16 research outputs found
Public key cryptography and error correcting codes as Ising models
We employ the methods of statistical physics to study the performance of
Gallager type error-correcting codes. In this approach, the transmitted
codeword comprises Boolean sums of the original message bits selected by two
randomly-constructed sparse matrices. We show that a broad range of these codes
potentially saturate Shannon's bound but are limited due to the decoding
dynamics used. Other codes show sub-optimal performance but are not restricted
by the decoding dynamics. We show how these codes may also be employed as a
practical public-key cryptosystem and are of competitive performance to modern
cyptographical methods.Comment: 6 page
Thouless-Anderson-Palmer Approach for Lossy Compression
We study an ill-posed linear inverse problem, where a binary sequence will be
reproduced using a sparce matrix. According to the previous study, this model
can theoretically provide an optimal compression scheme for an arbitrary
distortion level, though the encoding procedure remains an NP-complete problem.
In this paper, we focus on the consistency condition for a dynamics model of
Markov-type to derive an iterative algorithm, following the steps of
Thouless-Anderson-Palmer's. Numerical results show that the algorithm can
empirically saturate the theoretical limit for the sparse construction of our
codes, which also is very close to the rate-distortion function.Comment: 10 pages, 3 figure
One step RSB scheme for the rate distortion function
We apply statistical mechanics to an inverse problem of linear mapping to
investigate the physics of the irreversible compression. We use the replica
symmetry breaking (RSB) technique with a toy model to demonstrate the Shannon's
result. The rate distortion function, which is widely known as the theoretical
limit of the compression with a fidelity criterion, is derived using the Parisi
one step RSB scheme. The bound can not be achieved in the sparsely-connected
systems, where suboptimal solutions dominate the capacity.Comment: 8 pages, 2 figure
Typical Performance of Gallager-type Error-Correcting Codes
The performance of Gallager's error-correcting code is investigated via
methods of statistical physics. In this approach, the transmitted codeword
comprises products of the original message bits selected by two
randomly-constructed sparse matrices; the number of non-zero row/column
elements in these matrices constitutes a family of codes. We show that
Shannon's channel capacity is saturated for many of the codes while slightly
lower performance is obtained for others which may be of higher practical
relevance. Decoding aspects are considered by employing the TAP approach which
is identical to the commonly used belief-propagation-based decoding.Comment: 6 pages, latex, 1 figur
Cryptographical Properties of Ising Spin Systems
The relation between Ising spin systems and public-key cryptography is
investigated using methods of statistical physics. The insight gained from the
analysis is used for devising a matrix-based cryptosystem whereby the
ciphertext comprises products of the original message bits; these are selected
by employing two predetermined randomly-constructed sparse matrices. The
ciphertext is decrypted using methods of belief-propagation. The analyzed
properties of the suggested cryptosystem show robustness against various
attacks and competitive performance to modern cyptographical methods.Comment: 4 pages, 2 figure
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