on the invertible cellular automata 150 over f p in this work the reversibility problem for cellular automata with rule number 150 over the finite field f p is tackled it is shown that when null boundary conditions are stated the reversibility is independent of the state set f p and it appears when the number of cells of the cellular space satisfies some conditions furthermore the explicit expressions of the inverse cellular automata are computed

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

nondeterministic cellular automata abstract using purely topological concepts we characterize the class of nondeterministic mappings between shift spaces that give rise to nondeterministic sliding block codes we use such topological notions to investigate basic properties of surjective and reversible nondeterministic cellular automata

when and how can a cellular automaton be rewritten as a lattice gas both cellular automata ca and lattice gas automata lg provide finite algorithmic presentations for certain classes of infinite dynamical systems studied by symbolic dynamics it is customary to use the terms cellular automaton and lattice gas for a dynamic system itself as well as for its presentation the two kinds of presentation share many traits but also display profound differences on issues ranging from decidability to modeling convenience and physical implementability following a conjecture by toffoli and margolus it had been proved by kari that any invertible ca at least up to two dimensions can be rewritten as an isomorphic lg but until now it was not known whether this is possible in general for noninvertible ca which comprise almost all ca and represent the bulk of examples in theory and applications even circumstantial evidence whether in favor or against was lacking here for noninvertible ca a we prove that an lg presentation is out of the question for the vanishingly small class of surjective ones we then turn our attention to all the rest noninvertible and nonsurjective which comprise all the typical ones including conway s game of life for these b we prove by explicit construction that all the one dimensional ones are representable as lg and c we present and motivate the conjecture that this result extends to any number of dimensions the tradeoff between dissipation rate and structural complexity implied by the above results have compelling implications for the thermodynamics of computation at a microscopic scale

an adaptive secret image sharing with a new bitwise steganographic property recently numerous studies have been conducted in the area of secret image sharing and steganography the main objective of these schemes is to produce high quality tampering resistant stego images a number of these methods are based on least significant bit embedding and hence the presence of hidden data can be detected by well designed steganalysis algorithms this paper proposes a new sharing scheme for critical images so that stego images are obtained with better visual quality and at the same time the authentication ability can be adjusted to any desired level adaptive in order to achieve these objectives the construction of cellular automata is modified so that visual quality is improved although authentication bits are hidden in cover images furthermore authentication bits are computed in such a way that any tampering with one block affects more than one block for steganographic purposes least significant bit replacement is substituted with a new blockwise xor operation so that the scheme can withstand steganalysis attacks the other interesting property of the proposed method is that the size of blocks is determined dynamically therefore our approach can be adapted with secret cover images of different sizes

how to turn a second order cellular automaton into a lattice gas a new inversion scheme in second order cellular automata the signals made available to the local transition function include besides the current state of a site s neighbors also the previous state of the site itself some similarities had been noted especially in physics simulations between certain second order cellular automata and certain lattice gases here we show how to construct for any second order cellular automaton a lattice gas having isomorphic functional behavior paradoxically this isomorphism of function is achieved by compromising on the isomorphism of structure namely the group of translation symmetries of the resulting lattice gas is coarser than that of the original cellular automaton the advantage of our construction is that while invertibility in cellular automata is not directly deducible from the local map indeed it is in general undecidable in the second order case the invertibility of the cellular automaton goes hand in hand with that of the corresponding lattice gas and for lattices gases invertibility is trivially decidable on the basis of the local structure from a physical viewpoint our construction illustrates a trade off between two ways of achieving a force of a given range in a cellular automaton multiple copies of a signal are made and distributed in parallel to several sites explicitly providing the desired fanout width in a lattice gas with our construction one can let a single signal serially service a number of sites along an appropriate spacetime route the latter method is better suited to a nondissipative implementation of the dynamics

invertible cellular automata a review abstract in the light of recent developments in the theory of invertible cellular automata we attempt to give a unified presentation of the subject and discuss its relevance to computer science and mathematical physics

ward type identities for the two dimensional anderson model at weak disorder using the particular momentum conservation laws in dimension d 2 we rewrite the anderson model in terms of low momentum long range fields at the price of introducing electron loops the corresponding loops satisfy a ward type identity hence are much smaller than expected this fact should be useful for the study of the weak coupling model in the middle of the spectrum of the free hamiltonian

from quantum cellular automata to quantum lattice gases a natural architecture for nanoscale quantum computation is that of a quantum cellular automaton motivated by this observation we begin an investigation of exactly unitary cellular automata after proving that there can be no nontrivial homogeneous local unitary scalar cellular automaton in one dimension we weaken the homogeneity condition and show that there are nontrivial exactly unitary partitioning cellular automata we find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton this model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the dirac equation we describe two generalizations of this automaton the second of which to multiple interacting particles is the correct definition of a quantum lattice gas

a lattice boltzmann equation for diffusion the formulation of lattice gas automata lga for given partial differential equations is not straightforward and still requires some sort of magic lattice boltzmann equation lbe models are much more flexible than lga because of the freedom in choosing equilibrium distributions with free parameters which can be set after a multiscale expansion according to certain requirements here a lbe is presented for diffusion in an arbitrary number of dimensions the model is probably the simplest lbe which can be formulated it is shown that the resulting algorithm with relaxation parameter 1 is identical to an explicit finite difference efd formulation at its stability limit underrelaxation 0 1 allows stable integration beyond the stability limit of efd the time step of the explicit lbe integration is limited by accuracy and not by stability requirements

