Electronic Colloquium on Computational Complexity 

About: Electronic Colloquium on Computational Complexity is an academic journal. The journal publishes majorly in the area(s): Upper and lower bounds & Polynomial. It has an ISSN identifier of 1433-8092. Over the lifetime, 2467 publications have been published receiving 48160 citations.

Trapdoors for Hard Lattices and New Cryptographic Constructions.

Abstract: We show how to construct a variety of "trapdoor" cryptographic tools assuming the worst-case hardness of standard lattice problems (such as approximating the length of the shortest nonzero vector to within certain polynomial factors). Our contributions include a new notion of trapdoor function with preimage sampling, simple and efficient "hash-and-sign" digital signature schemes, and identity-based encryption. A core technical component of our constructions is an efficient algorithm that, given a basis of an arbitrary lattice, samples lattice points from a discrete Gaussian probability distribution whose standard deviation is essentially the length of the longest Gram-Schmidt vector of the basis. A crucial security property is that the output distribution of the algorithm is oblivious to the particular geometry of the given basis.

Networks of Spiking Neurons: The Third Generation of Neural Network Models

Abstract: -The computational power of formal models for networks of spiking neurons is compared with that of other neural network models based on McCulloch Pitts neurons (i.e., threshold gates), respectively, sigmoidal gates. In particular it is shown that networks of spiking neurons are, with regard to the number of neurons that are needed, computationally more powerful than these other neural network models. A concrete biologically relevant function is exhibited which can be computed by a single spiking neuron (for biologically reasonable values o f its parameters), but which requires hundreds of hidden units on a sigmoidal neural net. On the other hand, it is known that any function that can be computed by a small sigmoidal neural net can also be computed by a small network of spiking neurons. This article does not assume prior knowledge about spiking neurons, and it contains an extensive list o f references to the currently available literature on computations in networks of spiking neurons and relevant results from neurobiology. © 1997 Elsevier Science Ltd. All rights reserved. Keywords--Spiking neuron, Integrate-and-fire neutron, Computational complexity, Sigmoidal neural nets, Lower bounds. 1. D E F I N I T I O N S AND M O T I V A T I O N S If one classifies neural network models according to their computational units, one can distinguish three different generations. The f irst generation is based on M c C u l l o c h P i t t s neurons as computational units. These are also referred to as perceptrons or threshold gates. They give rise to a variety of neural network models such as multilayer perceptrons (also called threshold circuits), Hopfield nets, and Boltzmann machines. A characteristic feature of these models is that they can only give digital output. In fact they are universal for computations with digital input and output, and every boolean function can be computed by some multilayer perceptron with a single hidden layer. The second generation is based on computational units that apply an "activation function" with a continuous set of possible output values to a weighted sum (or polynomial) of the inputs. Common activation functions are the s igmoid func t ion a(y) = 1/(1 + e -y) and the linear Acknowledgements: I would like to thank Eduardo Sontag and an anonymous referee for their helpful comments. Written under partial support by the Austrian Science Fund. Requests for reprints should be sent to W. Maass, Institute for Theoretical Computer Science, Technische Universit~it Graz, Klosterwiesgasse 32/2, A-8010, Graz, Austria; tel. +43 316 873-5822; fax: +43 316 873-5805; e-mail: maass@igi,tu-graz.ac.at saturated function 7r with 7r(y) = y for 0 --< y --< 1, 7r(y) = 0 for y < 0, lr(y) = 1 for y > 1. Besides piecewise polynomial activation functions we consider in this paper also "piecewise exponential" activation functions, whose pieces can be defined by expressions involving exponentiation (such as the definition of a). Typical examples for networks from this second generation are feedforward and recurrent sigmoidal neural nets, as well as networks of radial basis function units. These nets are also able to compute (with the help of thresholding at the network output) arbitrary boolean functions. Actually it has been shown that neural nets from the second generation can compute certain boolean functions with f e w e r gates than neural nets from the first generation (Maass, Schnitger, & Sontag, 1991; DasGupta & Schnitger, 1993). In addition, neural nets from the second generation are able to compute functions with analog input and output. In fact they are universal for analog computations in the sense that any continuous function with a compact domain and range can be approximated arbitrarily well (with regard to uniform convergence, i.e., the L= norm) by a network of this type with a single hidden layer. Another characteristic feature of this second generation of neural network models is that they support learning algorithms that are based on gradient descent such as backprop.

Property Testing and its connection to Learning and Approximation

Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Generating Hard Instances of Lattice Problems

Abstract: We give a random class of lattices in Z n so that, if there is a probabilistic polynomial time algorithm which nds a short vector in a random lattice with a probability of at least 1 2 then there is also a probabilistic polynomial time algorithm which solves the following three lattice problems in every lattice in Z n with a probability exponentially close to one. (1) Find the length of a shortest nonzero vector in an n-dimensional lattice, approximately, up to a polynomial factor. (2) Find the shortest nonzero vector in an n-dimensional lattice L where the shortest vector v is unique in the sense that any other vector whose length is at most n c kvk is parallel to v, where c is a suuciently large absolute constant. (3) Find a basis b 1 ; :::; b n in the n-dimensional lattice L whose length, deened as max n i=1 kb i k, is the smallest possible up to a polynomial factor. A large number of the existing techniques of cryptography include the generation of a speciic instance of a problem in NP (together with a solution) which for some reason is thought to be diicult to solve. As an example we may think about factor-ization. Here a party of a cryptographic protocol is supposed to provide a composite number m so that the factorization of m is known to her but she has some serious reason to believe that nobody else will be able to factor m. The most compelling reason for such a belief would be a mathematical proof of the fact that the prime factors of m cannot be found in less then k step in some realistic model of computation, where k is a very large number. For the moment we do not have any proof of this type, neither for speciic numerical values of m and k, nor in some assymptotic sense. In spite of the lack of mathematical proofs, in two cases at least, we may expect that a problem will be diicult to solve. One is the class of NP-complete problems. Here we may say that if there is a problem at all which is diicult to solve, then an NP-complete problem will provide such an example. The other case is, if the problem is a very famous question (e.g. prime factorization), which for a long time were unsuccesfully attacked by …

Quantum measurements and the Abelian Stabilizer Problem

Abstract: We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.