Random Number Generation

Introduction Random numbers play a large and significant role in several fields and applications from cryptography to gaming to gambling. As a result, random number generators (RNGs) occupy a similarly important role. By their very nature, however, random numbers are difficult to fabricate, especially in the large quantities that they are often needed. In this paper I will discuss various topics related to random number generation. First, methods of generating “random” numbers will be explained, comparing and contrasting the two types: True Random Number Generators (TRNGs) and Pseudo-Random Number Generators (PRNGs). Methods for testing these generators will be discussed and 3 of the tests will be used to compare the output of several RNGs. Finally, attacks on random number generators will be explored, including examples of combating these attacks.

[1]  Pierre L'Ecuyer,et al.  Random numbers for simulation , 1990, CACM.

[2]  P. L'Ecuyer,et al.  About polynomial-time “unpredictable” generators , 1989, WSC '89.

[3]  R. Bailey.  Polar generation of random variates with the t -distribution , 1994 .

[4]  Pierre L'Ecuyer,et al.  Combined Multiple Recursive Random Number Generators , 1995, Oper. Res..

[5]  J. Ziv.  Compression, tests for randomness and estimating the statistical model of an individual sequence , 1990 .

[6]  P. L’Ecuyer,et al.  Structural properties for two classes of combined random number generators , 1990 .

[7]  Makoto Matsumoto,et al.  Strong deviations from randomness in m-sequences based on trinomials , 1996, TOMC.

[8]  Lih-Yuan Deng,et al.  Generation of Uniform Variates from Several Nearly Uniformly Distributed Variables , 1990 .

[9]  Anna Gavling,et al.  The ART at , 2008 .

[10]  Pei-Chi Wu,et al.  Multiplicative, congruential random-number generators with multiplier ± 2k1 ± 2k2 and modulus 2p - 1 , 1997, TOMS.

[11]  Bruce W. Schmeiser,et al.  Random variate generation , 1981, WSC '81.

[12]  Scott R. Fluhrer,et al.  Statistical Analysis of the Alleged RC4 Keystream Generator , 2000, FSE.

[13]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[14]  Pierre L'Ecuyer,et al.  Recent Advances in Randomized Quasi-Monte Carlo Methods , 2002 .

[15]  Peter Hellekalek,et al.  Empirical evidence concerning AES , 2003, TOMC.

[16]  Andreas Reuter,et al.  Transaction Processing: Concepts and Techniques , 1992 .

[17]  Paul Bratley,et al.  A guide to simulation , 1983 .

[18]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[19]  Makoto Matsumoto,et al.  Getting rid of correlations among pseudorandom numbers: discarding versus tempering , 1999, TOMC.

[20]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[21]  W. Hörmann,et al.  Automatic random variate generation for simulation input , 2000, 2000 Winter Simulation Conference Proceedings (Cat. No.00CH37165).

[22]  Pierre L'Ecuyer,et al.  Maximally equidistributed combined Tausworthe generators , 1996, Math. Comput..

[23]  Pierre L'Ecuyer,et al.  An Implementation of the Lattice and Spectral Tests for Multiple Recursive Linear Random Number Generators , 1997, INFORMS J. Comput..

[24]  Sibylle Mund,et al.  Liv-Lempel Complexity for Periodic Sequences and its Cryptographic Application , 1991, EUROCRYPT.

[25]  G. Marsaglia,et al.  A New Class of Random Number Generators , 1991 .

[26]  Albert J. Kinderman,et al.  Computer Generation of Random Variables Using the Ratio of Uniform Deviates , 1977, TOMS.

[27]  Peter Wright,et al.  Formula 1 Technology , 2001 .

[28]  Pierre L'Ecuyer,et al.  Tables of linear congruential generators of different sizes and good lattice structure , 1999, Math. Comput..

[29]  Pierre L'Ecuyer,et al.  A random number generator based on the combination of four LCGs , 1997 .

[30]  Voratas Kachitvichyanukul,et al.  Binomial random variate generation , 1988, CACM.

[31]  Pierre L'Ecuyer,et al.  Implementing a random number package with splitting facilities , 1991, TOMS.

[32]  Jovan Dj. Golic,et al.  Linear Statistical Weakness of Alleged RC4 Keystream Generator , 1997, EUROCRYPT.

