Multiplicative Congruential Random Number Generators Research Articles

Implementasi Algoritma Multiplicative Congruential Random Number Generator Pada Aplikasi Seleksi Mojang Jajaka

Mojang Jajaka (MOKA) is one of icons of tourism ambassadors who come from most districts/cities in West Java. The selection of tourism ambassadors in Kuningan was organized by the Department of Youth, Sports and Tourism (DISPORAPAR) of Kuningan, and the Mojang Jajaka Association of Kuningan. One of the series of activities carried out is participant registration and written test selection. Currently the registration process and the written test selection are still carried out conventionally by collecting hardcopy files and writing tests using a sheet of paper. This is considered less effective and efficient because using paper often causes problems, namely tucked, damaged, and easily lost. In the written test activities, cheating often occurs when filling out answers. This study aims to apply the Multiplicative Congruential Random Number Generator Algorithm in randomizing the questions on the MOKA selection test. The selection process begins with participants registering on the application. Then the participant logs in and after that the admin sees the participant data and then verifies the participant data. After that the admin will do question management, where the admin will input 30 questions. Then the participants will carry out the exam process. The Multiplicative Congruential Random Number Generator algorithm will work to randomize the questions from 30 questions and 25 questions will be issued. This application is made using the Java programming language, PHP, Javascript and MySQL database. This study uses the RUP (Rational Unified Process) methodology which consists of 4 stages, namely Inception, Elaboration, Construction, and Transition. This study results a MOKA Selection Application that can simplify the registration process and written test selection for both participants and the Mojang Jajaka selection committee, Kuningan.

TERAPAN MCRNG SEBAGAI GENERATOR ACAK UNTUK GAME

Most game s are used as a means of entertainment, but can also function as a means of training, education and simulation. In the field of education, educational game s can be applied by adopting educational materials to be informed in an interesting way to users. To provide a challenge to the user, the concept of level and scoring is applied in the game . The type of game made is a quiz-based educational game . Elements of educational game s made in the form of text, images, sounds and videos regarding knowledge about the classification of vertebrate and invertebrate animals. The data and answers used in the quiz are also displayed in a random way. The randomization process uses a random generator that applies the concept of Multiplicative Congruential Random Number Generator (CRNG). The use of this generator is intended so that users get different questions and answer compositions at each level. This is so that the user does not get bored quickly and the game becomes more interesting to play. So that the user indirectly conducts learning activities because the information conveyed is in the form of knowledge that is packaged in the form of a game.

The construction of fast portable multiplicative congruential random number generators

The construction of fast portable multiplicative congruential random number generators

An exhaustive analysis of prime modulus multiplicative congruential random number generators with modulus smaller than 215

This paper presents the results of an exhaustive analysis of all of the prime modulus multiplicative congruential random number (RN) generators with moduli smaller than 215. In an amount of around 20 million multipliers which are able to produce a full period of RNs, 239 multipliers have a good lattice structure. Among which 52 multipliers further pass a comprehensive battery of empirical tests. These 52 multipliers thus possess good local and global statistical properties. It is worthwhile to note that some empirically tested multipliers recommended in some previous studies are not on this list. The conclusion is that both theoretical and empirical tests are mandatory to sieve out good multipliers. To generate RNs of very long period, many existing techniques can be applied without further effort.

Symmetry property of multiplicative congruential random number generator in chi-square test

The k-term prime modulus multiplicative congruential generator: Rn = (a1Rn-1 +...,+ akRn-k ) mod m, is able to produce numbers (RNs) of full period mk-1 when the multipliers a1....,ak are chosen properly. In testing uniformity, the full period of RNs is usually divided into segments to calculate the chi-square statistics of the segments and test subsequently whether these statistics conform to a chi-square distribution. A symmetry property is that if an even number of segments, say 2s, is divided, then the chi-square statistic calculated from the ith segment of the first s segments is the same as that of the ith segment of the last s segments. Based on this property, the computational effort usually needed in testing uniformity is reduced by half.

