estimation in spin glasses a first step the sherrington kirkpatrick model of spin glasses the hopfield model of neural networks and the ising spin glass are all models of binary data belonging to the one parameter exponential family with quadratic sufficient statistic under bare minimal conditions we establish the n consistency of the maximum pseudolikelihood estimate of the natural parameter in this family even at critical temperatures since very little is known about the low and critical temperature regimes of these extremely difficult models the proof requires several new ideas the author s version of stein s method is a particularly useful tool we aim to introduce these techniques into the realm of mathematical statistics through an example and present some open questions

on the singularity of adjacency matrices for random regular digraphs we prove that the non symmetric adjacency matrix of a uniform random d regular directed graph on n vertices is asymptotically almost surely invertible assuming min d n d ge c log 2n for a sufficiently large constant c 0 the proof makes use of a coupling of random regular digraphs formed by shuffling the neighborhood of a pair of vertices as well as concentration results for the distribution of edges proved in cook random struct algorithms arxiv 1410 5595 2014 we also apply our general approach to prove asymptotically almost surely invertibility of hadamard products varsigma mathrm circ varxi where varxi is a matrix of iid uniform pm 1 signs and varsigma is a 0 1 matrix whose associated digraph satisfies certain expansion properties

lattice basis reduction improved practical algorithms and solving subset sum problems we report on improved practical algorithms for lattice basis reduction we present a variant of the l3 algorithm with deep insertions and a practical algorithm for blockwise korkine zolotarev reduction a concept extending l3 reduction that has been introduced by schnorr 1987 empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a unisys 6000 70 or within a couple of minutes on a sparc 2 computer

lower bounds of shortest vector lengths in random ntru lattices finding the shortest vector of a lattice is one of the most important problems in computational lattice theory for a random lattice one can estimate the length of the shortest vector using the gaussian heuristic however no rigorous proof can be provided for some classes of lattices as the gaussian heuristic may not hold for them in this paper we propose a general method to estimate lower bounds of the shortest vector lengths for random integral lattices in certain classes which is based on the incompressibility method from the theory of kolmogorov complexity as an application we can prove that for a random ntru lattice with an overwhelming probability the ratio between the length of the shortest vector and the length of the target vector which corresponds to the secret key is at least a constant independent of the rank of the lattice

a knapsack based probabilistic encryption scheme knapsack based cryptosystems had been viewed as the most attractive and the most promising asymmetric cryptographic algorithms for a long time due to their np completeness nature and high speed in encryption decryption unfortunately most of them are broken for the low density feature of the underlying knapsack problems in this paper we investigate a new easy compact knapsack problem and propose a novel knapsack based probabilistic public key cryptosystem in which the cipher text is non linear with the plaintext for properly chosen parameters the underlying knapsack problem enjoys a high density larger than 1 06 in the worst case hence it is secure against the low density subset sum attacks our scheme can also defeat other potential attacks such as the brute force attacks and the simultaneous diophantine approximation attack compared with previous knapsack based cryptosystems our scheme is efficient and practical

stationary distribution and cover time of sparse directed configuration models we consider sparse digraphs generated by the configuration model with given in degree and out degree sequences we establish that with high probability the cover time is linear up to a poly logarithmic correction for a large class of degree sequences we determine the exponent gamma ge 1 of the logarithm and show that the cover time grows as n log gamma n where n is the number of vertices the results are obtained by analysing the extremal values of the stationary distribution of the digraph in particular we show that the stationary distribution pi is uniform up to a poly logarithmic factor and that for a large class of degree sequences the minimal values of pi have the form frac 1 n log 1 gamma n while the maximal values of pi behave as frac 1 n log 1 kappa n for some other exponent kappa in 0 1 in passing we prove tight bounds on the diameter of the digraphs and show that the latter coincides with the typical distance between two vertices

