Discrete Normal Distribution Research Articles

Comparison of the quality of sampling algorithms from discrete normal distribution on NTRU lattices

Post-quantum cryptography is a field of research that studies cryptographic transformations protected against attacks using quantum computers. For many years, lattice-based cryptography has become one of the most promising solutions to protect against the threat of quantum computing. An important feature of the post-quantum period in cryptography is the significant uncertainty about the source data for cryptanalysis and countermeasures in the capabilities of quantum computers, their mathematical support and software, as well as the application of quantum cryptanalysis to existing cryptocurrencies and cryptoprotocol. The main methods are mathematical methods of electronic signature, which have undergone significant analysis and justification in the process of extensive research by cryptologists and mathematicians at the highest level. The security of signature schemes depends strongly on the standard deviation of the discrete Gaussian distribution, which has a sampling algorithm. In this paper, the most common variants of sampling algorithms were considered and analyzed, because the quality of all algorithms depends significantly on the structure of the lattice for which sampling takes place. A comparison of the quality of lattice sampling algorithms is highlighted. In particular, the paper considers Klein's algorithms (its modification is the Thomas Prest and Dukas algorithm), Peikert's algorithm and the floating-point sampling algorithm. Klein's sampling algorithm, in particular its modification, namely, the Dukas-Prest algorithm, gives the smallest vectors. Theoretically, it is much better than Klein's algorithm on NTRU lattices, but it requires the use of floating-point arithmetic, which complicates greatly the analysis of its security and tocreation of software or hardware implementation.

Approximation of the Binomial Probability Function Using the Discrete Normal Distribution

A new method of approximating the Binomial probability function is introduced. The method is based on the discrete normal distribution. In particular, the discrete normal probability function is used to approximate the binomial probability function. The new approximation is compared with the exact values and the approximation based on Central limit theorem. The maximum absolute error of the approximation is used to measure the accuracy of the method. It turned out that this method of approximation is useful and easy to use in practice. Also, the result can be an important theoretical statistical result that can be used in educational statistics.

The Kullback–Leibler Divergence Between Lattice Gaussian Distributions

Discrete normal distributions are defined as the distributions with prescribed means and covariance matrices which maximize entropy on the integer lattice support. The set of discrete normal distributions form an exponential family with cumulant function related to the Riemann theta function. In this paper, we present several formula for common statistical divergences between discrete normal distributions including the Kullback-Leibler divergence. In particular, we describe an efficient approximation technique for calculating the Kullback-Leibler divergence between discrete normal distributions via the R\'enyi $\alpha$-divergences or the projective $\gamma$-divergences.

Analysis of ways to increase stability of cryptographic algorithms on algebraic lattices against time attacks

The aim of this work is to study the algorithms, the stability of which is based on the search for a short lattice vector, as well as to obtain time-resistant parameters of these algorithms. Existing methods for generating keys and choosing parameters for cryptographic transformations on algebraic lattices resistant to time attacks are considered. It is shown that the uniform distribution of coefficients for generating the NTRU algorithm keys has certain shortages, namely, a limited number of parameters suitable for use in cryptographical transformations. This is due to the vulnerability of this algorithm to time attacks. The possibility of using a discrete normal (Gaussian) distribution to form a key pair, which will prevent the sensitivity of the algorithm to time attacks, is considered. This method of generation requires checking the obtained sample for compliance with the properties of the normal distribution. The usage of SAGA tests has been proposed. They make it possible to check the Gaussian samples obtained using the discrete normal distribution. The verification result shows whether or not the sample has properties that are inherent in the normal distribution. The application of the SAGA statistical tests to the NTRU cryptographic transformation polynomials allowed us to conclude that the discrete Gaussian sample makes it possible to generate time-resistant parameters using the norm or the length of the short basis (vector) of the lattice as the mean-square deviation.

A note on inconsistent families of discrete multivariate distributions

We construct a d-dimensional discrete multivariate distribution for which any proper subset of its components belongs to a specific family of distributions. However, the joint d-dimensional distribution fails to belong to that family and in other words, it is ‘inconsistent’ with the distribution of these subsets. We also address preservation of this ‘inconsistency’ property for the symmetric Binomial distribution, and some discrete distributions arising from the multivariate discrete normal distribution.

Nonintersecting Brownian motions on the unit circle

We consider an ensemble of $n$ nonintersecting Brownian particles on the unit circle with diffusion parameter $n^{-1/2}$, which are conditioned to begin at the same point and to return to that point after time $T$, but otherwise not to intersect. There is a critical value of $T$ which separates the subcritical case, in which it is vanishingly unlikely that the particles wrap around the circle, and the supercritical case, in which particles may wrap around the circle. In this paper, we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as $n\to\infty$, and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution. We also give a streamlined approach to identifying the Pearcey and tacnode processes in scaling limits. The formula of the tacnode correlation kernel is new and involves a solution to a Lax system for the Painlev\'{e} II equation of size 2 $\times$ 2. The proofs are based on the determinantal structure of the ensemble, asymptotic results for the related system of discrete Gaussian orthogonal polynomials, and a formulation of the correlation kernel in terms of a double contour integral.

