extension of the berlekamp massey algorithm to n dimensions abstract we present an algorithm for finding a minimal set of linear recurring relations which are valid for a given n dimensional array over any field where the minimality is defined with respect to the partial order over the n dimensional lattice the algorithm is an extension of our two dimensional version of the berlekamp massey algorithm to more than two dimensions the n dimensional theory is based on more general concepts which can be reduced into those of the two dimensional theory in the previous paper in a typical case the resulting set of polynomials characterizing the minimal linear recurring relations proves to be a groebner basis of the ideal defined by the array and consequently the structure of an n dimensional linear feedback shift register with the minimum number of storage devices which can generate the array is determined by it

decoding binary 2 d cyclic codes by the 2 d berlekamp massey algorithm a method of decoding two dimensional 2 d cyclic codes by applying the 2 d berlekamp massey algorithm is proposed to explain this decoding method the author introduces a subclass of 2 d cyclic codes which are called 2 d bch codes due to their similarity with bch codes it is shown that there are some short 2 d cyclic codes with a better cost parameter value the merit of the approach is verified by showing several simple examples of 2 d cyclic codes

finding recursions for multidimensional arrays we produce a concise description of sakata s algorithm for finding recursion relations that are valid for an n dimensional array the new analysis extends to limit points which are excluded by previous authors we show that under a natural hypothesis the algorithm becomes stationary below each limit point and in particular constructs a full set of recursion relations for a recursive array at some finite point the method used to justify the algorithm stresses the duality between the fundamental extension operation of sakata and the s polynomial operation of buchberger

a chaos based symmetric image encryption scheme using a bit level permutation in recent years a variety of chaos based digital image encryption algorithms have been suggested most of these algorithms implement permutations and diffusions at the pixel level by considering the pixel as the smallest atomic element of an image in fact a permutation at the bit level not only changes the position of the pixel but also alters its value here we propose an image cryptosystem employing the arnold cat map for bit level permutation and the logistic map for diffusion simulations have been carried out and analyzed in detail demonstrating the superior security and high efficiency of our cryptosystem

a learning result for continuous time recurrent neural networks abstract the following learning problem is considered for continuous time recurrent neural networks having sigmoidal activation functions given a black box representing an unknown system measurements of output derivatives are collected for a set of randomly generated inputs and a network is used to approximate the observed behavior it is shown that the number of inputs needed for reliable generalization the sample complexity of the learning problem is upper bounded by an expression that grows polynomially with the dimension of the network and logarithmically with the number of output derivatives being matched

2d sine logistic modulation map for image encryption because of the excellent properties of unpredictability ergodicity and sensitivity to their parameters and initial values chaotic maps are widely used in security applications in this paper we introduce a new two dimensional sine logistic modulation map 2d slmm which is derived from the logistic and sine maps compared with existing chaotic maps it has the wider chaotic range better ergodicity hyperchaotic property and relatively low implementation cost to investigate its applications we propose a chaotic magic transform cmt to efficiently change the image pixel positions combining 2d slmm with cmt we further introduce a new image encryption algorithm simulation results and security analysis demonstrate that the proposed algorithm is able to protect images with low time complexity and a high security level as well as to resist various attacks

2d sudoku associated bijections for image scrambling sudoku refers to a two dimensional 2d number placement puzzle with simple constraints that there are no repeated digits in each row each column or each block motivated by this sudoku configuration we introduce a number of sudoku associated matrix element representations besides the conventional representation using matrix row column pair specifically they are representations via sudoku matrix row digit pair digit row pair column digit pair digit column pair block digit pair and digit block pair this means we can secretly represent matrix elements via a sudoku matrix and furthermore develop new sudoku associated 2d parametric bijections to demonstrate the effectiveness and randomness of bijections we introduce a simple but effective sudoku associated image scrambler only using 2d sudoku associated bijections for image scrambling without bandwidth expansion simulations and comparisons demonstrate that the proposed method outperforms several state of the art methods

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

lur e systems with multilayer perceptron and recurrent neural networks absolute stability and dissipativity sufficient conditions for absolute stability and dissipativity of continuous time recurrent neural networks with two hidden layers are presented in the autonomous case this is related to a lur e system with multilayer perceptron nonlinearity such models are obtained after parametrizing general nonlinear models and controllers by a multilayer perceptron with one hidden layer and representing the control scheme in standard plant form the conditions are expressed as matrix inequalities and can be employed for nonlinear h sub spl infin control and imposing closed loop stability in dynamic backpropagation

algebraic geometric codes and multidimensional cyclic codes a unified theory and algorithms for decoding using grobner bases it is proved that any algebraic geometric ag code can be expressed as a cross section of an extended multidimensional cyclic code both ag codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets ag codes are defined over the points of an algebraic curve and an m dimensional cyclic code is defined over the points in m dimensional space the power of the unified theory is in its description of decoding techniques using grobner bases in order to fit an ag code into this theory a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position for curves in special position all computations can be performed with polynomials and this also makes it possible to use the theory of grobner bases next a transform is defined for ag codes which generalizes the discrete fourier transform the transform is also related to a grobner basis and is useful in setting up the decoding problem in the decoding problem a key step is finding a grobner basis for an error locator ideal for ag codes multidimensional cyclic codes and indeed any cross section of an extended multidimensional cyclic code sakata s algorithm can be used to find linear recursion relations which hold on the syndrome array in this general context the authors give a self contained and simplified presentation of sakata s algorithm and present a general framework for decoding algorithms for this family of codes in which the use of sakata s algorithm is supplemented by a procedure for extending the syndrome array