the optimal lll algorithm is still polynomial in fixed dimension in this paper we consider the open problem of the complexity of the lll algorithm in the case when the approximation parameter of the algorithm has as its extreme value 1 this case is of interest because the output is then the strongest lovasz reduced basis experiments reported by lagarias and odlyzko j acm 32 1 1985 229 seem to show that the algorithm remains polynomial in average however no bound better than a naive exponential order one is established for the worst case complexity of the optimal lll algorithm even for fixed small dimension higher than 2 here we prove that for any fixed dimension n the number of iterations of the lll algorithm is linear with respect to the size of the input it is easy to deduce from vallee j algorithms 12 1991 556 that the linear order is optimal moreover in 3 dimensions we give a tight bound for the maximum number of iterations and we characterize precisely the output basis our bound also improves the known one for the usual non optimal lll algorithm

generalized monte carlo significance tests summary simple monte carlo significance testing has many applications particularly in the preliminary analysis of spatial data the method requires the value of the test statistic to be ranked among a random sample of values generated according to the null hypothesis however there are situations in which a sample of values can only be conveniently generated using a markov chain initiated by the observed data so that independence is violated this paper describes two methods that overcome the problem of dependence and allow exact tests to be carried out the methods are applied to the rasch model to the finite lattice ising model and to the testing of association between spatial processes power is discussed in a simple case

bounded size bias coupling a gamma function bound and universal dickman function behavior under the assumption that the distribution of a nonnegative random variable x admits a bounded coupling with its size biased version we prove simple and strong concentration bounds in particular the upper tail probability is shown to decay at least as fast as the reciprocal of a gamma function guaranteeing a moment generating function that converges everywhere the class of infinitely divisible distributions with finite mean whose levy measure is supported on an interval contained in 0 c for some c infty forms a special case in which this upper bound is logarithmically sharp in particular the asymptotic estimate for the dickman function that rho u approx u u for large u is shown to be universal for this class a special case of our bounds arises when x is a sum of independent random variables each admitting a 1 bounded size bias coupling in this case our bounds are comparable to chernoff hoeffding bounds however ours are broader in scope sharper for the upper tail and equal for the lower tail we discuss bounded and monotone couplings give a sandwich principle and show how this gives an easy conceptual proof that any finite positive mean sum of independent bernoulli random variables admits a 1 bounded coupling with the same conditioned to be nonzero

solving a system of linear diophantine equations with lower and upper bounds on the variables we develop an algorithm for solving a system of diophantine equations with lower and upper bounds on the variables the algorithm is based on lattice basis reduction it first finds a short vector satisfying the system of diophantine equations and a set of vectors belonging to the nullspace of the constraint matrix due to basis reduction all these vectors are relatively short the next step is to branch on linear combinations of the null space vectors which either yields a vector that satisfies the bound constraints or provides a proof that no such vector exists the research was motivated by the need for solving constrained diophantine equations as subproblems when designing integrated circuits for video signal processing our algorithm is tested with good results on real life data and on instances from the literature

analysis of the efficiency of the chor rivest cryptosystem implementation in a safe parameter range the chor rivest cryptosystem based on a high density knapsack problem on a finite field f q h was broken by vaudenay for q 200 h 24 and h admitting a factor s verifying a certain condition a new set of parameters q and h which prevent this cryptosystem against vaudenay s attack is presented and the computational aspects of its implementation in the magma computational algebra system are analyzed

on the equivalence of mceliece s and niederreiter s public key cryptosystems it is shown that mceliece s and niederreiter s public key cryptosystems are equivalent when set up for corresponding choices of parameters a security analysis for the two systems based on this equivalence observation is presented

efron stein inequalities for random matrices this paper establishes new concentration inequalities for random matrices constructed from independent random variables these results are analogous with the generalized efron stein inequalities developed by boucheron et al the proofs rely on the method of exchangeable pairs

hardness of approximating the shortest vector problem in lattices let p 1 be any fixed real we show that assuming np n rp there is no polynomial time algorithm that approximates the shortest vector problem svp in e p norm within a constant factor under the stronger assumption np n rtime 2poly log n we show that there is no polynomial time algorithm with approximation ratio 2 log n 1 2 e where n is the dimension of the lattice and e 0 is an arbitrarily small constant we first give a new randomized reduction from closest vector problem cvp to svp that achieves some constant factor hardness the reduction is based on bch codes its advantage is that the svp instances produced by the reduction behave well under the augmented tensor product a new variant of tensor product that we introduce this enables us to boost the hardness factor to 2 log n 1 2 e

