778 research outputs found
Global Orbit Patterns for One Dimensional Dynamical Systems
In this article, we study the behaviour of discrete one-dimensional dynamical
systems associated to functions on finite sets. We formalise the global orbit
pattern formed by all the periodic orbits (gop) as the ordered set of periods
when the initial value thumbs the finite set in the increasing order. We are
able to predict, using closed formulas, the number of gop for the set
of all the functions on . We also explore by computer
experiments special subsets of in which interesting patterns of
gop are found.Comment: 33 pages, 1 figur
Deterministic Chaos in Digital Cryptography
This thesis studies the application of deterministic chaos to digital
cryptography. Cryptographic systems such as pseudo-random generators
(PRNG), block ciphers and hash functions are regarded as a dynamic
system (X, j), where X is a state space (Le. message space)
and f : X -+ X is an iterated function. In both chaos theory and
cryptography, the object of study is a dynamic system that performs
an iterative nonlinear transformation of information in an apparently
unpredictable but deterministic manner. In terms of chaos theory, the
sensitivity to the initial conditions together with the mixing property
ensures cryptographic confusion (statistical independence) and diffusion
(uniform propagation of plaintext and key randomness into cihertext).
This synergetic relationship between the properties of chaotic and
cryptographic systems is considered at both the theoretical and practical
levels: The theoretical background upon which this relationship is
based, includes discussions on chaos, ergodicity, complexity, randomness,
unpredictability and entropy.
Two approaches to the finite-state implementation of chaotic systems
(Le. pseudo-chaos) are considered: (i) floating-point approximation of
continuous-state chaos; (ii) binary pseudo-chaos. An overview is given
of chaotic systems underpinning cryptographic algorithms along with
their strengths and weaknesses. Though all conventional cryposystems
are considered binary pseudo-chaos, neither chaos, nor pseudo-chaos are
sufficient to guarantee cryptographic strength and security.
A dynamic system is said to have an analytical solution Xn = (xo)
if any trajectory point Xn can be computed directly from the initial
conditions Xo, without performing n iterations. A chaotic system with an
analytical solution may have a unpredictable multi-valued map Xn+l =
f(xn). Their floating-point approximation is studied in the context of
pseudo-random generators.
A cryptographic software system E-Larm ™ implementing a multistream
pseudo-chaotic generator is described. Several pseudo-chaotic
systems including the logistic map, sine map, tangent- and logarithm feedback
maps, sawteeth and tent maps are evaluated by means of floating point
computations. Two types of partitioning are used to extract
pseudo-random from the floating-point state variable: (i) combining the
last significant bits of the floating-point number (for nonlinear maps);
and (ii) threshold partitioning (for piecewise linear maps). Multi-round
iterations are produced to decrease the bit dependence and increase non-linearity.
Relationships between pseudo-chaotic systems are introduced
to avoid short cycles (each system influences periodically the states of
other systems used in the encryption session).
An evaluation of cryptographic properties of E-Larm is given using
graphical plots such as state distributions, phase-space portraits, spectral
density Fourier transform, approximated entropy (APEN), cycle length
histogram, as well as a variety of statistical tests from the National Institute
of Standards and Technology (NIST) suite. Though E-Larm passes
all tests recommended by NIST, an approach based on the floating-point
approximation of chaos is inefficient in terms of the quality/performance
ratio (compared with existing PRNG algorithms). Also no solution is
known to control short cycles.
In conclusion, the role of chaos theory in cryptography is identified;
disadvantages of floating-point pseudo-chaos are emphasized although
binary pseudo-chaos is considered useful for cryptographic applications.Durand Technology Limite
Global Orbit Patterns for Dynamical Systems On Finite Sets
In this paper, the study of the global orbit pattern (gop) formed by all the
periodic orbits of discrete dynamical systems on a finite set allows us to
describe precisely the behaviour of such systems. We can predict by means of
closed formulas, the number of gop of the set of all the function from to
itself. We also explore, using the brute force of computers, some subsets of
locally rigid functions on , for which interesting patterns of periodic
orbits are found.Comment: 29 pages, 3 figures. to appear in Proceedings of the International
Conference On Modeling of Engineering & Technological Problems (ICMETP), Agra
(India), 2009. Published by the American Institute of Physics, U.S.
Preperiodicity and systematic extraction of periodic orbits of the quadratic map
Iteration of the quadratic map produces sequences of polynomials whose degrees explode as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree 4020, while for the 52 377 period-20 orbits the degree rises already to 1 047 540. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, without any need for iteration. Exact orbital equations provide valuable insight about the arithmetic structure and nesting properties of towers of algebraic numbers which define orbital points and bifurcation cascades of the map
Generating quantum channels from functions on discrete sets
Using the recent ability of quantum computers to initialize quantum states
rapidly with high fidelity, we use a function operating on a discrete set to
create a simple class of quantum channels. Fixed points and periodic orbits,
that are present in the function, generate fixed points and periodic orbits in
the associated quantum channel. Phenomenology such as periodic doubling is
visible in a 6 qubit dephasing channel constructed from a truncated version of
the logistic map. Using disjoint subsets, discrete function-generated channels
can be constructed that preserve coherence within subspaces. Error correction
procedures can be in this class as syndrome detection uses an initialized
quantum register. A possible application for function-generated channels is in
hybrid classical/quantum algorithms. We illustrate how these channels can aid
in carrying out classical computations involving iteration of non-invertible
functions on a quantum computer with the Euclidean algorithm for finding the
greatest common divisor of two integers
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