reversible boolean networks i distribution of cycle lengths abstract we consider a class of models describing the dynamics of n boolean variables where the time evolution of each depends on the values of k of the other variables previous work has considered models with dissipative dynamics here we consider time reversible models which necessarily have the property that every possible point in the state space is an element of one and only one cycle the orbits can be classified by their behavior under time reversal the orbits that transform into themselves under time reversal have properties quite different from those that do not in particular a significant fraction of latter type orbits have lengths enormously longer than orbits that are time reversal symmetric for large k and moderate n the vast majority of points in the state space are on one of the time reversal singlet orbits and a random hopping model gives an accurate description of orbit lengths however for any finite k the random hopping approximation fails qualitatively when n is large enough n 22k as in the dissipative case when k is large typical orbit lengths grow exponentially with n whereas for small enough k typical orbit lengths grow much more slowly with n the numerical data are consistent with the existence of a phase transition at which the average orbit length grows as a power of n at a value of k between 1 4 and 1 7 however in the reversible models the interplay between the discrete symmetry and quenched randomness can lead to enormous fluctuations of orbit lengths and other interesting features that are unique to the reversible case

universality of a reversible two counter machine abstract a k counter machine cm k is an automaton having k counters as an auxiliary memory it has been shown by minsky that a cm 2 can simulate any turing machine and thus it is universal in this paper we investigate the computing ability of reversible i e backward deterministic cms we first show that any irreversible cm k can be simulated by a reversible cm k 2 in this simulation however the reversible cm k 2 leaves a large number as a garbage in some counter when it halts we then show that if k more counters are added this garbage information is erased reversibly finally we prove that any reversible cm k k 1 2 3 can be simulated by a reversible cm 2 from these results computation universality of a reversible cm 2 is established

efficient exhaustive listings of reversible one dimensional cellular automata this paper looks at an algebraic formulation of one dimensional cellular automata using the formulation connections to combinatorial structures and graph theory become clear strong results about uniqueness and isomorphism allows us to outline effective algorithms for the generation of exhaustive lists of reversible one dimensional cellular automata and to count the number of distinct examples that exist these algorithms use the orderly algorithm methods to avoid the pitfalls of brute force searches

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

a new data hiding scheme for binary image authentication with small image distortion a new data hiding scheme for binary image authentication that has a small distortion of the cover image is proposed in this paper using the data embedding algorithm that is based on hamming codes the proposed scheme embeds authentication information into the cover image with flipping only a small number of pixels a special type of the pixels are selected and flipped by a new algorithm to minimize visual distortion this new algorithm is based on elssm edge line segment similarity measure randomly shuffling the bit order of the authentication information to be embedded the information can only be extracted by the designated receiver who has the symmetric key we employ two measurement metrics miss detection rates for the degree of security and psnr peak signal to noise ratio and elssm for the degree of the image distortion to demonstrate the feasibility of the proposed scheme using these metrics we analyze the proposed scheme and the previous schemes the analysis reveals that the proposed scheme requires less image distortion than the previous schemes whilst achieving the same level of the miss detection rate experimental results demonstrate that the proposed scheme is more resilient against recent steganalysis attacks than the previous schemes

graph theoretical characterization of invertible cellular automata abstract a cellular automaton is defined to be invertible if its induced global rule is surjective on the set of all finite configurations it is shown that all one dimensional invertible cellular automata are obtainable from invertible nearest pair ones i e cellular automata given by a rule matrix for these an algorithm is proposed which runs in a time which is a polynomial function of the number of letters of the alphabet the result is expressed in a graph theoretical way the cellular automaton is invertible if and only if two graphs constructed by the algorithm have no edges in common it follows that if the number of letters in the alphabet is a prime the only invertible cellular automata are those for which every row or column of the rule matrix is a permutation of the alphabet for composite alphabet cardinalities several methods analytical and computational ones to obtain more complicated invertible cellular automata are proposed and used to construct examples all four letter invertible automata are constructed

cellular automata as an alternative to rather than an approximation of differential equations in modeling physics abstract cellular automata are models of distributed dynamical systems whose structure is particularly well suited to ultrafast exact numerical simulation on the other hand they constitute a radical departure from the traditional partial differential equation approach to distributed dynamics here we discuss the problem of encoding the state variables and evolution laws of a physical system into this new setting and of giving suitable correspondence rules for interpreting the model s behavior

convergence in higher mean of a random schrodinger to a linear boltzmann evolution we study the macroscopic scaling and weak coupling limit for a random schrodinger equation on mathbb z 3 we prove that the wigner transforms of a large class of macroscopic solutions converge in rth mean to solutions of a linear boltzmann equation for any 1 r this extends previous results where convergence in expectation was established