beyond the regret minimization barrier optimal algorithms for stochastic strongly convex optimization we give novel algorithms for stochastic strongly convex optimization in the gradient oracle model which return a o 1 t approximate solution after t iterations the first algorithm is deterministic and achieves this rate via gradient updates and historical averaging the second algorithm is randomized and is based on pure gradient steps with a random step size his rate of convergence is optimal in the gradient oracle model this improves upon the previously known best rate of o log t t which was obtained by applying an online strongly convex optimization algorithm with regret o log t to the batch setting we complement this result by proving that any algorithm has expected regret of log t in the online stochastic strongly convex optimization setting this shows that any online to batch conversion is inherently suboptimal for stochastic strongly convex optimization this is the first formal evidence that online convex optimization is strictly more difficult than batch stochastic convex optimization

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

concentration of haar measures with an application to random matrices abstract we show that the mixing times of random walks on compact groups can be used to obtain concentration inequalities for the respective haar measures as an application we derive a concentration inequality for the empirical distribution of eigenvalues of sums of random hermitian matrices with possible applications in free probability the advantage over existing techniques is that the new method can deal with functions that are non lipschitz or even discontinuous with respect to the usual metrics

solving low density subset sum problems the subset sum problem is to decide whether or not the 0 l integer programming problem s n i l a i x i m i x i 0 or 1 has a solution where the a i and m are given positive integers this problem is np complete and the difficulty of solving it is the basis of public key cryptosystems of knapsack type an algorithm is proposed that searches for a solution when given an instance of the subset sum problem this algorithm always halts in polynomial time but does not always find a solution when one exists it converts the problem to one of finding a particular short vector v in a lattice and then uses a lattice basis reduction algorithm due to a k lenstra h w lenstra jr and l lovasz to attempt to find v the performance of the proposed algorithm is analyzed let the density d of a subset sum problem be defined by d n log 2 max i a i then for almost all problems of density d d n it is proved that the lattice basis reduction algorithm locates v extensive computational tests of the algorithm suggest that it works for densities d d c n where d c n is a cutoff value that is substantially larger than 1 n this method gives a polynomial time attack on knapsack public key cryptosystems that can be expected to break them if they transmit information at rates below d c n as n

commutation relations and markov chains it is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain markov chains examples studied include random walk on irreducible representations a local random walk on partitions whose stationary distribution is the ewens distribution and some birth death chains

the hardness of the closest vector problem with preprocessing we give a new simple proof of the np hardness of the closest vector problem in addition to being much simpler than all previously known proofs the new proof yields new interesting results about the complexity of the closest vector problem with preprocessing this is a variant of the closest vector problem in which the lattice is specified in advance and can be preprocessed for an arbitrarily long amount of time before the target vector is revealed we show that there are lattices for which the closest vector problem remains hard regardless of the amount of preprocessing

iterative simulation methods abstract the standard methods of generating sample from univariate distributions often become hopelessly inefficient when applied to realizations of stochastic processes iterative methods are not widely known amongst statisticians but some are standard practice in statistical physics and chemistry the methods are surveyed and compared with particular reference to their convergence properties

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

linear regression with random projections we investigate a method for regression that makes use of a randomly generated subspace gp f of finite dimension p of a given large possibly infinite dimensional function space f for example l2 0 1 d r gp is defined as the span of p random features that are linear combinations of a basis functions of f weighted by random gaussian i i d coefficients we show practical motivation for the use of this approach detail the link that this random projections method share with rkhs and gaussian objects theory and prove both in deterministic and random design approximation error bounds when searching for the best regression function in gp rather than in f and derive excess risk bounds for a specific regression algorithm least squares regression in gp this paper stresses the motivation to study such methods thus the analysis developed is kept simple for explanations purpose and leaves room for future developments