Sampling Exactly from the Normal Distribution

An algorithm for sampling exactly from the normal distribution is given. The algorithm reads some number of uniformly distributed random digits in a given base and generates an initial portion of the representation of a normal deviate in the same base. Thereafter, uniform random digits are copied directly into the representation of the normal deviate. Thus, in contrast to existing methods, it is possible to generate normal deviates exactly rounded to any precision with a mean cost that scales linearly in the precision. The method performs no extended precision arithmetic, calls no transcendental functions, and uses no floating point arithmetic whatsoever; it uses only simple integer operations. It can easily be adapted to sample exactly from the discrete normal distribution whose parameters are rational numbers.

The discrete asymmetric Laplace distribution

The three-parameter asymmetric Laplace distribution is modified by using the discretization method of Roy based on the survival function. The new modified distribution is called the discrete asymmetric Laplace distribution. It is the distribution over integers. We show some of its mathematical properties: the probability mass function, moment, the probability generating function, the characteristic function, the reliability measure, and the quantile function. Moreover, the estimation of parameters for this distribution based on maximum likelihood method is discussed. An application to the change of the stock price data is presented by comparing the proposed distribution with the discrete normal distribution.

On a family of q-binomial distributions

Abstract We introduce a family of q-analogues of the binomial distribution, which generalises the Stieltjes-Wigert-, Rogers-Szegö-, and Kemp-distribution. Basic properties of this family are provided and several convergence results involving the classical binomial, Poisson, discrete normal distribution, and a family of q-analogues of the Poisson distribution are established. These results generalize convergence properties of Kemp’s-distribution, and some of them are q-analogues of classical convergence properties.

A Skew Laplace Distribution on Integers

We propose a discrete version of the skew Laplace distribution. In contrast with the discrete normal distribution, here closed form expressions are available for the probability density function, the distribution function, the characteristic function, the mean, and the variance. We show that this distribution on integers shares many properties of the skew Laplace distribution on the real line, including unimodality, infinite divisibility, closure properties with respect to geometric compounding, and a maximum entropy property. We also discuss statistical issues of estimation under this model.

The Discrete Normal Distribution

The normal distribution has been playing a key role in stochastic modeling for a continuous setup. But its distribution function does not have an analytical form. Moreover, the distribution of a complex multicomponent system made of normal variates occasionally poses derivational difficulties. It may be worth exploring the possibility of developing a discrete version of the normal distribution so that the same can be used for modeling discrete data. Keeping in mind the above requirement we propose a discrete version of the continuous normal distribution. The Increasing Failure Rate property in the discrete setup has been ensured. Characterization results have also been made to establish a direct link between the discrete normal distribution and its continuous counterpart. The corresponding concept of a discrete approximator for the normal deviate has been suggested. An application of the discrete normal distributions for evaluating the reliability of complex systems has been elaborated as an alternative to simulation methods.

Discrete Normal distribution and its relationship with Jacobi Theta functions

We introduce new, natural parameters in a formula defining a family of discrete Normal distributions. One of the parameters is closely related to the expectation and the other to the variance of that family. We show that under such a parametrization, uniformly for all sufficiently large variances and all expectations, discrete Normal distributions and their first two moments are given by very simple formulae. We indicate the relation between our results and Jacobi Theta functions and Jacobi summation formulae.

Monotone empirical Bayes tests for a discrete normal distribution

Abstract We study the two-action problem in a discrete normal distribution via the empirical Bayes approach. An empirical Bayes test δ n ∗ is constructed based on an estimator a n ∗ of the critical point aG of the Bayes test. The empirical Bayes test δ n ∗ possesses the monotonicity and the asymptotic optimality, and its regret converges to zero at an exponential-type rate of order O(exp(−nτG)), where τG is a positive number, depending on the unknown prior distribution G.

Characterizations of a discrete normal distribution

The paper obtains a discrete analogue of the normal distribution as the distribution that is characterized by maximum entropy, specified mean and variance, and integer support on (− ∞, ∞). Two alternative characterizations are given, firstly as the distribution of the difference of two related Heine distributions, and secondly as a weighted distribution.

Letter: Cell level modelling of H.261 coded video sequences at different bit rates

AbstractA video source, with a low to medium activity, coded with an H.261 type video codec can be modelled at the frame level as an M state Markov chain. For cell level modelling, the generated cells within the frame are distributed using a modulated Poisson process with interarrivals of Gamma function. Sequences with scene cuts use an. additional state in the Markov chain to represent the scene cut bit rate. The multiplexed model of N homogeneous sources is represented by a linear Markov model of N × M states. The transition probabilities in the multiplexed model are approximated by a truncated discrete normal distribution bounded within ±N states. The proposed single and multiplexed models exhibit similar cell loss performance with the real coded data at both cell and burst scale levels. The multiplexed model is then generalised to represent the aggregate traffic of several homogeneous sources at different bit rates.

Growth models of cultures with two liquid phases. IV. Cell adsorption, drop size distribution, and batch growth.

AbstractIn this work, a mathematical model which can be used to describe butch growth in fermentations with two liquid phases present is developed for systems in which the growth limiting substrate is dissolved in the dispersed phase. The model takes into account the drop size distribution, the rate of adsorption of cells on the drop surface, the rate of desorption of cells from the drop surface, substrate transport between phases, phase equilibrium, and growth kinetics. The model also considers the effect, of coalescence and redispersion of oil drops in the system. It is assumed that the composition of the dispersed phase is such that substrate utilization from it causes little or no change in the interfacial area. A discrete uniform distribution and a discrete normal distribution which is obtained from an experimental distribution curve are used as drop size distributions. Simulation results are obtained for a wide range of parameter values using the IBM S/360 Continuous System Modeling Program.