
larger than that of SPS and initial energy density of the fluid is only 5%
higher than the one in SPS. The solution well describes the HBT radii at SPS
energy but shows some deviations from the ones at RHIC.Comment: 28 pages, 21 figures, REVTeX4, one figure is added and some figures
are replace
Improving the Berlekamp Algorithm for Binomials x n âââa
In this paper, we describe an improvement of the Berlekamp algorithm, a method for factoring univariate polynomials over finite fields, for binomials xn âa over finite fields Fq. More precisely, we give a deterministic algorithm for solving the equation h(x)qâĄh(x) (mod xnâa) directly without applying the sweeping-out method to the corresponding coefficient matrix. We show that the factorization of binomials using the proposed method is performed in OË, (n log q) operations in Fq if we apply a probabilistic version of the Berlekamp algorithm after the first step in which we propose an improvement. Our method is asymptotically faster than known methods in certain areas of q, n and as fast as them in other areas
Solving a 676-Bit Discrete Logarithm Problem in GF(36n )
Pairings on elliptic curves over finite fields are crucial for constructing various cryptographic schemes. The \eta_T pairing on supersingular curves over GF(3^n) is particularly popular since it is efficiently implementable. Taking into account the Menezes-Okamoto-Vanstone (MOV) attack, the discrete logarithm problem (DLP) in GF(3^{6n}) becomes a concern for the security of cryptosystems using \eta_T pairings in this case. In 2006, Joux and Lercier proposed a new variant of the function field sieve in the medium prime case, named JL06-FFS. We have, however, not yet found any practical implementations on JL06-FFS over GF(3^{6n}). Therefore, we first fulfilled such an implementation and we successfully set a new record for solving the DLP in GF(3^{6n}), the DLP in GF(3^{6 \cdot 71}) of 676-bit size. In addition, we also compared JL06-FFS and an earlier version, named JL02-FFS, with practical experiments. Our results confirm that the former is several times faster than the latter under certain conditions
Algebraic Cryptanalysis of Curry and Flurry using Correlated Messages
In \cite{BPW}, Buchmann, Pyshkin and Weinmann have described two families of
Feistel and SPN block ciphers called Flurry and Curry
respectively. These two families of ciphers are fully parametrizable and have
a sound design strategy against basic statistical attacks; i.e. linear and
differential attacks. The encryption process can be easily described by a set
of algebraic equations. These ciphers are then targets of choices for
algebraic attacks. In particular, the key recovery problem has been reduced to
changing the order of a Groebner basis \cite{BPW,BPWext}. This attack -
although being more efficient than linear and differential attacks - remains
quite limited. The purpose of this paper is to overcome this limitation by
using a small number of suitably chosen pairs of message/ciphertext for
improving algebraic attacks. It turns out that this approach permits to go one
step further in the (algebraic) cryptanalysis of Flurry and
\textbf{Curry}. To explain the behavior of our attack, we have established an
interesting connection between algebraic attacks and high order differential
cryptanalysis \cite{Lai}. From extensive experiments, we estimate that our
approach, that we can call an ``algebraic-high order
differential cryptanalysis, is polynomial when the Sbox is a power function.
As a proof of concept, we have been able to break Flurry -- up to
rounds -- in few hours