stein s method and plancherel measure of the symmetric group we initiate a stein s method approach to the study of the plancherel measure of the symmetric group a new proof of kerov s central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained the proof gives an error term the construction of an exchangeable pair needed for applying stein s method arises from the theory of harmonic functions on bratelli diagrams we also find the spectrum of the markov chain on partitions underlying the construction of the exchangeable pair this yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group

on normal approximations of distributions in terms of dependency graphs l auteur interprete les bornes de l erreur introduites par stein dans l approximation normale de sommes de variables aleatoires dependantes en termes de graphes de dependance ceci mene a des ameliorations d un theoreme de limite centrale de petrovskaya et leontovich et de recentes applications de baldi et rinatt en particulier on obtient des bornes pour des taux de convergence on etudie ensuite l approximation normale du nombre de maximums locaux d une fonction aleatoire sur un graphe

wishart planted ensemble a tunably rugged pairwise ising model with a first order phase transition we propose the wishart planted ensemble a class of zero field ising models with tunable algorithmic hardness and specifiable or planted ground state the problem class arises from a simple procedure for generating a family of random integer programming problems with specific statistical symmetry properties but turns out to have intimate connections to a sign inverted variant of the hopfield model the hamiltonian contains only 2 spin interactions with the coupler matrix following a type of wishart distribution the class exhibits a classical first order phase transition in temperature for some parameter settings the model has a locally stable paramagnetic state a feature which correlates strongly with difficulty in finding the ground state and suggests an extremely rugged energy landscape we analytically probe the ensemble thermodynamic properties by deriving the thouless anderson palmer equations and free energy and corroborate the results with a replica and annealed approximation analysis extensive monte carlo simulations confirm our predictions of the first order transition temperature the class exhibits a wide variation in algorithmic hardness as a generation parameter is varied with a pronounced easy hard easy profile and peak in solution time towering many orders of magnitude over that of the easy regimes by deriving the ensemble averaged energy distribution and taking into account finite precision representation we propose an analytical expression for the location of the hardness peak and show that at fixed precision the number of constraints in the integer program must increase with system size to yield truly hard problems the wishart planted ensemble is interesting for its peculiar physical properties and provides a useful and analytically transparent set of problems for benchmarking optimization algorithms

applications of stein s method for concentration inequalities stein s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and gibbs measures in this paper we provide some extensions of the theory and three applications 1 we obtain a concentration inequality for the magnetization in the curie weiss model at critical temperature where it obeys a nonstandard normalization and super gaussian concentration 2 we derive exact large deviation asymptotics for the number of triangles in the erd s renyi random graph g n p when p 0 31 similar results are derived also for general subgraph counts 3 we obtain some interesting concentration inequalities for the ising model on lattices that hold at all temperatures

stein s method for diffusion approximations stein s method of obtaining distributional approximations is developed in the context of functional approximation by the wiener process and other gaussian processes an appropriate analogue of the one dimensional stein equation is derived and the necessary properties of its solutions are established the method is applied to the partial sums of stationary sequences and of dissociated arrays to a process version of the wald wolfowitz theorem and to the empirical distribution function

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

on the convergence of monte carlo maximum likelihood calculations summary monte carlo maximum likelihood for normalized families of distributions can be used for an extremely broad class of models given any family he 0 e 0 of non negative integrable functions maximum likelihood estimates in the family obtained by normalizing the functions to integrate to 1 can be approximated by monte carlo simulation the only regularity conditions being a compactification of the parameter space such that the evaluation maps 0 h0 x remain continuous then with probability 1 the monte carlo approximant to the log likelihood hypoconverges to the exact log likelihood its maximizer converges to the exact maximum likelihood estimate approximations to profile likelihoods hypoconverge to the exact profile and level sets of the approximate likelihood support regions converge to the exact sets in painleve kuratowski set convergence the same results hold when there are missing data if a wald type integrability condition is satisfied asymptotic normality of the monte carlo error and convergence of the monte carlo approximation to the observed fisher information are also shown

