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Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ¹ minimization
 PROC. NATL ACAD. SCI. USA 100 2197–202
, 2002
"... Given a ‘dictionary’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered ..."
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Cited by 626 (37 self)
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Given a ‘dictionary’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered the special case where D is an overcomplete system consisting of exactly two orthobases, and has shown that, under a condition of mutual incoherence of the two bases, and assuming that S has a sufficiently sparse representation, this representation is unique and can be found by solving a convex optimization problem: specifically, minimizing the ℓ¹ norm of the coefficients γ. In this paper, we obtain parallel results in a more general setting, where the dictionary D can arise from two or several bases, frames, or even less structured systems. We introduce the Spark, ameasure of linear dependence in such a system; it is the size of the smallest linearly dependent subset (dk). We show that, when the signal S has a representation using less than Spark(D)/2 nonzeros, this representation is necessarily unique. We
Uncertainty principles and ideal atomic decomposition
 IEEE Transactions on Information Theory
, 2001
"... Suppose a discretetime signal S(t), 0 t<N, is a superposition of atoms taken from a combined time/frequency dictionary made of spike sequences 1ft = g and sinusoids expf2 iwt=N) = p N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every d ..."
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Cited by 588 (19 self)
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Suppose a discretetime signal S(t), 0 t<N, is a superposition of atoms taken from a combined time/frequency dictionary made of spike sequences 1ft = g and sinusoids expf2 iwt=N) = p N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discretetime signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time/frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the `1 norm of the coe cients among all decompositions. Here \highly sparse " means that Nt + Nw < p N=2 where Nt is the number of time atoms, Nw is the number of frequency atoms, and N is the length of the discretetime signal.
Symmetric Ciphers Based On TwoDimensional Chaotic Maps
 Int. J. Bifurcation and Chaos
, 1998
"... this paper, methods are shown how to adapt invertible twodimensional chaotic maps on a ..."
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Cited by 104 (0 self)
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this paper, methods are shown how to adapt invertible twodimensional chaotic maps on a
Convergence rates of Markov chains
, 1995
"... this paper, we attempt to describe various mathematical techniques which have been used to bound such rates of convergence. In particular, we describe eigenvalue analysis, random walks on groups, coupling, and minorization conditions. Connections are made to modern areas of research wherever possibl ..."
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Cited by 76 (4 self)
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this paper, we attempt to describe various mathematical techniques which have been used to bound such rates of convergence. In particular, we describe eigenvalue analysis, random walks on groups, coupling, and minorization conditions. Connections are made to modern areas of research wherever possible. Elements of linear algebra, probability theory, group theory, and measure theory are used, but efforts are made to keep the presentation elementary and accessible. Acknowledgements. I thank Eric Belsley for comments and corrections, and thank Persi Diaconis for introducing me to this subject and teaching me so much. 1. Introduction and motivation.
Improving Lattice Based Cryptosystems Using the Hermite Normal Form
 In Silverman [Sil01
"... We describe a simple technique that can be used to substantially reduce the key and ciphertext size of various lattice based cryptosystems and trapdoor functions of the kind proposed by Goldreich, Goldwasser and Halevi (GGH). The improvement is signi cant both from the theoretical and practical poin ..."
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Cited by 39 (8 self)
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We describe a simple technique that can be used to substantially reduce the key and ciphertext size of various lattice based cryptosystems and trapdoor functions of the kind proposed by Goldreich, Goldwasser and Halevi (GGH). The improvement is signi cant both from the theoretical and practical point of view, reducing the size of both key and ciphertext by a factor n equal to the dimension of the lattice (i.e., several hundreds for typical values of the security parameter.) The eciency improvement is obtained without decreasing the security of the functions: we formally prove that the new functions are at least as secure as the original ones, and possibly even better as the adversary gets less information in a strong information theoretical sense. The increased eciency of the new cryptosystems allows the use of bigger values for the security parameter, making the functions secure against the best cryptanalytic attacks, while keeping the size of the key even below the smallest key size for which lattice cryptosystems were ever conjectured to be hard to break.
