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In typical stochastic simulations, randomness is produced by generating a sequence of independent uniform variates (usually real-valued between 0 and 1, or integer-valued in some interval) and transforming them in an appropriate way. In this paper, we examine practical ways of generating (deterministic approximations to) such uniform variates on a computer. We compare them in terms of ease of implementation, efficiency, theoretical support, and statistical robustness. We look in particular at several classes of generators, such as linear congruential, multiple recursive, digital multistep, Tausworthe, lagged-Fibonacci, generalized feedback shift register, matrix, linear congruential over fields of formal series, and combined generators, and show how all of them can be analyzed in terms of their lattice structure. We also mention other classes of generators, like non-linear generators, discuss other kinds of theoretical and empirical statistical tests, and give a bibliographic survey of recent papers on the subject.

Content uploaded by Pierre L’Ecuyer

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... Example 15. In a box there are two coins, the first is fair and the second is biased so that p(H) = 1. ...

... Example 15. Let X and Y be two d. ...

... Definition 1. [15] The pseudo-random number generator (PRNG) is a structure (S, P 0 , f, U, g) such that: -S : a finite set of states (states space) s 0 , s 1 , · · · , s n -P 0 : probability distribution on S to select the initial state s 0 (seed) f : transition function f : S → S -U : space of generated numbers (output) (often [0, 1]) g : output function g : S → U The elements of this structure make it possible to perform the steps of random numbers generation, which are : 1-Select s 0 using P 0 , then generate the first random number u 0 = g(s 0 ) 2-At each step i ≥ 1, the transition function changes the state of the generator and then the output function gives the random number associated with the new state. ...

... The problem faced in the design of a noise generator system is how to generate a random signal that has a certain distribution (uniform or gaussian) efficiently. From the literature search, the authors get a lot of literature that writes about methods of generating random signals either purely (pure random) or artificial (pseudo random) [3]. ...

... The artificial noise generation method generally uses a mathematical algorithm (to generate a series of random numbers) which is then converted into an analog signal using a DAC (Digital to Analog Converter). The main weakness of the noise generation method using this mathematical algorithm is the appearance of periodization in the resulting signal [3][4] [5]. There is a lot of literature that focuses its research on developing algorithms to generate a series of random numbers close to the characteristics of pure random numbers. ...

... From Figure 1, it can be seen that to generate Gaussian distributed random numbers using the Box-Muller method, a uniformly distributed random number is required. To generate random numbers with uniform distribution, the Lehmer method [3][10] can be used. Mathematically, the Lehmer method for generating random numbers can be written as: ...

The random noise signal is widely used as a test signal to identify a physical or biological system. In particular, the Gaussian distributed white noise signal (Gaussian White Noise) is popularly used to simulate environmental noise in telecommunications system testing, input noise in testing ADC (Analog to Digital Converter) devices as well as testing other digital systems. Random noise signal generation can be done using resistors or diodes. The weakness of the noise generator system using physical components is the statistical distribution. An alternative solution is to use a Pseudo-Random System that can be adjusted for distribution and other statistical parameters. In this study, the implementation of the Gaussian distributed pseudo noise generation algorithm based on the Enhanced Box-Muller method is described. Prototype of noise generation system using a minimum system board based on Cortex Microcontroller or MCU-STM32F4. From the test results, it was found that the Enhanced Box-Muller (E Box-Muller) method can be applied to the MCU-STM32F4 efficiently, producing signal noise with Gaussian distribution. The resulting noise signal has an amplitude of ±1Volt, is Gaussian distributed and has a relatively wide frequency spectrum. The noise signal can be used as a jamming device in a certain frequency band using an Analog modulator.

... Définition 1. [15] Le générateur des nombres pseudo-aleatoires (PRNG) est la structure (S, P 0 , f, U, g) tel que : -S : un ensemble fini d'états (Espace d'états) s 0 , s 1 , · · · , s n -P 0 : distribution de probabilité sur S pour séléctioner l'état initial s 0 (seed) f : fonction de transition f : S → S -U : espace des nombres générés (sortie) (souvent [0, 1]) g : fonction des sorties g : S → U Les éléments de cette structure permettent de réaliser les étapes de la génération des nombres aléatoires qui sont : 1-Sélectionner s 0 en utilisant P 0 , puis générer le premier nombre aléatoire u 0 = g(s 0 ) 2-À chaque étape i ≥ 1, la fonction de transition change l'état du générateur et ensuite la fonction des sorties donne le nombre aléatoire associé au nouvel état. ...

... Output ______________________________________ # Expected value: 3.9628993353781383 # Variance: 15.587130335771537 # Standard deviation: 3.9480539935228265 # Before normalization. ...

... Considering a vector of elements and the quicksort algorithm [53], sorting is of average time complexity log( ). Meanwhile, the random sequences used in the proposed algorithm are derived using a PRNG, the complexity cost of which can be computed by considering the well-known linear congruential generator (LCG) as a uniform random number generator [54]. The operations related to (15)-(18) represent the complexity of exchanging the sum of image pixels . ...

