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List of random number generators

October 21, 2023

Random number generators are important in many kinds of technical applications, including physics, engineering or mathematical computer studies (e.g., Monte Carlo simulations), cryptography and gambling (on game servers).

This list includes many common types, regardless of quality or applicability to a given use case.

Pseudorandom number generators (PRNGs) Edit

The following algorithms are pseudorandom number generators.

Generator Date First proponents References Notes
Middle-square method 1946 J. von Neumann [1] In its original form, it is of poor quality and of historical interest only.
Lehmer generator 1951 D. H. Lehmer [2] One of the very earliest and most influential designs.
Linear congruential generator (LCG) 1958 W. E. Thomson; A. Rotenberg [3][4] A generalisation of the Lehmer generator and historically the most influential and studied generator.
Lagged Fibonacci generator (LFG) 1958 G. J. Mitchell and D. P. Moore [5]
Linear-feedback shift register (LFSR) 1965 R. C. Tausworthe [6] A hugely influential design. Also called Tausworthe generators.
Wichmann–Hill generator 1982 B. A. Wichmann and D. I. Hill [7] A combination of three small LCGs, suited to 16-bit CPUs. Widely used in many programs, e.g. it is used in Excel 2003 and later versions for the Excel function RAND[8] and it was the default generator in the language Python up to version 2.2.[9]
Rule 30 1983 S. Wolfram [10] Based on cellular automata.
Inversive congruential generator (ICG) 1986 J. Eichenauer and J. Lehn [11]
Blum Blum Shub 1986 M. Blum, L. Blum and M. Shub [12] Blum-Blum-Shub is a PRNG algorithm that is considered cryptographically secure. Its base is based on prime numbers.
Park-Miller generator 1988 S. K. Park and K. W. Miller [13] A specific implementation of a Lehmer generator, widely used because it is included in C++ as the function minstd_rand0 from C++11 onwards.[14]
ACORN generator (discovered 1984) 1989 R. S. Wikramaratna [15][16] The Additive Congruential Random Number generator.

Simple to implement, fast, but not widely known. With appropriate initialisations, passes all current empirical test suites, and is formally proven to converge. Easy to extend for arbitrary period length and improved statistical performance over higher dimensions and with higher precision.

MIXMAX generator 1991 G. K. Savvidy and N. G. Ter-Arutyunyan-Savvidy [17] It is a member of the class of matrix linear congruential generator, a generalisation of LCG. The rationale behind the MIXMAX family of generators relies on results from ergodic theory and classical mechanics.
Add-with-carry (AWC) 1991 G. Marsaglia and A. Zaman [18] A modification of Lagged-Fibonacci generators.
Subtract-with-borrow (SWB) 1991 G. Marsaglia and A. Zaman [18] A modification of Lagged-Fibonacci generators. A SWB generator is the basis for the RANLUX generator,[19] widely used e.g. for particle physics simulations.
Maximally periodic reciprocals 1992 R. A. J. Matthews [20] A method with roots in number theory, although never used in practical applications.
KISS 1993 G. Marsaglia [21] Prototypical example of a combination generator.
Multiply-with-carry (MWC) 1994 G. Marsaglia; C. Koç [22][23]
Complementary-multiply-with-carry (CMWC) 1997 R. Couture and P. L’Ecuyer [24]
Mersenne Twister (MT) 1998 M. Matsumoto and T. Nishimura [25] Closely related with LFSRs. In its MT19937 implementation is probably the most commonly used modern PRNG. Default generator in R and the Python language starting from version 2.3.
Xorshift 2003 G. Marsaglia [26] It is a very fast sub-type of LFSR generators. Marsaglia also suggested as an improvement the xorwow generator, in which the output of a xorshift generator is added with a Weyl sequence. The xorwow generator is the default generator in the CURAND library of the nVidia CUDA application programming interface for graphics processing units.
Well equidistributed long-period linear (WELL) 2006 F. Panneton, P. L'Ecuyer and M. Matsumoto [27] A LFSR closely related with Mersenne Twister, aiming at remedying some of its shortcomings.
A small noncryptographic PRNG (JSF) 2007 Bob Jenkins [28]
Advanced Randomization System (ARS) 2011 J. Salmon, M. Moraes, R. Dror and D. Shaw [29] A simplified version of the AES block cipher, leading to very fast performance on systems supporting the AES-NI.
Threefry 2011 J. Salmon, M. Moraes, R. Dror and D. Shaw [29] A simplified version of the Threefish block cipher, suitable for GPU implementations.
Philox 2011 J. Salmon, M. Moraes, R. Dror and D. Shaw [29] A simplification and modification of the block cipher Threefish with the addition of an S-box.
WELLDOC 2013 L. Balkova, M. Bucci, A. de Luca, J. Hladky, S. Puzynina [30] Aperiodic pseudorandom number generators based on infinite words technique.
SplitMix 2014 G. L. Steele, D. Lea and C. H. Flood [31] Based upon the final mixing function of MurmurHash3. Included in Java Development Kit 8 and above.
Permuted Congruential Generator (PCG) 2014 M. E. O'Neill [32] A modification of LCG.
Random Cycle Bit Generator (RCB) 2016 R. Cookman [33] RCB is described as a bit pattern generator made to overcome some of the shortcomings with Mersenne Twister and short periods/bit length restriction of shift/modulo generators.
Middle-Square Weyl Sequence RNG (see also middle-square method) 2017 B. Widynski [34][35] A variation on John von Neumann's original middle-square method, this generator may be the fastest RNG that passes all the statistical tests.
Xoroshiro128+ 2018 D. Blackman, S. Vigna [36] A modification of Marsaglia's Xorshift generators, one of the fastest generators on modern 64-bit CPUs. Related generators include xoroshiro128**, xoshiro256+ and xoshiro256**.
64-bit MELG (MELG-64) 2018 S. Harase, T. Kimoto [37] An implementation of 64-bit maximally equidistributed F2-linear generators with Mersenne prime period.
Squares RNG 2020 B. Widynski [38] A counter-based version of Middle-Square Weyl Sequence RNG. Similar to Philox in design but significantly faster.