complete controllability of continuous time recurrent neural networks this paper studies controllability for the class of control systems commonly called continuous time recurrent neural networks it is shown that under a generic condition on the input matrix the system is controllable for every possible state matrix the result holds when the activation function is the hyperbolic tangent

gray code permutation algorithm for high dimensional data encryption abstract we present a novel permutation algorithm for fast encryption of a large amount of data such as 3d images and real time videos the proposed p box algorithm takes advantage of gray code properties and allows fast encryption with high information diffusion the algorithm is optimized for integer q bit operations q 8 16 32 allowing a direct implementation in almost any hardware platform while avoiding rounding errors of floating point operations by combining the p box with chaotic s box based on the logistic map we design a complete highly secure and fast cryptosystem

a novel lossless color image encryption scheme using 2d dwt and 6d hyperchaotic system this paper proposes a new lossless encryption algorithm for color images based on a six dimensional 6d hyperchaotic system and the two dimensional 2d discrete wavelet transform dwt different from the current image encryption methods our image encryption scheme is constructed using the 2d dwt and 6d hyperchaotic system in both the frequency domain and the spatial domain where the key streams depend on not only the hyperchaotic system but the plain image in the presented algorithm the plain image is firstly divided into four image sub bands by means of the 2d dwt secondly the sub bands are permutated by a key stream and then the size of them is decreased by a constant factor thirdly the 2d inverse dwt is employed to reconstruct an intermediate image by the four encrypted image sub bands finally to further enhance the security the pixel values of the intermediate image are modified by using another key stream experimental results and security analysis demonstrate that the proposed algorithm has a high security fast speed and can resist various attacks

syndrome generation and error location search for the decoding of algebraic geometry codes on plane garcia stichtenoth curves in this paper the basic function field of an asymptotically optimal tower of function fields by garcia and stichtenoth is studied where bases of one point linear systems can be explicitly constructed a representation of finite gf q 2 rational points on the associated plane curve is derived in this paper this representation is exploited to extend the use of horner s loops and the mechanism of chien search in the decoding of reed solomon codes for syndrome generation and error location search respectively to algebraic geometry codes built on the plane garcia stichtenoth curve

architectures for syndrome generation and error location search in the decoding of hermitian codes when a hermitian curve is put in a special position with respect to its infinite rational point the x y coordinates of all its finite rational points have regular algebraic properties based on these properties the use of horner s rule and the mechanism of chien search in the decoding of reed solomon codes can be extended to render architectures for syndrome generation and error location search in the decoding of codes constructed from the hermitian curve

decoding algebraic geometric codes up to the designed minimum distance a simple decoding procedure for algebraic geometric codes c sub omega d g is presented this decoding procedure is a generalization of peterson s decoding procedure for the bch codes it can be used to correct any d 1 2 or fewer errors with complexity o n sup 3 where d is the designed minimum distance of the algebraic geometric code and n is the codelength

sample complexity for learning recurrent perceptron mappings recurrent perceptron classifiers generalize the usual perceptron model they correspond to linear transformations of input vectors obtained by means of autoregressive moving average schemes or infinite impulse response filters and take into account those correlations and dependences among input coordinates which arise from linear digital filtering this paper provides tight bounds on the sample complexity associated to the fitting of such models to experimental data the results are expressed in the context of the theory of probably approximately correct pac learning

a symmetric image encryption algorithm based on mixed linear nonlinear coupled map lattice abstract we propose a new image encryption algorithm based on the spatiotemporal chaos of the mixed linear nonlinear coupled map lattices this spatiotemporal chaotic system has more outstanding cryptography features in dynamics than the logistic map or the system of coupled map lattices does in the proposed image encryption we employ the strategy of bit level pixel permutation which enables the lower bit planes and higher bit planes of pixels permute mutually without any extra storage space simulations have been carried out and the results demonstrate the superior security and high efficiency of the proposed algorithm

further results on controllability of recurrent neural networks this paper studies controllability properties of recurrent neural networks the new contributions are 1 an extension of a previous result to a slightly different model 2 a formulation and proof of a necessary and sufficient condition and 3 an analysis of a low dimensional case for which the hypotheses made in previous work do not apply

on the decoding of algebraic geometric codes this paper provides a survey of the existing literature on the decoding of algebraic geometric codes definitions theorems and cross references will be given we show what has been done discuss what still has to be done and pose some open problems

decoding geometric goppa codes using an extra place decoding geometric goppa codes can be reduced to solving the key congruence of a received word in an affine ring if the codelength is smaller than the number of rational points on the curve then this method can correct up to 1 2 d l 2 s errors where d is the designed minimum distance of the code and s is the clifford defect the affine ring with respect to a place p is the set of all rational functions which have no poles except at p and it is somehow similar to a polynomial ring for a special kind of geometric goppa code namely c sub omega d mp the decoding algorithm is reduced to solving the key equation in the affine ring which can be carried out by the subresultant sequence in the affine ring with complexity o n sup 3 where n is the length of codewords