understanding the related key security of feistel ciphers from a provable perspective we initiate the provable related key security treatment for models of practical feistel ciphers in detail we consider feistel networks with four whitening keys w i k i 0 1 2 3 and round functions of the form f gamma j k oplus x where k is the master key w i and gamma j are efficient transformations and f is a public ideal function or permutation accessible by the adversary we investigate the key schedule conditions that are sufficient for security against xor induced related key attacks up to 2 n 2 adversarial queries when the key schedules are non linear we prove security for four rounds when only affine key schedules are used we prove security for six rounds these also imply secure tweakable feistel ciphers in the random oracle model by shuffling the key schedules our model unifies both the des like structure known as feistel 2 scheme in the cryptanalytic community also known as key alternating feistel due to lampe and seurin and the lucifer like model previously analyzed by guo and lin this allows us to derive concrete implications on these two more common models and helps understanding their related key security difference

physics like models of computation abstract reversible cellular automata are computer models that embody discrete analogues of the classical physics notions of space time locality and microscopic reversibility they are offered as a step towards models of computation that are closer to fundamental physics

matrix methods and local properties of reversible one dimensional cellular automata the local behaviour of reversible one dimensional cellular automata is analysed based on perron frobenius theory it is proved by means of irreducible components that the connectivity matrices of reversible automata have a single eigenvalue equal to 1 the idempotent behaviour of such matrices is also proved by faddeev s algorithm the decomposition of these matrices into triangular factors is used to find the inverse rule for a given reversible automaton

on the size of the inverse neighborhoods for one dimensional reversible cellular automata in this paper we investigate the possible neighborhood size of the inverse automaton of some types of one dimensional reversible cellular automata considering only the case when the local function is a size two map we give a quadratic upper bound for the neighborhood size of the inverse automaton we show that this bound can be lowered in some particular cases and give an algorithm for computing these better bounds

decay of the fourier transform of surfaces with vanishing curvature we prove lp bounds on the fourier transform of measures supported on two dimensional surfaces our method allows to consider surfaces whose gauss curvature vanishes on a one dimensional submanifold under a certain non degeneracy condition we prove that hat mu in l 4 beta 0 and we give a logarithmically divergent bound on the l4 norm we use this latter bound to estimate almost singular integrals involving the dispersion relation e p sum 1 3 1 cos p j of the discrete laplace operator on the cubic lattice we briefly explain our motivation for this bound originating in the theory of random schrodinger operators

a hierarchical classification of cellular automata abstract a central issue in the theory of cellular automata is classification a classification is a structure imposed on the space of automata which groups together cellular automata with related properties in this paper a classification based on the action of cellular automata on n step markov measures is presented this classification is hierarchical and parametric an algorithm is described which efficiently constructs all rules in a given class at a given level of the hierarchy the utility of the classification is explored in the study of the invariant statistical properties of next nearest neighbor cellular automata on two states per cell

finite trotter approximation to the averaged mean square distance in the anderson model we prove that a finite trotter approximation to the averaged mean square distance traveled by a particle in a disordered system on a lattice d exhibits at most a diffusive behavior in dimensions d 3 as long as the fourier transform of the single site probability hat mu is in l2

the structure of rank 3 permutation modules for o2n 2 and um 2 acting on nonsingular points abstract we study the odd characteristic structure of permutation modules for the rank 3 natural actions of o 2 n 2 n 3 and u m 2 m 4 on nonsingular points of their standard modules

dynamical delocalization in random landau hamiltonians we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero in this article we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level more precisely we prove that for these two dimensional hamiltonians there exists at least one energy e near each landau level such that e 1 where e the local transport exponent introduced in gk5 is a measure of the rate of transport in wave packets with spectral support near e since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero random landau hamiltonians are the subject of intensive study due to their links with the integer quantum hall e ect kli for which von klitzing received the 1985 nobel prize in physics they describe an electron moving in a very thin flat conductor with impurities under the influence of a constant magnetic field perpendicular to the plane of the conductor and play an important role in the understanding of the quantum hall e ect l aoa t h nt ku be avss bees laughlin s argument l as pointed out by halperin h uses the assumption that under weak disorder and strong magnetic field the energy spectrum consists of bands of extended states separated

classifying circular cellular automata we introduce a hierarchy of linear cellular automata based on their limiting behavior on spatially periodic configurations we show that it is undecidable to which class in the hierarchy a cellular automaton belongs in particular it is undecidable whether all spatially periodic configurations evolve to a fixed point furthermore there is no computable bound on the period lengths of these configurations our arguments are based on a non standard simulation of turing machines on circular cellular automata

continuity of information transport in surjective cellular automata we introduce a local version of the shannon entropy in order to describe information transport in spatially extended dynamical systems and to explore to what extent information can be viewed as a local quantity using an appropriately defined information current this quantity is shown to obey a local conservation law in the case of one dimensional reversible cellular automata with arbitrary initial measures the result is also shown to apply to one dimensional surjective cellular automata in the case of shift invariant measures bounds on the information flow are also shown

relaxation properties of elementary reversible cellular automata abstract the relaxation properties of a number of elementary reversible cellular automata are studied different rules have different types of relaxation behavior possible relationships between the thermodynamic behavior and phase space structure are discussed

inverse rules of eca with rule number 150 the explicit expressions of local transition functions of the inverse cellular automata of elementary cellular automata with rule number 150 and periodic boundary conditions are given the procedure to obtain such local transition functions is based on the algebraic properties of elementary cellular automata

