346 research outputs found
Asymmetric quantum error correcting codes
The noise in physical qubits is fundamentally asymmetric: in most devices,
phase errors are much more probable than bit flips. We propose a quantum error
correcting code which takes advantage of this asymmetry and shows good
performance at a relatively small cost in redundancy, requiring less than a
doubling of the number of physical qubits for error correction
Analysis of reaction and timing attacks against cryptosystems based on sparse parity-check codes
In this paper we study reaction and timing attacks against cryptosystems
based on sparse parity-check codes, which encompass low-density parity-check
(LDPC) codes and moderate-density parity-check (MDPC) codes. We show that the
feasibility of these attacks is not strictly associated to the quasi-cyclic
(QC) structure of the code but is related to the intrinsically probabilistic
decoding of any sparse parity-check code. So, these attacks not only work
against QC codes, but can be generalized to broader classes of codes. We
provide a novel algorithm that, in the case of a QC code, allows recovering a
larger amount of information than that retrievable through existing attacks and
we use this algorithm to characterize new side-channel information leakages. We
devise a theoretical model for the decoder that describes and justifies our
results. Numerical simulations are provided that confirm the effectiveness of
our approach
Statistical mechanical analysis of a hierarchical random code ensemble in signal processing
We study a random code ensemble with a hierarchical structure, which is
closely related to the generalized random energy model with discrete energy
values. Based on this correspondence, we analyze the hierarchical random code
ensemble by using the replica method in two situations: lossy data compression
and channel coding. For both the situations, the exponents of large deviation
analysis characterizing the performance of the ensemble, the distortion rate of
lossy data compression and the error exponent of channel coding in Gallager's
formalism, are accessible by a generating function of the generalized random
energy model. We discuss that the transitions of those exponents observed in
the preceding work can be interpreted as phase transitions with respect to the
replica number. We also show that the replica symmetry breaking plays an
essential role in these transitions.Comment: 24 pages, 4 figure
Properties of Classical and Quantum Jensen-Shannon Divergence
Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the
most important divergence measure of information theory, Kullback divergence.
As opposed to Kullback divergence it determines in a very direct way a metric;
indeed, it is the square of a metric. We consider a family of divergence
measures (JD_alpha for alpha>0), the Jensen divergences of order alpha, which
generalize JD as JD_1=JD. Using a result of Schoenberg, we prove that JD_alpha
is the square of a metric for alpha lies in the interval (0,2], and that the
resulting metric space of probability distributions can be isometrically
embedded in a real Hilbert space. Quantum Jensen-Shannon divergence (QJD) is a
symmetrized and smoothed version of quantum relative entropy and can be
extended to a family of quantum Jensen divergences of order alpha (QJD_alpha).
We strengthen results by Lamberti et al. by proving that for qubits and pure
states, QJD_alpha^1/2 is a metric space which can be isometrically embedded in
a real Hilbert space when alpha lies in the interval (0,2]. In analogy with
Burbea and Rao's generalization of JD, we also define general QJD by
associating a Jensen-type quantity to any weighted family of states.
Appropriate interpretations of quantities introduced are discussed and bounds
are derived in terms of the total variation and trace distance.Comment: 13 pages, LaTeX, expanded contents, added references and corrected
typo
Statistical mechanics of typical set decoding
The performance of ``typical set (pairs) decoding'' for ensembles of
Gallager's linear code is investigated using statistical physics. In this
decoding, error happens when the information transmission is corrupted by an
untypical noise or two or more typical sequences satisfy the parity check
equation provided by the received codeword for which a typical noise is added.
We show that the average error rate for the latter case over a given code
ensemble can be tightly evaluated using the replica method, including the
sensitivity to the message length. Our approach generally improves the existing
analysis known in information theory community, which was reintroduced by
MacKay (1999) and believed as most accurate to date.Comment: 7 page
Information Theory based on Non-additive Information Content
We generalize the Shannon's information theory in a nonadditive way by
focusing on the source coding theorem. The nonadditive information content we
adopted is consistent with the concept of the form invariance structure of the
nonextensive entropy. Some general properties of the nonadditive information
entropy are studied, in addition, the relation between the nonadditivity
and the codeword length is pointed out.Comment: 9 pages, no figures, RevTex, accepted for publication in Phys. Rev.
E(an error in proof of theorem 1 was corrected, typos corrected
The Statistical Physics of Regular Low-Density Parity-Check Error-Correcting Codes
A variation of Gallager error-correcting codes is investigated using
statistical mechanics. In codes of this type, a given message is encoded into a
codeword which comprises Boolean sums of message bits selected by two randomly
constructed sparse matrices. The similarity of these codes to Ising spin
systems with random interaction makes it possible to assess their typical
performance by analytical methods developed in the study of disordered systems.
The typical case solutions obtained via the replica method are consistent with
those obtained in simulations using belief propagation (BP) decoding. We
discuss the practical implications of the results obtained and suggest a
computationally efficient construction for one of the more practical
configurations.Comment: 35 pages, 4 figure
Properties of sparse random matrices over finite fields
Typical properties of sparse random matrices over finite (Galois) fields are studied, in the limit of large matrices, using techniques from the physics of disordered systems. For the case of a finite field GF(q) with prime order q, we present results for the average kernel dimension, average dimension of the eigenvector spaces and the distribution of the eigenvalues. The number of matrices for a given distribution of entries is also calculated for the general case. The significance of these results to error-correcting codes and random graphs is also discussed
- …