Cryptographic algorithms Edit

Cipher algorithms and cryptographic hashes can be used as very high-quality pseudorandom number generators. However, generally they are considerably slower (typically by a factor 2–10) than fast, non-cryptographic random number generators.

These include:

A few cryptographically secure pseudorandom number generators do not rely on cipher algorithms but try to link mathematically the difficulty of distinguishing their output from a `true' random stream to a computationally difficult problem. These approaches are theoretically important but are too slow to be practical in most applications. They include:

Random number generators that use external entropy Edit

These approaches combine a pseudo-random number generator (often in the form of a block or stream cipher) with an external source of randomness (e.g., mouse movements, delay between keyboard presses etc.).

See also Edit

References Edit

  1. ^ Some of von Neumann's 1949 papers were printed only in 1951. John von Neumann, “Various techniques used in connection with random digits,” in A.S. Householder, G.E. Forsythe, and H.H. Germond, eds., Monte Carlo Method, National Bureau of Standards Applied Mathematics Series, vol. 12 (Washington, D.C.: U.S. Government Printing Office, 1951): pp. 36–38.
  2. ^ Lehmer, Derrick H. (1951). "Mathematical methods in large-scale computing units". Proceedings of 2nd Symposium on Large-Scale Digital Calculating Machinery: 141–146.
  3. ^ Thomson, W. E. (1958). "A Modified Congruence Method of Generating Pseudo-random Numbers". The Computer Journal. 1 (2): 83. doi:10.1093/comjnl/1.2.83.
  4. ^ Rotenberg, A. (1960). "A New Pseudo-Random Number Generator". Journal of the ACM. 7 (1): 75–77. doi:10.1145/321008.321019. S2CID 16770825.
  5. ^ D. E. Knuth, The Art of Computer Programming, Vol. 2 Seminumerical Algorithms, 3rd ed., Addison Wesley Longman (1998); See pag. 27.
  6. ^ Tausworthe, R. C. (1965). "Random Numbers Generated by Linear Recurrence Modulo Two" (PDF). Mathematics of Computation. 19 (90): 201–209. doi:10.1090/S0025-5718-1965-0184406-1.
  7. ^ Wichmann, Brian A.; Hill, David I. (1982). "Algorithm AS 183: An Efficient and Portable Pseudo-Random Number Generator". Journal of the Royal Statistical Society. Series C (Applied Statistics). 31 (2): 188–190. doi:10.2307/2347988. JSTOR 2347988.
  8. ^ "Microsoft Support - Description of the RAND function in Excel". Apr 17, 2018.
  9. ^ "Documentation » The Python Standard Library » 9. Numeric and Mathematical Modules » 9.6. random — Generate pseudo-random numbers".
  10. ^ Wolfram, S. (1983). "Statistical mechanics of cellular automata". Rev. Mod. Phys. 55 (3): 601–644. Bibcode:1983RvMP...55..601W. doi:10.1103/RevModPhys.55.601.
  11. ^ Eichenauer, Jürgen; Lehn, Jürgen (1986). "A nonlinear congruential pseudorandom number generator". Statistische Hefte. 27: 315–326. doi:10.1007/BF02932576. S2CID 122052399.
  12. ^ Blum, L.; Blum, M.; Shub, M. (1986-05-01). "A Simple Unpredictable Pseudo-Random Number Generator". SIAM Journal on Computing. 15 (2): 364–383. doi:10.1137/0215025. ISSN 0097-5397.
  13. ^ Park, Stephen K.; Miller, Keith W. (1988). "Random Number Generators: Good Ones Are Hard To Find" (PDF). Communications of the ACM. 31 (10): 1192–1201. doi:10.1145/63039.63042. S2CID 207575300.
  14. ^ "Pseudo-random number generation". cppreference.com. Retrieved 14 November 2021.
  15. ^ Wikramaratna, R. S. (1989). "ACORN — A new method for generating sequences of uniformly distributed Pseudo-random Numbers". Journal of Computational Physics. 83 (1): 16–31. Bibcode:1989JCoPh..83...16W. doi:10.1016/0021-9991(89)90221-0.
  16. ^ Wikramaratna, R.S. Theoretical and empirical convergence results for additive congruential random number generators, Journal of Computational and Applied Mathematics (2009), doi:10.1016/j.cam.2009.10.015