a randomness test for block ciphers this paper describes a randomness test which can be used to measure the cryptographic strength of a block cipher or its underlying cryptographic primitive s cryptographic strength in the context of this paper is related to the ability of the round function to produce a random output which in turn is defined as the distance between a theoretical calculation and an experimental measure the measurements are based on the diffusion characteristic of the cipher potentially the test for randomness proposed in this paper could be used as a distinguisher based on diffusion

a random np complete problem for inversion of 2d cellular automata in this paper we prove the co rnp completeness rnp random np of the following decision problem given a 2 dimensional cellular automaton a is a reversible when restricted to finite configurations extending a given row in order to prove this result we introduce a polynomial reduction from problems concerning finite tilings into problems concerning cellular automata then we add to tile sets and cellular automata two probability functions and we prove that these problems are not only co np complete but co rnp complete too

on the semantic security of cellular automata based pseudo random permutation using results from the luby rackoff construction this paper proposes a semantically secure construction of pseudo random permutations using second order reversible cellular automata we show that the proposed construction is equivalent to the luby rackoff model if it is built using non uniform transition rules and we prove that the construction is strongly secure if an adequate number of iterations is performed moreover a corresponding symmetric block cipher is constructed and analysed experimentally in comparison with popular ciphers obtained results approve robustness and efficacy of the construction while achieved performances overcome those of some existing block ciphers

a multisecret sharing scheme for color images based on cellular automata in this work a new multisecret sharing scheme for secret color images among a set of users is proposed the protocol allows that each participant in the scheme to share a secret color image with the rest of participants in such a way that all of them can recover all the secret color images only if the whole set of participants pools their shadows the proposed scheme is based on the use of bidimensional reversible cellular automata with memory the security of the scheme is studied and it is proved that the protocol is ideal and perfect and that it resists the most important statistical attacks

predecessors of cellular automata states ii pre images of finite sequences abstract graph theoretic techniques are used to derive algorithms which determine all pre images of any given finite sequence s under one dimensional cellular automata rules defined over z 2 it is also shown that for a given rule q the number of pre images of any finite sequence is determined by an algebraic form in the components of q this extends results of jen on the enumeration of pre images we also show that the set of k site rules for which a given sequence s has no pre images is determined by the solution set of a non linear diophantine equation

distinguishing properties and applications of higher order derivatives of boolean functions abstract higher order differential cryptanalysis is based on a property of higher order derivatives of boolean functions such that derivative of a boolean function reduces its degree at least 1 and continuously taking derivatives eventually yields a zero function a quicker degree reduction means a lower data complexity in cryptanalysis which can be determined by fast point at which the derivative reduces the degree at least 2 in this paper we show that the set of the fast points of a boolean function constitutes a linear subspace and its dimension plus the degree of the function is at most the size of the function we also show that non zero fast point exists in every n variable boolean function of degree n 1 every symmetric boolean function of degree d where n d mod 2 or every quadratic boolean function of odd number variables which help us distinguish a few block ciphers and propose a new design principle about degree for block cipher

quantum diffusion of the random schrodinger evolution in the scaling limit ii the recollision diagrams we consider random schrodinger equations on mathbb r d for d 3 with a homogeneous anderson poisson type random potential denote by the coupling constant and t the solution with initial data 0 the space and time variables scale as x sim lambda 2 kappa 2 t sim lambda 2 kappa with 0 0 d we prove that in the limit 0 the expectation of the wigner distribution of t converges weakly to the solution of a heat equation in the space variable x for arbitrary l2 initial data the proof is based on a rigorous analysis of feynman diagrams in the companion paper 10 the analysis of the non repetition diagrams was presented in this paper we complete the proof by estimating the recollision diagrams and showing that the main terms i e the ladder diagrams with renormalized propagator converge to the heat equation

number theoretic constructions of efficient pseudo random functions we describe efficient constructions for various cryptographic primitives in private key as well as public key cryptography our main results are two new constructions of pseudo random functions we prove the pseudo randomness of one construction under the assumption that factoring blum integers is hard while the other construction is pseudo random if the decisional version of the diffie hellman assumption holds computing the value of our functions at any given point involves two subset products this is much more efficient than previous proposals furthermore these functions have the advantage of being in tc0 the class of functions computable by constant depth circuits consisting of a polynomial number of threshold gates this fact has several interesting applications the simple algebraic structure of the functions implies additional features such as a zero knowledge proof for statements of the form y e fs x and y n fs x given a commitment to a key s of a pseudo random function fs

two dimensional cellular automata a largely phenomenological study of two dimensional cellular automata is reported qualitative classes of behavior similar to those in one dimensional cellular automata are found growth from simple seeds in two dimensiona cellular automata can produce patterns with complicated boundaries characterized by a variety of growth dimensions evolution from disordered states can give domains with boundaries that execute effectively continuous motions some global properties of cellular automata can be described by entropies and lyapunov exponents others are undecidable

a construction of the simplest super pseudorandom permutation generator in this paper we consider one of the two open problems proposed by pieprzyk 1 i e whether a permutation generator which is designed using five rounds of the data encryption standard and a single pseudorandom function is a super pseudorandom permutation generator first we show that f f f2 f is pseudorandom then using the pseudorandomness of f2 f f f and f f f2 f we show f f2 f f f is super pseudorandom