Decomposition with simulated division for efficiently generating random numbers

Most portable implementations of the prime modulus multiplicative congruential random number (RN) generator have restrictions on the multiplier. The decomposition method which decomposes a multiplier into a = a 0 + a 12 b 2 − 1 + a 22 b − 2 , where b is the size of the computer word, on the other hand, is applicable to all multipliers. With the special structure of the decomposition, the technique of simulated division is considered in this paper to facilitate the computation for RN generators with modulus 2 31 − 1. Some empirical speed comparisons reveal that the reduction in the execution time is around 1 4 to 1 3 for various computers.

Several extensively tested random number generators

In searching for statistically sound multipliers for the prime modulus multiplicative congruential random number generator, both theoretical and empirical tests are conducted extensively over one hundred thousand primitive roots of the ten largest primes in 2 15. Sixty-two multipliers pass all the tests to possess good global and local statistical properties. It is worthwhile to note that the empirically tested multipliers recommended in some previous studies are not on this list. The conclusion is that both theoretical and empirical tests are mandatory to sieve out good multipliers.

Some good multipliers for random number generator for 16-bit microcomputers

In searching for statistically sound multipliers for the prime modulus multiplicative congruential random number generator for 16-bit microcomputers, an exhaustive analysis is conducted over the 10,912 primitive roots of the modulus 2 15 − 19 = 32,749. The result shows that there are only 227 good multipliers among the 10,912 candidate multipliers. Here a multiplier is said to be good if its corresponding generator has good statistical properties and computational efficiency.

Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators

The work is concerned with the first-order linear autoregressive process which has a rectangular stationary marginal distribution. A derivation is given of the result that the time-reversed version is deterministic, with a first-order recursion function of the type used in multiplicative congruential random number generators, scaled to the unit interval. The uniformly distributed sequence generated is chaotic, giving an instance of a chaotic process which when reversed has a linear causal and non-chaotic structure. An mk-valued discrete process is then introduced which resembles a first-order linear autoregressive model and uses k-adic arithmetic. It is a particular form of moving-average process, and when reversed approximates in m a non-linear discrete-valued process which has the congruential generator function as its deterministic part, plus a discrete-valued noise component. The process is illustrated by scatter plots of adjacent values, time series plots and directed scatter plots (phase diagrams). The behaviour very much depends on the adic number, with k = 2 being very distinctly non-linear and k = 10 being virtually indistinguishable from independence.

Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators

The work is concerned with the first-order linear autoregressive process which has a rectangular stationary marginal distribution. A derivation is given of the result that the time-reversed version is deterministic, with a first-order recursion function of the type used in multiplicative congruential random number generators, scaled to the unit interval. The uniformly distributed sequence generated is chaotic, giving an instance of a chaotic process which when reversed has a linear causal and non-chaotic structure. An mk -valued discrete process is then introduced which resembles a first-order linear autoregressive model and uses k-adic arithmetic. It is a particular form of moving-average process, and when reversed approximates in m a non-linear discrete-valued process which has the congruential generator function as its deterministic part, plus a discrete-valued noise component. The process is illustrated by scatter plots of adjacent values, time series plots and directed scatter plots (phase diagrams). The behaviour very much depends on the adic number, with k = 2 being very distinctly non-linear and k = 10 being virtually indistinguishable from independence.

Discussion on the Meeting on Chaos

Discussion on the Meeting on Chaos

Multiplicative congruential random number generators with modulus 2^{𝛽}: an exhaustive analysis for 𝛽=32 and a partial analysis for 𝛽=48

This paper presents the results of a search to find optimal maximal period multipliers for multiplicative congruential random number generators with moduli 2 32 {2^{32}} and 2 48 {2^{48}} . Here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which k-tuples lie does not exceed the minimal achievable distance by more than 25 percent for k = 2 , … , 6 k = 2, \ldots ,6 . This criterion is considerably more stringent than prevailing standards of acceptability and leads to a total of only 132 multipliers out of the more than 536 million candidate multipliers that exist for modulus 2 32 {2^{32}} and to only 42 multipliers in a sample of about 67.1 million tested among the more than 351 × 10 11 351 \times {10^{11}} candidate multipliers for modulus 2 48 {2^{48}} . Section 1 reviews the basic properties of multiplicative congruential generators and § 2 \S 2 describes worst case performance measures. These include the maximal distance between adjacent parallel hyperplanes, the minimal number of parallel hyperplanes, the minimal distance between k-tuples and the discrepancy. For modulus 2 32 {2^{32}} , § 3 \S 3 presents the ten best multipliers and compares their performances with those of two multipliers that have been recommended in the literature. Comparisons using packing measures in the space of k-tuples and in the dual space are also made. For modulus 2 48 {2^{48}} , § 4 \S 4 also presents analogous results for the five best multipliers and for two multipliers suggested in the literature.