[33]  Averill M. Law,et al.  Simulation Modeling and Analysis , 1982 .

[34]  Stefan Wegenkittl,et al.  A survey of quadratic and inversive congruential pseudorandom numbers , 1998 .

[35]  Pierre L'Ecuyer,et al.  Beware of linear congruential generators with multipliers of the form a = ±2q ±2r , 1999, TOMS.

[36]  Pierre L'Ecuyer,et al.  Bad Lattice Structures for Vectors of Nonsuccessive Values Produced by Some Linear Recurrences , 1997, INFORMS J. Comput..

[37]  J. P. R. Tootill,et al.  An Asymptotically Random Tausworthe Sequence , 1973, JACM.

[38]  Barry L. Nelson,et al.  Input modeling tools for complex problems , 1998, 1998 Winter Simulation Conference. Proceedings (Cat. No.98CH36274).

[39]  Elaine B. Barker,et al.  A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications , 2000 .

[40]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[41]  Philip Zimmermann,et al.  PGP source code and internals , 1995 .

[42]  Wolfgang Hörmann,et al.  Variants of Transformed Density Rejection and Correlation Induction , 2002 .

[43]  U. Fincke,et al.  Improved methods for calculating vectors of short length in a lattice , 1985 .

[44]  Jean-Sébastien Coron,et al.  An Accurate Evaluation of Maurer's Universal Test , 1998, Selected Areas in Cryptography.

[45]  Pierre L'Ecuyer,et al.  On the Deng-Lin random number generators and related methods , 2004, Stat. Comput..

[46]  Takuji Nishimura,et al.  Tables of 64-bit Mersenne twisters , 2000, TOMC.

[47]  A. J. Walker.  New fast method for generating discrete random numbers with arbitrary frequency distributions , 1974 .

[48]  Pierre L'Ecuyer,et al.  On the performance of birthday spacings tests with certain families of random number generators , 2001 .

[49]  Lih-Yuan Deng,et al.  A system of high-dimensional, efficient, long-cycle and portable uniform random number generators , 2003, TOMC.

[50]  Pierre L'Ecuyer,et al.  Randomized Polynomial Lattice Rules for Multivariate Integration and Simulation , 2001, SIAM J. Sci. Comput..

[51]  Tatu Ylonen,et al.  SSH: secure login connections over the internet , 1996 .

[52]  Joachim H. Ahrens,et al.  Computer Generation of Poisson Deviates from Modified Normal Distributions , 1982, TOMS.

[53]  Pierre L'Ecuyer,et al.  Random Number Generators Based on Linear Recurrences in F 2 w , 2003 .

[54]  Pierre L'Ecuyer,et al.  Random Number Generators: Selection Criteria and Testing , 1998 .

[55]  Tim B. Swartz,et al.  Approximating Integrals Via Monte Carlo and Deterministic Methods , 2000 .

[56]  Philip A. Bernstein,et al.  Principles of Transaction Processing , 1996 .

[57]  Pierre L'Ecuyer,et al.  An Object-Oriented Random-Number Package with Many Long Streams and Substreams , 2002, Oper. Res..

[58]  Su. Centre-Ville,et al.  Construction of Equidistributed Generators based on linear recurrences modulo 2 , 2000 .

[59]  Ronald L. Wasserstein,et al.  Monte Carlo: Concepts, Algorithms, and Applications , 1997 .

[60]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[61]  Pierre L'Ecuyer,et al.  Sparse Serial Tests of Uniformity for Random Number Generators , 1998, SIAM J. Sci. Comput..

[62]  Ueli Maurer,et al.  A universal statistical test for random bit generators , 1990, Journal of Cryptology.

[63]  Adi Shamir,et al.  A Practical Attack on Broadcast RC4 , 2001, FSE.

[64]  Mark Goresky,et al.  Efficient multiply-with-carry random number generators with maximal period , 2003, TOMC.

[65]  Makoto Matsumoto,et al.  Twisted GFSR generators , 1992, TOMC.

[66]  P. Hellekalek,et al.  Random and Quasi-Random Point Sets , 1998 .