integer sets with distinct subset sums in section 1 we introduce the problem of finding minimal height sets of n natural numbers with distinct subset sums ssd and in section 2 review the well known conway guy sequence u conjectured to yield a minimal ssd set for every n we go on section 3 to prove that u certainly cannot be improved upon by any greedy sequence to verify numerically section 4 that it does yield ssd sets for n 80 and section 5 by direct search to show that these are minimal for n s 8 there is a brief interlude section 6 on the problem of decoding the subset from its sum in section 7 generalizations of u are constructed which are asymptotically smaller defining the limit ratio of a sequence w to be a 1imn wn 2l the atkinson negro santoro sequence v known to give ssd sets has a 0 6334 conwayguy conjectured to has a 0 4703 and our best generalization has a 0 4419 we also section 8 discuss when such sequences have the same a and section 9 how a may efficiently be computed to high accuracy 1 the distinct subset sum problem a well known problem in combinatorial number theory 2 pp 64 65 involves the construction for given n of a set p pi i 1 n of natural numbers possessing the property we shall call subset sum distinctness or ssd for short that is l l f pi p e pi s t c l n i s i t or distinct subsets of p have distinct sums evidently choosing pi 21 satisfies 1 1 the interest lies in how much the maximum element pn can be reduced below 2n 1 in order to explore the ssd property it is convenient to consider a more general concept we say that x has a representation by p with length k and signature 1 when for some s and t as before 1 2 x e pe po ies iet where s n t 0 si it k isi iti 1 that is x is the difference of two disjoint subset sums of which the first has 1 terms more than the second and the two together have k terms alternatively n 1 3 x eip i l received june 23 1986 revised october 20 1986 1980 mathematics subject classification 1985 revision primary 11b13 05 04 05a99 11j13 94a60 65b05

on the existence of maximum likelihood estimates in logistic regression models summary the problems of existence uniqueness and location of maximum likelihood estimates in log linear models have received special attention in the literature haberman 1974 chapter 2 wedderburn 1976 silvapulle 1981 for multinomial logistic regression models we prove existence theorems by considering the possible patterns of data points which fall into three mutually exclusive and exhaustive categories complete separation quasicomplete separation and overlap our results suggest general rules for identifying infinite parameter estimates in log linear models for frequency tables

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

intrinsic autoregressions and related models on the two dimensional lattice summary stationary autoregressions on a two dimensional lattice are generalized to intrinsic models where only increments are assumed to be stationary prediction formulae and the asymptotic behaviour of the semivariogram are derived for parameter estimation we propose an approximate maximum likelihood estimator a generalization of whittle s estimator it is derived also for general intrinsic models

random projections for classification a recovery approach random projection has been widely used in data classification it maps high dimensional data into a low dimensional subspace in order to reduce the computational cost in solving the related optimization problem while previous studies are focused on analyzing the classification performance in the low dimensional space in this paper we consider the recovery problem i e how to accurately recover the optimal solution to the original high dimensional optimization problem based on the solution learned after random projection we present a simple algorithm termed dual random projection which uses the dual solution of the low dimensional optimization problem to recover the optimal solution to the original problem our theoretical analysis shows that with a high probability the proposed algorithm is able to accurately recover the optimal solution to the original problem provided that the data matrix is approximately low rank and or optimal solution is approximately sparse we further show that the proposed algorithm can be applied iteratively to reducing the recovery error exponentially

sparser johnson lindenstrauss transforms we give two different and simple constructions for dimensionality reduction in e2 via linear mappings that are sparse only an o e fraction of entries in each column of our embedding matrices are non zero to achieve distortion 1 e with high probability while still achieving the asymptotically optimal number of rows these are the first constructions to provide subconstant sparsity for all values of parameters improving upon previous works of achlioptas 2003 and dasgupta et al 2010 such distributions can be used to speed up applications where e2 dimensionality reduction is used

an improved balanced algorithm for the subset sum problem abstract we present balsublast which is an improved balanced algorithm for the subset sum problem that was designed to solve benchmarks that require the exhaustion of the search space and where there are many subsets with the same sum this new algorithm which spends time o n 2 w max and space o n w max where w max is the largest of n items has obtained excellent performance in comparison to other known algorithms

