Using linear congruential generators for parallel random number generation

Random number generators are integral components to Monte Carlo codes. They provide the pseudorandom number sequence used to actually sample the distributions of interest. As a result, they are one of the most important components to the software. The current recommended MCNP random number generator is a 63-bit linear congruential generator (LCG). This generator is quite fast, but it has some drawbacks. First, it only has a period of 2⁶³. Due to the necessarily non-optimal usage of random numbers to ensure parallel reproducibility, this amount is too few to guarantee random number sequences are not reused in all configurations the code runs under. As simulation size increases, users will need to be aware of the limitations of the generator and tune configuration variables to best suit their simulations, or they will need to assume that reuse is not negatively affecting their answers. Neither of these are optimal. Second, small LCGs are fairly weak in bit generation quality, and this can have an unknown impact on the quality of the simulation. This paper is an investigation into whether or not more modern random number generators can supersede the current ones. The goal is to find a generator that is similar or superior in speed to the LCGs, has a state space large enough to make strong guarantees about random number reuse, and passes all modern random number test suites. If such a generator is found, it would eliminate the need for the user to even be aware of the limitations of the random number generator and would simplify the use of the code. This paper will be broken into several parts. Sec. 2 will discuss the evolution of the random number generator within the MCNP code. Sec. 3 will go over what a Monte Carlo code needs from a generator to be reproducible and portable and how the current generator behaves in that light. Sec. 4 goes through how each generator was tested. Finally, Sec. 5 will discuss improvements that could be made to the code.

Many numerically intensive computations done in a scientific computing environment require uniformly distributed pseudorandom numbers in the range (0, 1) and (−1, 1). For multiplicative congruential generators with modulus 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> , k ≤ 52, and period 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k-2</sup> , we show that the cost per random number for these two distributions is 3 and 3.125 multiply–adds on RS/6000® processors. Our code, on the IBM POWER2 Model 590, produces more than 40 million uniformly distributed pseudorandom numbers per second for both ranges (0, 1) and (−1, 1). Additionally, our code sustains the 40 million per second rate for data out of cache. The Numerical Aerodynamic Simulation (NAS) parallel benchmarks use a linear congruential generator with modulus 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">46</sup> . Our result is about 50 times faster than the generic implementation given in the benchmarks. The extra-accuracy fused multiply-add instruction of RS/6000 machines combined with a few algorithmic innovations gives rise to the 50-fold increase. If IEEE 64-bit arithmetic is used with our Fortran code on POWER and PowerPC® architectures, the results we obtain are bit-wise identical to the generic algorithms. The paper gives several illustrations of a general technique called the Algorithm and Architecture approach. We demonstrate herein that programmer-controlled unrolling of loops is equivalent to “customized vectorization of RISC-type code.” Customized vectorization is more powerful than ordinary vectorization, and it is only possible on RISC-type machines. We illustrate its use to show that RS/6000 processors can compute the distribution (−1, 1) at the rate of 3.125 multiply–adds. We also specify a linear congruential generator that is related to the multiplicative congruential generator referred to above. It has a full period of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> , where 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> is the modulus. The cost per random number [in the range (0, 1)] for this generator is four multiply–adds on RS/6000 processors. Our code, on the IBM POWER2 Model 590, for this generator produces more than 30 million uniformly distributed pseudorandom numbers per second for the range (0, 1). We show that this generator is “embarrassingly parallel,” or EP. Using the Algorithm and Architecture approach, we describe a new concept called “generalized unrolling.” Finally, we present a multiplicative congruential generator for which the modulus is not a power of 2. Such a generator, as well as one with modulus 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> , is selectable as the generator used in the RANDOM_NUMBER intrinsic function of IBM XL Fortran and XL High Performance Fortran. All of the generators reported here are EP. Using an IBM SP2 machine with 250 wide nodes, it is possible to compute more than ten billion uniform random numbers in a second.

Parallel linear congruential generators with prime moduli

Parameterizing parallel multiplicative lagged-Fibonacci generators

Pseudo‐random numbers, uniformly distributed on the interval , are widely used in the development of statistical methods when analytic methods are intractable. The following random number generators are described: linear congruential generators, MT19937, multiple recursive generators (MRGs), some special classes of MRGs such as DX, DL, DS, and DT. The merits of these generators are discussed. Designs and implementations of parallel random number generators are also discussed.

This paper considers the problem of empirically analyzing the linear congruential generators (LCGs) with ten largest prime moduli smaller than 231. For each modulus, a computer exhaustive search is conducted to find the 20 good multipliers with respect to spectral value for the full period LCGs. Eleven two-level statistical tests are applied to evaluate and compare the local randomness behaviors of these good LCGs. It is shown that modulus does not significantly affect spectral value. The spectral value and the two-level statistical test are also uncorrelated.