substitution permutation networks pseudorandom functions and natural proofs this article takes a new step towards closing the gap between pseudorandom functions prf and their popular bounded input length counterparts this gap is both quantitative because these counterparts are more efficient than prf in various ways and methodological because these counterparts usually fit in the substitution permutation network paradigm spn which has not been used to construct prf we give several candidate prf fi that are inspired by the spn paradigm most of our candidates are more efficient than previous ones our main candidates are as follows f1 0 1 n 0 1 n is an spn whose s box is a random function on b bits given as part of the seed we prove that f1 resists attacks that run in time 2eb f2 0 1 n 0 1 n is an spn where the s box is patched field inversion a common choice in practical constructions we show that f2 is computable with boolean circuits of size n logo 1 n and that it has exponential security 2 n against linear and differential cryptanalysis f3 0 1 n 0 1 is a nonstandard variant on the spn paradigm where states grow in length we show that f3 is computable with tc0 circuits of size n1 p e for any e 0 and that it is almost 3 wise independent f4 0 1 n 0 1 uses an extreme setting of the spn parameters one round one s box no diffusion matrix the s box is again patched field inversion we show that f4 is computable by circuits of size n logo 1 n and that it fools all parity tests on 20 9n outputs assuming the security of our candidates our work narrows the gap between the natural proofs barrier and existing lower bounds in three models circuits tc0 circuits and turing machines

reversible cellular automata with memory two dimensional patterns from a single site seed standard cellular automata ca are ahistoric memoryless i e the new state of a cell depends on its neighborhood configuration only at the preceding time step historic memory of all past iterations can be incorporated into ca by featuring each cell by a weighted mean of all its past states in this paper a new kind of reversible ca which incorporates memory is introduced in a two dimensional scenario

kinetic limit for wave propagation in a random medium we study crystal dynamics in the harmonic approximation the atomic masses are weakly disordered in the sense that their deviation from uniformity is of the order e the dispersion relation is assumed to be a morse function and to suppress crossed recollisions we then prove that in the limit e 0 the disorder averaged wigner function on the kinetic scale time and space of order e 1 is governed by a linear boltzmann equation

representations of geometrical and topological quantities in cellular automata abstract this paper addresses the problem of finding novel representations of analogs of physical field variables in cellular automata several examples illustrate how this might be done 1 a manifestly covariant vector law for diffusion 2 a model for potential energy and exact one forms 3 a counting algorithm using a winding number density and 4 fields which support topological charges in order to combine the advantages of cellular automata and continuum analysis it is necessary to have representations which involve more than one cell

theory of cellular automata a survey this article surveys some theoretical aspects of cellular automata ca research in particular we discuss classical and new results on reversibility conservation laws limit sets decidability questions universality and topological dynamics of ca the selection of topics is by no means comprehensive and reflects the research interests of the author the main goal is to provide a tutorial of ca theory to researchers in other branches of natural computing to give a compact collection of known results with references to their proofs and to suggest some open problems

how to construct pseudorandom permutations from pseudorandom functions we show how to efficiently construct a pseudorandom invertible permutation generator from a pseudorandom function generator goldreich goldwasser and micali how to construct random functions p

on the implementation of huge random objects we initiate a general study of the feasibility of implementing huge random objects and demonstrate its applicability to a number of areas in which random objects occur naturally we highlight two types of measures of the quality of the implementation with respect to the desired specification the first type corresponds to various standard notions of indistinguishability applied to function ensembles whereas the second type is a novel notion that we call truthfulness intuitively a truthful implementation of a random object of type t must always be an object of type t and not merely be indistinguishable from a random object of type t our formalism allows for the consideration of random objects that satisfy some fixed property or have some fixed structure as well as the consideration of objects supporting complex queries for example we consider the truthful implementation of random hamiltonian graphs as well as supporting complex queries regarding such graphs e g providing the next vertex along a fixed hamiltonian path in such a graph

summability of connected correlation functions of coupled lattice fields we consider two nonindependent random fields psi and phi defined on a countable set z for instance z mathbb z d or z mathbb z d times i where i denotes a finite set of possible internal degrees of freedom such as spin we prove that if the cumulants of psi and phi enjoy a certain decay property then all joint cumulants between psi and phi are ell 2 summable in the precise sense described in the text the decay assumption for the cumulants of psi and phi is a restricted ell 1 summability condition called ell 1 clustering property one immediate application of the results is given by a stochastic process psi t x whose state is ell 1 clustering at any time t then the above estimates can be applied with psi psi t and phi psi 0 and we obtain uniform in t estimates for the summability of time correlations of the field the above clustering assumption is obviously satisfied by any ell 1 clustering stationary state of the process and our original motivation for the control of the summability of time correlations comes from a quest for a rigorous control of the green kubo correlation function in such a system a key role in the proof is played by the properties of non gaussian wick polynomials and their connection to cumulants