The attractive properties of ergodicity, unpredictability, and initial state sensitivity have made chaotic maps the go-to tools in many applications, including cryptography and cyber–physical systems. Despite this, two challenges arise when using chaos systems in cryptography: (i) some one-dimensional (1D) chaotic maps do not satisfy the unpredictability property, and (ii) although they exhibit a complex and chaotic behavior, high-dimensional (HD) chaotic maps incur higher computational complexity. To address these issues, this paper proposes a new 1D chaotic map dubbed the improved Sine-Tangent map (IST map) that is derived from of the Sine map, a tangent function, and a Chebyshev polynomial of the first kind. Relative to chaotic maps, the proposed IST map provides better unpredictability and ergodicity, a vast chaotic range, enhanced complex behavior, and competitive computational complexity. Based on the IST map, we also introduced an encryption scheme for securing medical images in telemedicine. It consists of two diffusion phases, i.e., a bitwise XOR operation and a bitwise expanded XOR (eXOR) operation, with an intermediated confusion one, i.e., random circular-shift. An overriding step is foremost first completed before performing these cryptography phases, i.e., key generation. The secret key of the IST map is updated using the sine and cosine values of the sum of pixels of the input image. This leads to a unique secret key for each image. That is, one-time chaotic sequences are produced for each input image. The cyclic pattern of the sine and cosine values of the sum of pixels provides a prominent sensibility to small changes in the input image. Thus, the proposed algorithm is capable of resisting any chosen/known plaintext attacks. A performance analysis shows that the proposed algorithm outperforms a set of state-of-the-art comparison algorithms and its variants based on Sine, SE, and ST maps since it allows the best performance/complexity trade-off.

In this paper we present, using the arithmetic of elliptic curves over finite fields, an algorithm for the efficient generation of a sequence of uniform pseudorandom vectors in high dimensions, that simulates a sample of a sequence of i.i.d. random variables, with values in the hypercube [0,1]d with uniform distribution. As an application, we obtain, in the discrete time simulation, an efficient algorithm to simulate, uniformly distributed sample path sequence of a sequence of independent standard Wiener processes. This could be employed for use, in the full history recursive multi-level Picard approximation method, for numerically solving the class of semilinear parabolic partial differential equations of the Kolmogorov type.

The Uniform distribution on the interval (0, 1) plays an important role in many statistical applications, such as, its role in simulation procedures when a sequence of random numbers needs to be generated from some parent population. This paper investigates the sampling distribution of the sample mean for random samples drawn from a uniform (0, 1) population. A complete derivation of the probability density function of the sample mean is presented. The normal approximation to the series form of the probability density function of the sample mean is also discussed. To avoid the use of the complicated series form of the probability density function of the sample mean for small sample sizes when the normal approximation is not advisable, the tables of the cumulative distribution function for sample sizes 2, 3, 4 and 5 are constructed. The Minitab statistical software is used throughout this paper.

We reverse-engineer, test and analyse hardware and firmware of the commercial quantum-optical random number generator Quantis from ID Quantique. We show that $>99\%$ > 99 % of its output data originates in physically random processes: random timing of photon absorption in a semiconductor material, and random growth of avalanche owing to impact ionisation. Under a strong assumption that these processes correspond to a measurement of an initially pure state of the components, our analysis implies the unpredictability of the generated randomness. We have also found minor non-random contributions from imperfections in detector electronics and an internal processing algorithm, specific to this particular device. Our work shows that the design quality of a commercial quantum-optical randomness source can be verified without cooperation of the manufacturer and without access to the engineering documentation.

Unemployment impairs individuals’ well-being and health and there is some empirical evidence showing that macroeconomic conditions can moderate these effects. This paper goes a step further and investigates differences in how macroeconomic indicators of European countries’ economic situation relate to individual subjective health and well-being, and also moderate the relationship between individual labour market exclusion and these outcomes across age groups: young individuals (aged 15–29), prime working age adults (aged 30–49, base category) and pre-retirement age adults (aged 50–64). We used two different macroeconomic indicators to define macroeconomic situation: country-level unemployment rate and gross domestic product (GDP). Both indicators were disaggregated into long-term economic trend and business cycle shocks using Hodrick–Prescott filtering to allow distinguishing between expected and unexpected change in macroeconomic circumstances. We used the European Social Survey individual-level data from 35 European countries for 2002–2014. Multi-level analysis with three levels were run for men and women separately. Results revealed differences in how individual-level unemployment related to well-being depending on the age group, with pre-retirement age group adults’ health and well-being suffering the most. Also, macroeconomic indicators were found to moderate the relationship between individual-level unemployment and subjective health and well-being with some noticeable differences between age groups, and with GDP trend having the most sizeable influence.

Fibonacci polynomials are defined in the context of the two-dimensional discrepancy of Tausworthe pseudorandom sequences as an analogue to Fibonacci numbers, which give the best figure of merit for the two-dimensional discrepancy of linear congruential sequences. We conduct an exhaustive search for the Fibonacci polynomials of degree less than 32 whose associated Tausworthe sequences can be easily implemented and very quickly generated.

We analyze a class of combined random number generators recently proposed by L'Ecuyer, which combines a set of linear congruential generators (LCG's) with distinct prime moduli. We show that the geometrical behavior of the vectors of points produced by the combined generator can be approximated by the lattice structure of an associated LCG, whose modulus is the product of the moduli of the individual components. The approximation is good if these individual moduli are near each other and if the dimension of the vectors is large enough. The associated LCG is also exactly equivalent to a slightly different combined generator of the form suggested by Wichmann and Hill. We give illustrations, for which we examine the approximation error and assess the quality of the lattice structure of the associated LCG.

We analyze the lattice structure of certain types of linear congruential generators (LCGs), which include close approximations to the add-withcarry and subtract-with-borrow (AWC/SWB) random number generators introduced by Marsaglia and Zaman, and also to combinations of the latter with ordinary LCGs. It follows from our results that all these generators have an unfavorable lattice structure in large dimensions.

Up to now, the rectangle discrepancy of linear congruential pseudorandom number generators could be exactly calculated only in some simple cases for a small number of generated points. Here an algorithm for the exact determination of the twodimensional rectangle discrepancy is presented which is practicable for large generators and requires less computation time. The algorithm is based on special properties of linear congruential generators.