Exchange of information through the medium of the internet allows information can be accessed by unauthorized parties. Techniques for securing information that is commonly used is Cryptography. This research will use a combination of two classic algorithms which are Columnar Transposition Cipher and a Caesar Cipher with a Linear Congruential Generator modification. Information in the form of images often has some of the same pixel value components. For saving the memory then the writers use the technique of compression with Lempel Ziv Welch algorithm, which is a dictionary-based compression algorithm. Stages of the process used in this study are that the pictures will be encrypted using the algorithm of Columnar Transposition Cipher, next step the file that has been encrypted will be encrypted with the Caesar Cipher algorithm using a combination of and Linear Congruential generators are modified. Then the image file that has been encrypted will be compressed using the second algorithm Lempel Ziv Welch. For the implementation of programming language C # systems used. The results of this study showed that the combination of Columnar Transposition Cipher and a Caesar Cipher with a Linear Congruential Generators could keep the confidentiality, integrity, and security of data. Lempel Ziv Welch algorithm produces the value of the average compression ratio (RC) 67.05%, in Compression Ratio (CR) aspect the value is 1.49, and in Space Saving (SS) parameter, the percentage is 32.93%.

This paper studies a life-cycle model in which the consumption good is assumed to be indivisible. This assumption requires the number of units of the good purchased in each period to be an integer. It is shown that, if the discount factor (β) is sufficiently small, the policy function takes the form of a pseudo-random number generator or, more precisely, a linear congruential generator. It is also shown that optimal plans are almost always asymptotically non-periodic regardless of the discount factor. Various numerical examples of the value function and the policy function are provided. JEL Classification Numbers: C61, D91, E32.

Recently, a genetic-based random key generator (GRKG) for the one-time pad (OTP) cryptosystem has been proposed in the literature which has certain limitations. In this paper, two main characteristics (speed and randomness) of the GRKG method are significantly improved by presenting the IGRKG method (improved genetic-based random key generator method). The proposed IGRKG method generates an initial pad by using linear congruential generator (LCG) and improves the randomness of the initial pad using genetic algorithm. There are three reasons behind the use of LCG: it is easy to implement, it can run efficiently on computer hardware, and it has good statistical properties. The experimental results show the superiority of the IGRKG over GRKG in terms of speed and randomness. Hereby we would like to mention that no prior experimental work has been presented in the literature which is directly related to the OTP key generation using evolutionary algorithms. Therefore, this work can be considered as a guideline for future research.

A Novel Approach for Image Steganography Based on Random LSB Insertion in Color Images

Mathematics is a subject studied from elementary education up to the university level. In general, learning mathematics uses a conventional approach using a whiteboard. This method has shortcomings in the process of visualization and simulation, especially in the material of geometry. This research develops a learning application by utilizing Augmented Reality technology with the content of geometry for sixth-grade primary school students. Augmented Reality can visualize and simulate geometry for students with an interaction model. Linear Congruential Generator (LCG) and Fuzzy Tsukamoto methods are used in this application as a randomization method and score determination method. The results obtained from the testing process show that the LCG and fuzzy Tsukamoto methods can run according to the design. In user testing using a questionnaire, the results obtained 4.6 of scale 5.0 indicating that respondents provided good responses for the Augmented Reality Geometry application.

The article is dedicated to a detailed review of pseudorandom number generators (PRNGs), their operating principles, features, advantages, limitations, and various implementation methods. Pseudorandom number generators have become an integral part of modern information security, leveraging advanced cryptographic algorithms to ensure reliability and resistance to attacks. Pseudorandom numbers are widely used in cryptography, simulation, computer games, and many other fields. The main goal of the article is to analyze various methods of generating pseudorandom numbers, their features, advantages, and disadvantages, as well as to study their impact on cryptographic strength. The article begins with the basic principles of PRNG operation, focusing on linear congruential generators (LCGs) as the simplest example. Although LCGs are easy to implement, they have significant limitations in the context of cryptographic security, such as short periods and relative predictability. For higher security requirements, cryptographically secure pseudorandom number generators (CSPRNGs) based on block ciphers, hash functions, or other complex mathematical algorithms are recommended. Additional attention is given to generators such as ChaCha20 and NORX, which demonstrate high speed and resistance to cryptanalytic attacks. Theoretical cryptographic strength is also analyzed for generators based on modified algorithms, such as BBS and BM, supported by the complexity of computational problems. In conclusion, the article emphasizes the critical importance of understanding the operation of various PRNGs for information security professionals, as the choice of generator can significantly impact the overall security level of a system. A comparison of the efficiency and cryptographic strength of different pseudorandom number generation methods is presented in a table, supplemented by program algorithms, their characteristics, and examples of their use in cryptography.

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.

Chapter 3 - Pseudorandom Number Generators

In brief, we describe how to use knowledge about the sequence period generated by the linear multiplicative congruential generator of pseudorandom number sequences to determine the sequence period generated by continuous piecewise-linear chaotic maps of the unit interval or by maps topologically conjugate to these maps. The method exploits a relation between a continuous piecewise-linear map and a piecewise-linear but noncontinuous map of the unit interval with uniformly distributed extrema points, as well as the fact that the latter map can be considered as the generalization of the map describing the linear multiplicative congruential generator.