local structure theory for cellular automata abstract the mean field theory for cellular automata wolfram and schulman and seiden is generalized to the local structure theory the local structure theory is a sequence of finitely parameterized models of the statistical features of a cellular automaton s evolution the n th model in the sequence takes into account correlations in terms of the probability of blocks of n states a class of measures the n block measures is introduced the local structure operator of order n maps n block measures to n block measures in a manner which reflects the cellular automaton map on blocks of states the fixed points of the map on measures approximate the invariant measures of the cellular automaton the ability of the local structure theory to model evolution from uncorrelated initial distributions is studied the theory gives exact results in simple cases in more complex cases monte carlo numerical experiments suggest that an accurate statistical portrait of cellular automaton evolution is obtained the invariant measures of a cellular automaton and the stability of these measures may be obtained from the local structure theory the local structure theory appears to be a powerful method for characterization and classification of cellular automata nearest neighbor cellular automata with two states per cell are studied using this method

rank 3 permutation modules of the finite classical groups abstract the cross characteristic permutation modules for the actions of the finite classical groups on singular 1 spaces of their natural modules are studied the composition factors and submodule lattices are determined

statistical mechanics of surjective cellular automata reversible cellular automata are seen as microscopic physical models and their states of macroscopic equilibrium are described using invariant probability measures we establish a connection between the invariance of gibbs measures and the conservation of additive quantities in surjective cellular automata namely we show that the simplex of shift invariant gibbs measures associated to a hamiltonian is invariant under a surjective cellular automaton if and only if the cellular automaton conserves the hamiltonian a special case is the well known invariance of the uniform bernoulli measure under surjective cellular automata which corresponds to the conservation of the trivial hamiltonian as an application we obtain results indicating the lack of non trivial gibbs or markov invariant measures for sufficiently chaotic cellular automata we discuss the relevance of the randomization property of algebraic cellular automata to the problem of approach to macroscopic equilibrium and pose several open questions as an aside a shift invariant pre image of a gibbs measure under a pre injective factor map between shifts of finite type turns out to be always a gibbs measure we provide a sufficient condition under which the image of a gibbs measure under a pre injective factor map is not a gibbs measure we point out a potential application of pre injective factor maps as a tool in the study of phase transitions in statistical mechanical models

theory and applications of cellular automata in cryptography this paper deals with the theory and application of cellular automata ca for a class of block ciphers and stream ciphers based on ca state transitions certain fundamental transformations are defined which are block ciphering functions of the proposed enciphering scheme these fundamental transformations are found to generate the simple alternating group of even permutations which in turn is a subgroup of the permutation group these functions are implemented with a class of programmable cellular automata pca built around rules 51 153 and 195 further high quality pseudorandom pattern generators built around rule 90 and 150 programmable cellular automata with a rule selector i e combining function has been proposed as running key generators in stream ciphers both the schemes provide better security against different types of attacks with a simple regular modular and cascadable structure of ca hardware implementation of such schemes idealy suit vlsi implementation

algebraic properties of cellular automata cellular automata are discrete dynamical systems of simple construction but complex and varied behaviour algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata the complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities the systems are usually irreversible and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations

the equations for the moduli space of n points on the line a central question in invariant theory is that of determining the relations among invariants geometric invariant theory git quotients come with a natural ample line bundle and hence often a natural projective embedding this question translates to determining the equations of the moduli space under this embedding this article deals with one of the most classical quotients the space of ordered points on the projective line we show that under any weighting of the points this quotient is cut out scheme theoretically by a particularly simple set of quadric relations with the single exception of the segre cubic threefold the space of six points with equal weight we also show that the ideal of relations is generated in degree at most four and we give an explicit description of the generators if all the weights are even e g in the case of equal weight for odd n then we show that the ideal of relations is generated by quadrics

secret image sharing based on cellular automata and steganography recently lin and tsai secret image sharing with steganography and authentication the journal of systems and software 73 2004 405 414 and yang et al improvements of image sharing with steganography and authentication the journal of systems and software 80 2007 1070 1076 proposed secret image sharing schemes combining steganography and authentication based on shamir s polynomials the schemes divide a secret image into some shadows which are then embedded in cover images in order to produce stego images for distributing among participants to achieve better authentication ability chang et al sharing secrets in stego images with authentication pattern recognition 41 2008 3130 3137 proposed in 2008 an improved scheme which enhances the visual quality of the stego images as well and the probability of successful verification for a fake stego block is 1 16 in this paper we employ linear cellular automata digital signatures and hash functions to propose a novel t n threshold image sharing scheme with steganographic properties in which a double authentication mechanism is introduced which can detect tampering with probability 255 256 employing cellular automata instead of shamir s polynomials not only improves computational complexity from o nlog 2n to o n but obviates the need to modify pixels of cover images unnecessarily compared to previous methods c lin w tsai secret image sharing with steganography and authentication the journal of systems and software 73 2004 405 414 c yang t chen k yu c wang improvements of image sharing with steganography and authentication the journal of systems and software 80 2007 1070 1076 c chang y hsieh c lin sharing secrets in stego images with authentication pattern recognition 41 2008 3130 3137 we use fewer number of bits in each pixel of cover images for embedding data so that a better visual quality is guaranteed we further present some experimental results

invariant in cellular automata for a subclass of the set of reversible cellular automata the author gives the form of an exact time invariant quantity which can be seen as a kind of energy with the ergodic assumption this is a model of two interacting ising spin systems