stein s method for concentration inequalities we introduce a version of stein s method for proving concentration and moment inequalities in problems with dependence simple illustrative examples from combinatorics physics and mathematical statistics are provided

the computational complexity of simultaneous diophantine approximation problems simultaneous diophantine approximation in d dimensions deals with the approximation of a vector bf alpha alpha 1 cdots alpha d of d real numbers by vectors of rational numbers all having the same denominator this paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector bf alpha of d rational numbers we measure the goodness of an approximation using the sup norm we show that a result of h w lenstra jr produces polynomial time algorithms to find sup norm best approximations to a given vector bf alpha when the dimension d is fixed we show that a recent algorithm of a k lenstra h w lenstra jr and l lovasz to find short vectors in an integral lattice can be used to find a good approximation to a given vector bf alpha in d dimensions with a denominator q satisfying 1 leqq q leqq 2 d 2 n which is within a factor sqrt 5d 2 d 1 2 of the best approximation with denominato

a new paradigm for public key identification the present paper investigates the possibility of designing zero knowledge identification schemes based on hard problems from coding theory zero knowledge proofs were introduced by goldwasser micali and rackoff 1985 their practical significance was soon demonstrated in the work of fiat and shamir 1986 who turned zero knowledge proofs of quadratic residuosity into efficient means of establishing user identities in the present paper we propose a new identification scheme based on error correcting codes which is zero knowledge and seems of practical value furthermore we describe several variants including one which has an identity based character the security of our schemes depends on the hardness of finding a word of given syndrome and prescribed small weight with respect to some randomly generated binary linear error correcting code this is of course not the first attempt to design a cryptographic scheme using tools from coding theory the difference is that identification protocols do not follow the public key paradigm based on trap door functions and described in the seminal diffie hellman paper 1976 rather they only require one way functions which opens the way to using in a rather direct manner simple combinatorial problems of the kind provided by coding theory the resulting schemes compare favorably to their number theoretic analogs

on concentration inequalities and their applications for gibbs measures in lattice systems we consider gibbs measures on the configuration space s mathbb z d where mostly d ge 2 and s is a finite set we start by a short review on concentration inequalities for gibbs measures in the dobrushin uniqueness regime we have a gaussian concentration bound whereas in the ising model and related models at sufficiently low temperature we control all moments and have a stretched exponential concentration bound we then give several applications of these inequalities whereby we obtain various new results amongst these applications we get bounds on the speed of convergence of the empirical measure in the sense of kantorovich distance fluctuation bounds in the shannon mcmillan breiman theorem fluctuation bounds for the first occurrence of a pattern as well as almost sure central limit theorems

inference for discrete markov fields the simplest nontrivial case abstract markov fields provide a general context for describing the strength and structure of spatial interactions the gibbs markov equivalence theorem preston 1974 parameterizes markov fields via their neighborhood structures yielding exponential families in canonical form likelihood inference is therefore apparently straightforward the requisite normalizing constants however are obstreperous even when asymptotic characterizations can be obtained substantial location errors arise during implementation moreover markov fields can exhibit phase transitions and long range interactions thereby creating identifiability problems these issues are illustrated in the simplest nontrivial case the classical ising model of ferromagnetism

matrix concentration inequalities via the method of exchangeable pairs this paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix the analysis requires a matrix extension of the scalar concentration theory developed by sourav chatterjee using stein s method of exchangeable pairs when applied to a sum of independent random matrices this approach yields matrix generalizations of the classical inequalities due to hoeffding bernstein khintchine and rosenthal the same technique delivers bounds for sums of dependent random matrices and more general matrix valued functions of dependent random variables

on the lagarias odlyzko algorithm for the subset sum problem we give a simple analysis of an algorithm for solving subset sum problems proposed by lagarias and odlyzko 2

concentration inequalities for random fields via coupling we present a new and simple approach to concentration inequalities in the context of dependent random processes and random fields our method is based on coupling and does not use information inequalities in case one has a uniform control on the coupling one obtains exponential concentration inequalities if such a uniform control is no more possible then one obtains polynomial or stretched exponential concentration inequalities our abstract results apply to gibbs random fields both at high and low temperatures and in particular to the low temperature ising model which is a concrete example of non uniformity of the coupling