  17. ^ Savvidy, G.K; Ter-Arutyunyan-Savvidy, N.G (1991). "On the Monte Carlo simulation of physical systems". Journal of Computational Physics. 97 (2): 566. Bibcode:1991JCoPh..97..566S. doi:10.1016/0021-9991(91)90015-D.
  18. ^ a b George, Marsaglia; Zaman, Arif (1991). "A new class of random number generators". Annals of Applied Probability. 1 (3): 462–480. doi:10.1214/aoap/1177005878.
  19. ^ Martin, Lüscher (1994). "A portable high-quality random number generator for lattice field theory simulations". Computer Physics Communications. 79 (1): 100–110. arXiv:hep-lat/9309020. Bibcode:1994CoPhC..79..100L. doi:10.1016/0010-4655(94)90232-1. S2CID 17608961.
  20. ^ Matthews, Robert A. J. (1992). "Maximally periodic reciprocals". Bull. Inst. Math. Appl. 28: 147–148.
  21. ^ Marsaglia, George; Zaman, Arif (1993). "The KISS generator". Technical Report, Department of Statistics, Florida State University, Tallahassee, FL, USA.
  22. ^ Post by George Marsaglia on the newsgroup sci.stat.math dated 1 August 2018 with title 'Yet another RNG'.
  23. ^ Koç, Cemal (1995). "Recurring-with-Carry Sequences". Journal of Applied Probability. 32 (4): 966–971. doi:10.2307/3215210. JSTOR 3215210. S2CID 123798320.
  24. ^ Couture, Raymond; L'Ecuyer, Pierre (1997). "Distribution properties of multiply-with-carry random number generators" (PDF). Mathematics of Computation. 66 Number. 218 (218): 591–607. Bibcode:1997MaCom..66..591C. doi:10.1090/S0025-5718-97-00827-2.
  25. ^ Matsumoto, M.; Nishimura, T. (1998). "MersenneTwister: A623-dimensionally Equidistributed Uniform Pseudo-Random Number Generator". ACM Transactions on Modeling and Computer Simulation. 8 (1): 3–30. CiteSeerX 10.1.1.215.1141. doi:10.1145/272991.272995. S2CID 3332028.
  26. ^ Marsaglia, George (July 2003). "Xorshift RNGs". Journal of Statistical Software. 8 (14). doi:10.18637/jss.v008.i14.
  27. ^ Panneton, François O.; l'Ecuyer, Pierre; Matsumoto, Pierre (March 2006). "Improved long-period generators based on linear recurrences modulo 2" (PDF). ACM Transactions on Mathematical Software. 32 (1): 1–16. CiteSeerX 10.1.1.73.5499. doi:10.1145/1132973.1132974. S2CID 7368302.
  28. ^ Jenkins, Bob (2009). "A small noncryptographic PRNG".
  29. ^ a b c Salmon, John; Moraes, Mark; Dror, Ron; Shaw, David (2011). "Parallel random numbers: as easy as 1, 2, 3". Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis, Article No. 16. doi:10.1145/2063384.2063405.
  30. ^ Balkova, Lubomira; Bucci, Michelangelo; De Luca, Alessandro; Hladky, Jiri; Puzynina, Svetlana (September 2016). "Aperiodic pseudorandom number generators based on infinite words". Theoretical Computer Science. 647: 85–100. arXiv:1311.6002. doi:10.1016/j.tcs.2016.07.042. S2CID 2175443.
  31. ^ Steele, Guy L. Jr.; Lea, Doug; Flood, Christine H. (2014). "Fast splittable pseudorandom number generators" (PDF). OOPSLA '14 Proceedings of the 2014 ACM International Conference on Object Oriented Programming Systems Languages & Applications.
  32. ^ O'Neill, Melissa E. (2014). "PCG: A Family of Simple Fast Space-Efficient Statistically Good Algorithms for Random Number Generation" (PDF). Technical Report.
  33. ^ Cookman, Richard (2016). "random cycle bit generator (rcb_generator)". Technical Report.
  34. ^ Widynski, Bernard (2017). "Middle-Square Weyl Sequence RNG". arXiv:1704.00358 [cs.CR].
  35. ^ Kneusel, Ron (2018). Random Numbers and Computers (1 ed.). Springer. pp. 13–14. ISBN 9783319776972.
  36. ^ Blackman, David; Vigna, Sebastiano (2018). "Scrambled Linear Pseudorandom Generators". arXiv:1805.01407 [cs.DS].
  37. ^ Harase, S.; Kimoto, T. (2018). "Implementing 64-bit Maximally Equidistributed F2-Linear Generators with Mersenne Prime Period". ACM Transactions on Mathematical Software. 44 (3): 30:1–30:11. arXiv:1505.06582. doi:10.1145/3159444. S2CID 14923086.
  38. ^ Widynski, Bernard (2020). "Squares: A Fast Counter-Based RNG". arXiv:2004.06278 [cs.DS].
  39. ^ True Random Number Generator using Corona Discharge: Indian Patent Office. Patent Application Number: 201821026766