guaranteeing the diversity of number generators a major problem in using iterative number generators of the form xi f xi 1 is that they can enter unexpectedly short cycles this is hard to analyze when the generator is designed hard to detect in real time when the generator is used and can have devastating cryptanalytic implications in this paper we define a measure of security called sequence diversity which generalizes the notion of cycle length for noniterative generators we then introduce the class of counter assisted generators and show how to turn any iterative generator even a bad one designed or seeded by an adversary into a counter assisted generator with a provably high diversity without reducing the quality of generators which are already cryptographically strong 2001 elsevier science

on reversibility of cellular automata with periodic boundary conditions reversibility of one dimensional cellular automata with periodic boundary conditions is discussed it is shown that there exist exactly 16 reversible elementary cellular automaton rules for infinitely many cell sizes by means of a correspondence between elementary cellular automaton and the de bruijn graph in addition a sufficient condition for reversibility of three valued and two neighbour cellular automaton is given

how to leak a secret theory and applications of ring signatures in this work we formalize the notion of a ring signature which makes it possible to specify a set of possible signers without revealing which member actually produced the signature unlike group signatures ring signatures have no group managers no setup procedures no revocation procedures and no coordination any user can choose any set of possible signers that includes himself and sign any message by using his secret key and the others public keys without getting their approval or assistance ring signatures provide an elegant way to leak authoritative secrets in an anonymous way to sign casual email in a way that can only be verified by its intended recipient and to solve other problems in multiparty computations our main contribution lies in the presentation of efficient constructions of ring signatures the general concept itself under different terminology was first introduced by cramer et al cds94 our constructions of such signatures are unconditionally signer ambiguous secure in the random oracle model and exceptionally efficient adding each ring member increases the cost of signing or verifying by a single modular multiplication and a single symmetric encryption we also describe a large number of extensions modifications and applications of ring signatures which were published after the original version of this work in asiacrypt 2001

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

reversible boolean networks ii phase transitions oscillations and local structures we continue our consideration of a class of models describing the reversible dynamics of n boolean variables each with k inputs we investigate in detail the behavior of the hamming distance as well as of the distribution of orbit lengths as n and k are varied we present numerical evidence for a phase transition in the behavior of the hamming distance at a critical value kc 1 62 and also an analytic theory that yields the exact bounds 1 5 k c 2 we also discuss the large oscillations that we observe in the hamming distance for k k c as a function of time as well as in the distribution of cycle lengths as a function of cycle length for moderate k both greater than and less than kc we propose that local structures or subsets of spins whose dynamics are not fully coupled to the other spins in the system play a crucial role in generating these oscillations the simplest of these structures are linear chains called linkages and rings called circuits we discuss the properties of the linkages in some detail and sketch the properties of circuits we argue that the observed oscillation phenomena can be largely understood in terms of these local structures 2001 published by elsevier science b v

resolution scalable image coding with reversible cellular automata in a resolution scalable image coding algorithm a multiresolution representation of the data is often obtained using a linear filter bank reversible cellular automata have been recently proposed as simpler nonlinear filter banks that produce a similar representation the original image is decomposed into four subbands such that one of them retains most of the features of the original image at a reduced scale in this paper we discuss the utilization of reversible cellular automata and arithmetic coding for scalable compression of binary and grayscale images in the binary case the proposed algorithm that uses simple local rules compares well with the jbig compression standard in particular for images where the foreground is made of a simple connected region for complex images more efficient local rules based upon the lifting principle have been designed they provide compression performances very close to or even better than jbig depending upon the image characteristics in the grayscale case and in particular for smooth images such as depth maps the proposed algorithm outperforms both the jbig and the jpeg2000 standards under most coding conditions

the phonon boltzmann equation properties and link to weakly anharmonic lattice dynamics for low density gases the validity of the boltzmann transport equation is well established the central object is the one particle distribution function f which in the boltzmann grad limit satisfies the boltzmann equation grad and much refined cercignani argue for the existence of this limit on the basis of the bbgky hierarchy for hard spheres at least for a short kinetic time span the argument can be made mathematically precise following the seminal work of lanford in this article a corresponding program is undertaken for weakly nonlinear both discrete and continuum wave equations our working example is the harmonic lattice with a weakly nonquadratic on site potential we argue that the role of the boltzmann f function is taken over by the wigner function which is a very convenient device to filter the slow degrees of freedom the wigner function so to speak labels locally the covariances of dynamically almost stationary measures one route to the phonon boltzmann equation is a gaussian decoupling which is based on the fact that the purely harmonic dynamics has very good mixing properties as a further approach the expansion in terms of feynman diagrams is outlined both methods are extended to the quantized version of the weakly nonlinear wave equation the resulting phonon boltzmann equation has been hardly studied on a rigorous level as one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal wigner functions for three phonon processes such a result requires extra conditions on the dispersion law we also outline the reasoning leading to fourier s law for heat conduction

additive cellular automata and global injectivity abstract we employ algebraic methods to investigate the global injectivity of additive cellular automata when the state alphabet is a finite commutative ring the set of local rules of such additive cellular automata in d 1 dimensions itself forms a ring the invertible elements in this ring correspond to the globally injective additive cellular automata we indicate how one determines the invertible elements and thus the globally injectibe cellular automata in particular we show that in contrast to the general situation the global injectivity of additive cellular automata in d 1 dimensions is decidable construction of local rules for the inverse cellular automata is illustrated through examples a strict upper bound on the radius of the inverse cellular automata is obtained in the special case of linear cellular automata with state alphabet the ring of integers modulo m we count the globally injective cellular automata of up to a given radius we show that in one dimension a non globally injective additive cellular automaton is either not injective on spatially periodic configurations of any period n n or the set of n 0 such that the cellular automation is injective on spatially periodic configurations of period n has positive density in n