random walk on sparse random digraphs a finite ergodic markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale originally discovered in the context of card shuffling aldous and diaconis in am math mon 93 333 348 1986 this remarkable phenomenon is now rigorously established for many reversible chains here we consider the non reversible case of random walks on sparse directed graphs for which even the equilibrium measure is far from being understood we work under the configuration model allowing both the in degrees and the out degrees to be freely specified we establish the cutoff phenomenon determine its precise window and prove that the cutoff profile approaches a universal shape we also provide a detailed description of the equilibrium measure

concentration of measures via size biased couplings let y be a nonnegative random variable with mean and finite positive variance 2 and let ys defined on the same space as y have the y size biased distribution characterized by e yf y mu e f y s quad rm for all functions f rm for which these expectations exist under a variety of conditions on y and the coupling of y and ys including combinations of boundedness and monotonicity one sided concentration of measure inequalities such as p left frac y mu sigma ge t right le rm exp left frac t 2 2 a bt right quad rm for all t 0 hold for some explicit a and b the theorem is applied to the number of bulbs switched on at the terminal time in the so called lightbulb process of rao et al sankhy 69 137 161 2007

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

cutoff for nonbacktracking random walks on sparse random graphs a finite ergodic markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale discovered in the context of card shuffling aldous diaconis 1986 this phenomenon is now believed to be rather typical among fast mixing markov chains yet establishing it rigorously often requires a challengingly detailed understanding of the underlying chain here we consider non backtracking random walks on random graphs with a given degree sequence under a general sparsity condition we establish the cutoff phenomenon determine its precise window and prove that the suitably rescaled cutoff profile approaches a remarkably simple universal shape

concentration inequalities using the entropy method we investigate a new methodology worked out by ledoux and massart to prove concentration of measure inequalities the method is based on certain modified logarithmic sobolev inequalities we provide some very simple and general ready to use inequalities one of these inequalities may be considered as an exponential version of the efron stein inequality the main purpose of this paper is to point out the simplicity and the generality of the approach we show how the new method can recover many of talagrand s revolutionary inequalities and provide new applications in a variety of problems including rademacher averages rademacher chaos the number of certain small subgraphs in a random graph and the minimum of the empirical risk in some statistical estimation problems

quadratic compact knapsack public key cryptosystem knapsack type cryptosystems were among the first public key cryptographic schemes to be invented their np completeness nature and the high speed in encryption decryption made them very attractive however these cryptosystems were shown to be vulnerable to the low density subset sum attacks or some key recovery attacks in this paper additive knapsack type public key cryptography is reconsidered we propose a knapsack type public key cryptosystem by introducing an easy quadratic compact knapsack problem the system uses the chinese remainder theorem to disguise the easy knapsack sequence the encryption function of the system is nonlinear about the message vector under the relinearization attack model the system enjoys a high density we show that the knapsack cryptosystem is secure against the low density subset sum attacks by observing that the underlying compact knapsack problem has exponentially many solutions it is shown that the proposed cryptosystem is also secure against some brute force attacks and some known key recovery attacks including the simultaneous diophantine approximation attack and the orthogonal lattice attack

from stein identities to moderate deviations stein s method is applied to obtain a general cramer type moderate deviation result for dependent random variables whose dependence is defined in terms of a stein identity a corollary for zero bias coupling is deduced the result is also applied to a combinatorial central limit theorem a general system of binary codes the anti voter model on a complete graph and the curie weiss model a general moderate deviation result for independent random variables is also proved

approximating svp to within almost polynomial factors is np hard abstract we show svp and cvp to be np hard to approximate to within n c log log n for some constant c 0 we show a direct reduction from sat to these problems that combines ideas from arora et al proc 34th ieee symp on foundations of computer science 1993 p 724 and dinur et al approximating cvp to within almost polynomial factors is np hard manuscript 1999 along with some modifications our result is obtained without relying on the pcp characterization of np although some of our techniques are derived from the proof of the pcp characterization itself stoc acm symposium on theory of computing stoc 1999