Equidistribution on the Sphere
 SIAM J. Sci. Stat. Comput
, 1997
"... A concept of generalized discrepancy, which involves pseudodifferential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy. With the hel ..."
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Cited by 32 (2 self)
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A concept of generalized discrepancy, which involves pseudodifferential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy. With the help of this formula five kinds of point systems on the sphere, namely lattices in polar coordinates, transformed 2dimensional sequences, rotations on the sphere, triangulation, and "sum of three squares sequence", are investigated. Quantitative tests are done, and the results are compared with each other. Our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems. 1 Introduction Of practical importance is the problem of generating equidistributed pointsets on the sphere. For that reason a concept of generalized discrepancy, which involves pseudodifferential operators to give a quantifying criterion of equidistributed pointsets...
Shiftregister synthesis (modulo m)
 SIAM J. Computing
, 1985
"... The BerlekampMassey algorithm takes a sequence of elements from a field and finds the shortest linear recurrence (or linear feedback shift register) that can generate the sequence. In this paper we extend the algorithm to the case when the elements of the sequence are integers modulo m, where m is ..."
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Cited by 21 (0 self)
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The BerlekampMassey algorithm takes a sequence of elements from a field and finds the shortest linear recurrence (or linear feedback shift register) that can generate the sequence. In this paper we extend the algorithm to the case when the elements of the sequence are integers modulo m, where m is an arbitrary integer with known prime decomposition.
Convergence of random walks on the circle generated by an irrational rotation
 Trans. Amer. Math. Soc
, 1998
"... Abstract. Fix α ∈ [0, 1). Consider the random walk on the circle S1 which proceeds by repeatedly rotating points forward or backward, with probability 1,byanangle2πα. This paper analyzes the rate of convergence of this walk 2 to the uniform distribution under “discrepancy ” distance. The rate depend ..."
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Cited by 12 (5 self)
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Abstract. Fix α ∈ [0, 1). Consider the random walk on the circle S1 which proceeds by repeatedly rotating points forward or backward, with probability 1,byanangle2πα. This paper analyzes the rate of convergence of this walk 2 to the uniform distribution under “discrepancy ” distance. The rate depends on the continued fraction properties of the number ξ =2α. We obtain bounds for rates when ξ is any irrational, and a sharp rate when ξ is a quadratic irrational. In that case the discrepancy falls as k − 1 2 (up to constant factors), where k is the number of steps in the walk. This is the first example of a sharp rate for a discrete walk on a continuous state space. It is obtained by establishing an interesting recurrence relation for the distribution of multiples of ξ which allows for tighter bounds on terms which appear in the ErdősTurán inequality. 1.
Tensorbased Trapdoors for CVP and their Application to Public Key Cryptography
, 2000
"... . We propose two trapdoors for the ClosestVectorProblem in lattices (CVP) related to the lattice tensor product. Using these trapdoors we set up a latticebased cryptosystem which resembles to the McEliece scheme. 1 Keywords. Public Key Cryptosystem, Closest Vector Problem, Lattice Reduction ..."
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Cited by 10 (0 self)
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. We propose two trapdoors for the ClosestVectorProblem in lattices (CVP) related to the lattice tensor product. Using these trapdoors we set up a latticebased cryptosystem which resembles to the McEliece scheme. 1 Keywords. Public Key Cryptosystem, Closest Vector Problem, Lattice Reduction, Trapdoor, McEliece 1 Introduction Since the invention of public key cryptography in 1976 by Di#e and Hellman [DH76] security of most cryptosystems is based on the (assumed) hardness of factoring or computing discrete logarithms. Only a few schemes based on other problems remain unbroken. Among which there is the McEliece scheme [St95] based on the computational di#culty of decoding a random code. It is still a challenge to develop new public key cryptosystem originating from the hardness of non numbertheoretic problems. In a pioneer work Ajtai [A96] constructed an e#ciently computable function which is hard to invert on the average if the underlying lattice problem is intractable in th...