localization lengths and boltzmann limit for the anderson model at small disorders in dimension 3 we prove lower bounds on the localization length of eigenfunctions in the three dimensional anderson model at weak disorders our results are similar to those obtained by schlag shubin and wolff j anal math 88 2002 for dimensions one and two we prove that with probability one most eigenfunctions have localization lengths bounded from below by o left lambda 2 over log 1 over lambda right where is the disorder strength this is achieved by time dependent methods which generalize those developed by erdos and yau commun pure appl math liii 667 753 2003 to the lattice and non gaussian case in addition we show that the macroscopic limit of the corresponding lattice random schrodinger dynamics is governed by a linear boltzmann equation

transformations of one dimensional cellular automation rules by translation invariant local surjective mappings abstract if m is a noninvertible translation invariant local surjective mapping it is shown that some local one dimensional deterministic cellular automaton rules f have a transform by m defined by m m f when it exists is local and its wolfram s class is the same as f the evolution of a cellular automaton according to rule is simply related to the evolution according to rule f in the case of class 3 rules the evolution to the attractor may often be viewed as particle like structures evolving in a regular background if the structure of these particles and their interactions for a rule f are known then the structure and interactions of the transformed particles for rule are also known if m is a nontrivial invertible translation invariant local surjective mapping always exists but it is in general site and time dependent

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

the complexity of reversible cellular automata we study the orbits of reversible one dimensional cellular automata it is shown that the turing degree structure of the orbits of these automata is the same as for general cellular automata in particular there are reversible cellular automata whose orbits have arbitrary recursively enumerable degree

efficient tweakable enciphering schemes from block wise universal hash functions this paper describes several constructions of tweakable strong pseudorandom permutations sprps built from different modes of operations of a block cipher and suitable universal hash functions for the electronic codebook ecb mode based construction an invertible blockwise universal hash function is required we simplify an earlier construction of such a function described by naor and reingold the other modes of operations considered are the output feedback ofb mode and a counter like mode all the constructions make the same number of block cipher calls and the same number of multiplications combined with a class of polynomials defined by bernstein the new constructions provide the currently best known algorithms for the important practical problem of disk encryption

nonconstructible blocks in 1d cellular automata minimal generators and natural systems the paper concerns minimal nonconstructible blocks of cellular automata i e minimal combinations of cell states that never appear in the evolution of an automaton using our previous work on the construction of predecessors of given configurations we analyse with numerical experiments the sets of nonconstructible blocks for one dimensional 1d cellular automata with binary and ternary neighbourhoods and binary cell states and compare our results with the classification made by voorhees we also analyse the relationship of the voorhees classes to the static parameter gl on applying our algorithm to cellular automata models of natural systems we provide common sense interpretations of unreachable states of some biological distributed processes

calculation of critical properties for the anisotropic two layer ising model on the kagome lattice cellular automata approach abstract the critical point k c of the two layer ising model on the kagome lattice has been calculated with a high precision using the probabilistic cellular automata with the glauber algorithm the critical point is calculated for different values of the inter and intra layer couplings k 1 k 2 k 3 k z where k 1 k 2 and k 3 are the nearest neighbor interactions within each layer in the 1 2 and 3 directions respectively and k z is the intralayer coupling a general ansatz equation for the critical point is given as a function of the inter and intra layer interactions k 3 k 1 k 2 k 1 and k z k 1 for the one and two layer ising models on the kagome lattice

cellular automaton fluids 1 basic theory continuum equations are derived for the large scale behavior of a class of cellular automaton models for fluids the cellular automata are discrete analogues of molecular dynamics in which particles with discrete velocities populate the links of a fixed array of sites kinetic equations for microscopic particle distributions are constructed hydrodynamic equations are then derived using the chapman enskog expansion slightly modified navier stokes equations are obtained in two and three dimensions with certain lattices viscosities and other transport coefficients are calculated using the boltzmann transport equation approximation some corrections to the equations of motion for cellular automaton fluids beyond the navier stokes order are given

a verifiable multi secret sharing scheme based on cellular automata in this paper we propose a verifiable t n threshold multi secret sharing scheme based on one dimensional cellular automata where the number of secrets is not restricted by n or t we show that our scheme can be used to solve an open problem proposed recently in alvarez et al 2008 g alvarez l hernandez encinas a martin del rey a multisecret sharing scheme for color images based on cellular automata information sciences 178 2008 4382 4395

quantum diffusion of the random schrodinger evolution in the scaling limit we consider random schrodinger equations on r d for d 3 with a homogeneous anderson poisson type random potential denote by the coupling constant and psi t the solution with initial data psi 0 the space and time variables scale as x sim lambda 2 varkappa 2 text and t sim lambda 2 varkappa text with 0 varkappa varkappa 0 left d right we prove that in the limit 0 the expectation of the wigner distribution of psi t converges weakly to the solution of a heat equation in the space variable x for arbitrary l 2 initial data