[67]  Bruce Schneier,et al.  Yarrow-160: Notes on the Design and Analysis of the Yarrow Cryptographic Pseudorandom Number Generator , 1999, Selected Areas in Cryptography.

[68]  Abraham Lempel,et al.  A universal algorithm for sequential data compression , 1977, IEEE Trans. Inf. Theory.

[69]  Donald Ervin Knuth,et al.  The Art of Computer Programming, Volume II: Seminumerical Algorithms , 1970 .

[70]  H. Solomon,et al.  On Combining Pseudorandom Number Generators , 1979 .

[71]  T. Ala‐Nissila,et al.  Physical models as tests of randomness. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[72]  James R. Wilson,et al.  Using Univariate Bezier Distributions to Model Simulation Input Processes , 1996, Proceedings of 1993 Winter Simulation Conference - (WSC '93).

[73]  Alastair J. Walker,et al.  An Efficient Method for Generating Discrete Random Variables with General Distributions , 1977, TOMS.

[74]  Voratas Kachitvichyanukul,et al.  Computer generation of hypergeometric random variates , 1985 .

[75]  G. Casella,et al.  Explaining the Gibbs Sampler , 1992 .

[76]  Aaldert Compagner,et al.  On the use of reducible polynomials as random number generators , 1993 .

[77]  Michael Luby,et al.  Pseudorandomness and cryptographic applications , 1996, Princeton computer science notes.

[78]  Dennis K. J. Lin,et al.  Random Number Generation for the New Century , 2000 .

[79]  L'EcuyerPierre,et al.  Beware of linear congruential generators with multipliers of the form a = 2q 2r , 1999 .

[80]  S. Tezuka.  Uniform Random Numbers: Theory and Practice , 1995 .

[81]  Pierre L'Ecuyer,et al.  Lattice computations for random numbers , 2000, Math. Comput..

[82]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[83]  Russell C. H. Cheng.  Generating beta variates with nonintegral shape parameters , 1978, CACM.

[84]  Ramarathnam Venkatesan,et al.  Design of practical and provably good random number generators , 1995, SODA '95.

[85]  Pierre L'Ecuyer,et al.  Software for uniform random number generation: distinguishing the good and the bad , 2001, Proceeding of the 2001 Winter Simulation Conference (Cat. No.01CH37304).

[86]  Pierre L'Ecuyer,et al.  Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators , 1999, Oper. Res..

[87]  P. L’Ecuyer,et al.  Variance Reduction via Lattice Rules , 1999 .

[88]  Pierre L’Ecuyer,et al.  Polynomial Integration Lattices , 2004 .

[89]  Josef Leydold,et al.  Automatic sampling with the ratio-of-uniforms method , 2000, TOMS.

[90]  Ross Ihaka,et al.  Cryptographic Randomness from Air Turbulence in Disk Drives , 1994, CRYPTO.

[91]  Jiirgen Eichenauer-Herrmann,et al.  Pseudorandom Number Generation by Nonlinear Methods , 1995 .

[92]  Bruce Schneier,et al.  Cryptanalytic Attacks on Pseudorandom Number Generators , 1998, FSE.

[93]  P. L’Ecuyer,et al.  On the lattice structure of certain linear congruential sequences related to AWC/SWB generators , 1994 .

[94]  Pascal Junod,et al.  Cryptographic Secure Pseudo-Random Bits Generation : The Blum-Blum-Shub Generator , 1999 .

[95]  Oded Goldreich,et al.  Foundations of Cryptography (Fragments of a Book) , 1995 .

[96]  R. R. Hocking,et al.  Algorithm AS 53: Wishart Variate Generator , 1972 .

[97]  Huifen Chen,et al.  On Generating Random Variates from an Empirical Distribution , 1974 .

[98]  H. Niederreiter.  The Multiple-Recursive Matrix Method for Pseudorandom Number Generation , 1995 .

[99]  Pierre L'Ecuyer,et al.  Efficient and portable combined Tausworthe random number generators , 1990, TOMC.

[100]  Ian Goldberg,et al.  Randomness and the Netscape browser , 1996 .

[101]  M. Luescher,et al.  A Portable High-quality Random Number Generator for Lattice Field Theory Simulations , 1993 .