External links Edit

  • SP800-90 series on Random Number Generation, NIST
  • Random Number Generation in the GNU Scientific Library Reference Manual
  • Random Number Generation Routines in the NAG Numerical Library
  • Chris Lomont's overview of PRNGs, including a good implementation of the WELL512 algorithm
  • Source code to read data from a TrueRNG V2 hardware TRNG

October 21, 2023
list, random, number, generators, random, number, generators, important, many, kinds, technical, applications, including, physics, engineering, mathematical, computer, studies, monte, carlo, simulations, cryptography, gambling, game, servers, this, list, inclu. Random number generators are important in many kinds of technical applications including physics engineering or mathematical computer studies e g Monte Carlo simulations cryptography and gambling on game servers This list includes many common types regardless of quality or applicability to a given use case Contents 1 Pseudorandom number generators PRNGs 2 Cryptographic algorithms 3 Random number generators that use external entropy 4 See also 5 References 6 External linksPseudorandom number generators PRNGs EditThe following algorithms are pseudorandom number generators Generator Date First proponents References NotesMiddle square method 1946 J von Neumann 1 In its original form it is of poor quality and of historical interest only Lehmer generator 1951 D H Lehmer 2 One of the very earliest and most influential designs Linear congruential generator LCG 1958 W E Thomson A Rotenberg 3 4 A generalisation of the Lehmer generator and historically the most influential and studied generator Lagged Fibonacci generator LFG 1958 G J Mitchell and D P Moore 5 Linear feedback shift register LFSR 1965 R C Tausworthe 6 A hugely influential design Also called Tausworthe generators Wichmann Hill generator 1982 B A Wichmann and D I Hill 7 A combination of three small LCGs suited to 16 bit CPUs Widely used in many programs e g it is used in Excel 2003 and later versions for the Excel function RAND 8 and it was the default generator in the language Python up to version 2 2 9 Rule 30 1983 S Wolfram 10 Based on cellular automata Inversive congruential generator ICG 1986 J Eichenauer and J Lehn 11 Blum Blum Shub 1986 M Blum L Blum and M Shub 12 Blum Blum Shub is a PRNG algorithm that is considered cryptographically secure Its base is based on prime numbers Park Miller generator 1988 S K Park and K W Miller 13 A specific implementation of a Lehmer generator widely used because it is included in C as the function minstd rand0 from C 11 onwards 14 ACORN generator discovered 1984 1989 R S Wikramaratna 15 16 The Additive Congruential Random Number generator Simple to implement fast but not widely known With appropriate initialisations passes all current empirical test suites and is formally proven to converge Easy to extend for arbitrary period length and improved statistical performance over higher dimensions and with higher precision MIXMAX generator 1991 G K Savvidy and N G Ter Arutyunyan Savvidy 17 It is a member of the class of matrix linear congruential generator a generalisation of LCG The rationale behind the MIXMAX family of generators relies on results from ergodic theory and classical mechanics Add with carry AWC 1991 G Marsaglia and A Zaman 18 A modification of Lagged Fibonacci generators Subtract with borrow SWB 1991 G Marsaglia and A Zaman 18 A modification of Lagged Fibonacci generators A SWB generator is the basis for the RANLUX generator 19 widely used e g for particle physics simulations Maximally periodic reciprocals 1992 R A J Matthews 20 A