Multiplicative Congruential Random Number Generators with Modulus 2 β : An Exhaustive Analysis for β = 32 and a Partial Analysis for β = 48

Multiplicative Congruential Random Number Generators with Modulus 2 β : An Exhaustive Analysis for β = 32 and a Partial Analysis for β = 48

An Exhaustive Analysis of Multiplicative Congruential Random Number Generators with Modulus $2^{31} - 1$

This paper presents the results of an exhaustive search to find optimal full period multipliers for the multiplicative congruential random number generator with prime modulus $2^{31} - 1$. Here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which k-tuples lie does not exceed the minimal achievable distance by more than 25 percent for $k = 2, \cdots ,6$. This criterion is considerably more stringent than prevailing standards of acceptability and leads to a total of only 414 multipliers among the more than 534 million candidate multipliers. Section 1 reviews the basic properties of linear congruential generators and § 2 describes worst case performance measures. These include the maximal distance between adjacent parallel hyperplanes, the minimal number of parallel hyperplanes, the minimal distance between k-tuples, the lattice ratio and the discrepancy. Section 3 presents the five best multipliers and compares their performances with those of three commonly employed multipliers for all measures but the lattice test. Comparisons using packing measures in the space of k-tuples and in the dual space are also made. Section 4 presents the results of applying a battery of statistical tests to the best five to detect local departures from randomness. None were found. The Appendix contains a list of all optimal multipliers.

A Statistical Evaluation of Multiplicative Congruential Random Number Generators with Modulus 231— 1

Abstract This article presents the results of empirically testing 16 alternative multipliers for a multiplicative congruential random number generator with modulus 231 — 1. Two of the multipliers are in common use, six are the best of 50 candidate multipliers according to the theoretical spectral and lattice tests, and eight are the worst, with regard to 2-tuples, among the 50. The test results raise serious doubts about several of the multipliers, including one in common use. The tests were also applied to a well-known theoretically poor generator, RANDU, and gave strong empirical evidence of its inadequacy. Since comparison of the results of the First eight multipliers with those for the eight worst multipliers failed to show any apparent gross differences, one may want to relax the currently employed stringent criteria for acceptable performance on the lattice and spectral tests.

A Statistical Evaluation of Multiplicative Congruential Random Number Generators with Modulus 2 31 - 1

A Statistical Evaluation of Multiplicative Congruential Random Number Generators with Modulus 2 31 - 1

Autocorrelation Peaks in Congruential Pseudorandom Number Generators

One of the desirable properties of a pseudorandom number generator is that the sequence of numbers it generates should have very low autocorrelation for all shifts except for zero shift and those that are multiples of its cycle length. Due to the simple methods of constructing random numbers via modulo arithmetic, the ideal is often not quite fulfilled. The results of this paper were obtained by the simple method of examining the complete correlation structure of several generators of the same type and small cycle length. Once the regularities were discovered, the mathematical relationships were derived which describe the regular behavior for all generators of the same class. As examples, it is shown in this paper that high correlation exists in mixed and multiplicative congruential random number generators and prime moduli Lehmer generators for shifts a fraction of their cycle lengths.

Random number generators in cyclic queuing applications

Nine multiplicative congruential, shift register and composite random number generators are empirically compared on an IBM 360/65 computer. The basis for the comparison is the smulation of a Markov process which can also be analytically described. Any deviation of the simulation results from the theoretical indicates a non-random behaviour of the generator. The results, including the relative speeds of the generators, are presented and discussed