[102]  Russell C. H. Cheng,et al.  Some Simple Gamma Variate Generators , 1979 .

[103]  John Marsaglia,et al.  Rapid evaluation of the inverse of the normal distribution function , 1994 .

[104]  Masanori Fushimi.  Increasing the Orders of Equidistribution of the Leading Bits of the Tausworthe Sequence , 1983, Inf. Process. Lett..

[105]  Pierre L'Ecuyer,et al.  Orbits and lattices for linear random number generators with composite moduli , 1996, Math. Comput..

[106]  Robert W. Baldwin.  Preliminary Analysis of the BSAFE 3.x Pseudorandom Number Generators , 1998 .

[107]  Karl Entacher,et al.  A Collection of Selected Pseudorandom Number Generators With Linear Structures , 1997 .

[108]  Chris Lomont,et al.  Random Number Generation , 2008 .

[109]  Pierre L'Ecuyer,et al.  Tables of maximally equidistributed combined LFSR generators , 1999, Math. Comput..

[110]  R. Tausworthe.  Random Numbers Generated by Linear Recurrence Modulo Two , 1965 .

[111]  Simon W. Moore,et al.  The Frequency Injection Attack on Ring-Oscillator-Based True Random Number Generators , 2009, CHES.

[112]  A. C. Atkinson.  A family of switching algorithms for the computer generation of beta random variables , 1979 .

[113]  권태경,et al.  SSL Protocol 기반의 서버인증 , 2003 .

[114]  Pierre L'Ecuyer,et al.  Combined generators with components from different families , 2003, Math. Comput. Simul..

[115]  Thomas P. Turiel.  Quantum Random Bit Generators , 2007 .

[116]  Bruce Jay Collings,et al.  Compound Random Number Generators , 1987 .

[117]  Berk Sunar,et al.  A Provably Secure True Random Number Generator with Built-In Tolerance to Active Attacks , 2007, IEEE Transactions on Computers.

[118]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[119]  Pierre L'Ecuyer,et al.  Fast combined multiple recursive generators with multipliers of the form a = ±2q ±2r , 2000, WSC '00.

[120]  Andre Osterloh,et al.  A True Random Number Generator with Built-in Attack Detection , 2008, 2008 Third International Conference on Dependability of Computer Systems DepCoS-RELCOMEX.

[121]  Donald E. Eastlake,et al.  Randomness Recommendations for Security , 1994, RFC.

[122]  Makoto Matsumoto,et al.  Twisted GFSR generators II , 1994, TOMC.

[123]  Wietse Z. Venema,et al.  Murphy's Law and Computer Security , 1996, USENIX Security Symposium.

[124]  Wolfgang Hörmann,et al.  Continuous random variate generation by fast numerical inversion , 2003, TOMC.

[125]  M. Jöhnk.  Erzeugung von betaverteilten und gammaverteilten Zufallszahlen , 1964 .

[126]  Jovan Dj. Golic,et al.  High-Speed True Random Number Generation with Logic Gates Only , 2007, CHES.

[127]  Richard A. Kronmal,et al.  An acceptance-complement analogue of the mixture-plus-acceptance-rejection method for generating random variables , 1984, TOMS.

[128]  Samuel Kotz,et al.  New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics , 2001 .

[129]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[130]  Craig B. Borkowf,et al.  Random Number Generation and Monte Carlo Methods , 2000, Technometrics.

[131]  Abraham Lempel,et al.  Compression of individual sequences via variable-rate coding , 1978, IEEE Trans. Inf. Theory.

[132]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[133]  Pierre L'Ecuyer,et al.  A search for good multiple recursive random number generators , 1993, TOMC.

[134]  Pierre L'Ecuyer,et al.  Uniform random number generation , 1994, Ann. Oper. Res..

[135]  Peter Gutmann,et al.  Secure deletion of data from magnetic and solid-state memory , 1996 .

[136]  Igor E. Shparlinski,et al.  Recent Advances in the Theory of Nonlinear Pseudorandom Number Generators , 2002 .

[137]  A. Rukhin,et al.  Statistical Testing of Random Number Generators , 1999 .

[138]  Markus Jakobsson,et al.  A practical secure physical random bit generator , 1998, CCS '98.