A Hybrid Inversive Congruential Pseudorandom Number Generator with High Period
31 Jan 2021-European Journal of Pure and Applied Mathematics (New York Business Global LLC)-Vol. 14, Iss: 1, pp 1-18
TL;DR: In this paper, a hybrid inversive congruential generator (HICG) based on a second order recurrence using the inversive modulo M, a power of 2, was proposed.
Abstract: Though generating a sequence of pseudorandom numbers by linear methods (Lehmer generator) displays acceptable behavior under some conditions of the parameters, it also has undesirable features, which makes the sequence unusable for various stochastic simulations. An extension which showed promise for such applications is a generator obtained by using a first-order recurrence based upon the inversive modulo a prime or a prime power, called inversive congruential generator (ICG). A lot of work has been dedicated to investigate the periods (under some conditions of the parameters), the lattice test passing, discrepancy and other statistical properties of such a generator. Here, we propose a new method, which we call hybrid inversive congruential generator (HICG), based upon a second order recurrence using the inversive modulo M, a power of 2. We investigate the period of this pseudorandom numbers generator (PRNG) and give necessary and sufficient conditions for our PRNG to have periods M (thereby doubling the period of the classical ICG) and M/2 (matching the one of the ICG). Moreover, we show that the lattice test complexity for a binary sequence associated to (a full period) HICG is precisely M/2.
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01 Jan 2021
References
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01 Jan 2001
TL;DR: Here the authors haven’t even started the project yet, and already they’re forced to answer many questions: what will this thing be named, what directory will it be in, what type of module is it, how should it be compiled, and so on.
Abstract: Writers face the blank page, painters face the empty canvas, and programmers face the empty editor buffer. Perhaps it’s not literally empty—an IDE may want us to specify a few things first. Here we haven’t even started the project yet, and already we’re forced to answer many questions: what will this thing be named, what directory will it be in, what type of module is it, how should it be compiled, and so on.
6,547 citations
Book•
01 Jan 1992TL;DR: This chapter discusses Monte Carlo methods and Quasi-Monte Carlo methods for optimization, which are used for numerical integration, and their applications in random numbers and pseudorandom numbers.
Abstract: Preface 1. Monte Carlo methods and Quasi-Monte Carlo methods 2. Quasi-Monte Carlo methods for numerical integration 3. Low-discrepancy point sets and sequences 4. Nets and (t,s)-sequences 5. Lattice rules for numerical integration 6. Quasi- Monte Carlo methods for optimization 7. Random numbers and pseudorandom numbers 8. Nonlinear congruential pseudorandom numbers 9. Shift-Register pseudorandom numbers 10. Pseudorandom vector generation Appendix A. Finite fields and linear recurring sequences Appendix B. Continued fractions Bibliography Index.
3,815 citations
04 Oct 2019
TL;DR: A general algorithmic scheme for constructing polynomial-time deterministic algorithms that stretch a short secret random input into a long sequence of unpredictable pseudo-random bits is presented.
Abstract: Much effort has been devoted in the second half of this century to make precise the notion of Randomness. Let us informally recall one of these definitions due to Kolmogorov []. A sequence of bits A =all a2••.•• at is random if the length of the minimal program outputting A is at least k We remark that the above definition is highly non constructive and rules out the possibility of pseudo random number generators. Also. the length of a program, from a Complexity Theory point of view, is a rather unnatural measure. A more operative definition of Randomness should be pursued in the light of modern Complexity Theory.
1,216 citations
"A Hybrid Inversive Congruential Pse..." refers methods in this paper
...The PRNG {yn} derived from (5) is purely periodic with period M/2, regardless of the odd initial conditions y0, y1, and the set {y1, y2, . . . , yM 2 } = GM if and only if a ≡ 1 (mod 4), b ≡ 0 (mod M2 ) and c ≡ 2 (mod 4)....
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...Then, the PRNs {xn}, defined by (2), belong to the set HM = { 1 M , 3 M , . . . , M−1 M } ....
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...Also with the same parameters and 109 PRNs we ran the Dieharder Test which also showed promising results which are being visually displayed in Figure 2 (Here the 109 bits are partitioned into 31249999 32-bit numbers which are further converted to decimal numbers and then the Dieharder Test is run)....
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...In the spirit of the known Blum-Micali PRN [1], we define a pseudorandom binary sequence {zn} by zn = f(yn), where f : GM → {0, 1} is given by...
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...In the spirit of the known Blum-Micali PRN [1], we define a pseudorandom binary sequence {zn} by zn = f(yn), where f : GM → {0, 1} is given by f(x) = { 0 if x < M2 1 if x > M2 ....
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TL;DR: In this paper, the authors dealt exclusively with the problem of pseudorandom number generation in the interval [0, 1] and showed that for a large positive integer m and integers a, b, yo a linear congruential sequence (y, n),n-o of nonnegative integers less than m is defined by
Abstract: Uniform pseudorandom numbers in the interval [0, 1) are basic ingredients of any stochastic simulation. Their quality is of fundamental importance for the simulation outcome. The present paper deals exclusively with this problem. General background material on pseudorandom number generation can be found in the book of Knuth [21] and in the survey article of Niederreiter [26]. The classical standard method of generating uniform pseudorandom numbers in the interval [0, 1) is the linear congruential method. For a large positive integer m and integers a, b, yo a linear congruential sequence (y,),n-o of nonnegative integers less than m is defined by
89 citations
"A Hybrid Inversive Congruential Pse..." refers background in this paper
...The lattice test complexity is known for the inversive congruential generator (see [6, 19], for more on this generator)....
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TL;DR: A survey of recent work in the areas of uniform pseudorandom number and uniform pseudOrandom vector generation is presented and a progress report on the construction of quasirandom points for efficient multidimensional numerical integration is given.
Abstract: A survey of recent work in the areas of uniform pseudorandom number and uniform pseudorandom vector generation is presented. The emphasis is on methods for which a detailed theory is available. A progress report on the construction of quasirandom points for efficient multidimensional numerical integration is also given.
89 citations