a new method of normal approximation we introduce a new version of stein s method that reduces a large class of normal approximation problems to variance bounding exercises thus making a connection between central limit theorems and concentration of measure unlike skorokhod embeddings the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily as an application we derive a general clt for functions that are obtained as combinations of many local contributions where the definition of local itself depends on the data several examples are given including the solution to a nearest neighbor clt problem posed by p bickel

local rademacher complexities we propose new bounds on the error of learning algorithms in terms of a data dependent notion of complexity the estimates we establish give optimal rates and are based on a local and empirical version of rademacher averages in the sense that the rademacher averages are computed from the data on a subset of functions with small empirical error we present some applications to classification and prediction with convex function classes and with kernel classes in particular

an lll algorithm with quadratic complexity the lenstra lenstra lovasz lattice basis reduction algorithm called lll or rm l 3 is a fundamental tool in computational number theory and theoretical computer science which can be viewed as an efficient algorithmic version of hermite s inequality on hermite s constant given an integer d dimensional lattice basis with vectors of euclidean norm less than b in an n dimensional space the rm l 3 algorithm outputs a reduced basis in o d 3n rm log b cdot mathcal m d rm log b bit operations where mathcal m k denotes the time required to multiply k bit integers this worst case complexity is problematic for applications where d or and rm log b are often large as a result the original rm l 3 algorithm is almost never used in practice except in tiny dimension instead one applies floating point variants where the long integer arithmetic required by gram schmidt orthogonalization is replaced by floating point arithmetic unfortunately this is known to be unstable in the worst case the usual floating point rm l 3 algorithm is not even guaranteed to terminate and the output basis may not be rm l 3 reduced at all in this article we introduce the rm l 2 algorithm a new and natural floating point variant of the rm l 3 algorithm which provably outputs rm l 3 reduced bases in polynomial time o d 2n d rm log b rm log b cdot mathcal m d this is the first rm l 3 algorithm whose running time without fast integer arithmetic provably grows only quadratically with respect to rm log b like euclid s gcd algorithm and lagrange s two dimensional algorithm

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

cryptanalytic attacks on the multiplicative knapsack cryptosystem and on shamir s fast signature scheme the basic merkle hellman additive trapdoor knapsack public key cryptosystem was recently shown to be insecure and attacks have also been developed on stronger variants of it such as the graham shamir system and the iterated knapsack cryptosystem it is shown that some simple variants of another merkle hellman system the multiplicative knapsack cryptosystem are insecure it is also shown that the shamir fast signature scheme can be broken quickly similar attacks can also be used to break the schobi massey authentication scheme these attacks have not been rigorously proved to succeed but heuristic arguments and empirical evidence indicate that they work on systems of practical size

sequential monte carlo p values summary the assessment of statistical significance by monte carlo simulation may be costly in computer time this paper looks at a number of ways of calculating exact monte carlo p values by sequential sampling such p values are shown to have properties similar to those obtained by sampling with a fixed sample size both standard and generalized monte carlo procedures are discussed and in particular a sequential method is proposed for dealing with situations in which values can only be conveniently generated using a markov chain conditioned to pass through the observed data

likelihood analysis of spatial point patterns summary the likelihood procedure for estimating the pairwise interaction potential function is developed for statistical analysis of homogeneous spatial point patterns approximation methods of the normalizing factor of gibbs canonical distribution are discussed both to estimate a scale parameter and to measure the softness or hardness of repulsive interactions the approximations are useful up to a considerably high density the validity of our procedure is demonstrated by some computer experi ments some real data are analysed

safer parameters for the chor rivest cryptosystem vaudenay s cryptanalysis against chor rivest cryptosystem is applicable when the parameters p and h originally proposed by the authors are used nevertheless if p and h are both prime integers then vaudenay s attack is not applicable in this work a choice of these parameters resistant to the existing cryptanalytic attacks is presented the parameters are determined in a suitable range guaranteeing its security and the computational feasibility of implementation regrettably the obtained parameters are scarce in practice