method with roots in number theory although never used in practical applications KISS 1993 G Marsaglia 21 Prototypical example of a combination generator Multiply with carry MWC 1994 G Marsaglia C Koc 22 23 Complementary multiply with carry CMWC 1997 R Couture and P L Ecuyer 24 Mersenne Twister MT 1998 M Matsumoto and T Nishimura 25 Closely related with LFSRs In its MT19937 implementation is probably the most commonly used modern PRNG Default generator in R and the Python language starting from version 2 3 Xorshift 2003 G Marsaglia 26 It is a very fast sub type of LFSR generators Marsaglia also suggested as an improvement the xorwow generator in which the output of a xorshift generator is added with a Weyl sequence The xorwow generator is the default generator in the CURAND library of the nVidia CUDA application programming interface for graphics processing units Well equidistributed long period linear WELL 2006 F Panneton P L Ecuyer and M Matsumoto 27 A LFSR closely related with Mersenne Twister aiming at remedying some of its shortcomings A small noncryptographic PRNG JSF 2007 Bob Jenkins 28 Advanced Randomization System ARS 2011 J Salmon M Moraes R Dror and D Shaw 29 A simplified version of the AES block cipher leading to very fast performance on systems supporting the AES NI Threefry 2011 J Salmon M Moraes R Dror and D Shaw 29 A simplified version of the Threefish block cipher suitable for GPU implementations Philox 2011 J Salmon M Moraes R Dror and D Shaw 29 A simplification and modification of the block cipher Threefish with the addition of an S box WELLDOC 2013 L Balkova M Bucci A de Luca J Hladky S Puzynina 30 Aperiodic pseudorandom number generators based on infinite words technique SplitMix 2014 G L Steele D Lea and C H Flood 31 Based upon the final mixing function of MurmurHash3 Included in Java Development Kit 8 and above Permuted Congruential Generator PCG 2014 M E O Neill 32 A modification of LCG Random Cycle Bit Generator RCB 2016 R Cookman 33 RCB is described as a bit pattern generator made to overcome some of the shortcomings with Mersenne Twister and short periods bit length restriction of shift modulo generators Middle Square Weyl Sequence RNG see also middle square method 2017 B Widynski 34 35 A variation on John von Neumann s original middle square method this generator may be the fastest RNG that passes all the statistical tests Xoroshiro128 2018 D Blackman S Vigna 36 A modification of Marsaglia s Xorshift generators one of the fastest generators on modern 64 bit CPUs Related generators include xoroshiro128 xoshiro256 and xoshiro256 64 bit MELG MELG 64 2018 S Harase T Kimoto 37 An implementation of 64 bit maximally equidistributed F2 linear generators with Mersenne prime period Squares RNG 2020 B Widynski 38 A counter based version of Middle Square Weyl Sequence RNG Similar to Philox in design but significantly faster Cryptographic algorithms EditCipher algorithms and cryptographic hashes can be used as very high quality pseudorandom number generators However generally they are considerably slower typically by a factor 2 10 than fast non cryptographic random number generators These include Stream ciphers Popular choices are Salsa20 or ChaCha often with the number of rounds reduced to 8 for speed ISAAC HC 128 and RC4 Block ciphers in counter mode Common choices are AES which is very fast on systems supporting it in hardware TwoFish Serpent and Camellia Cryptographic hash functionsA few cryptographically secure pseudorandom number generators do not rely on cipher algorithms but try to link mathematically the difficulty of distinguishing their output from a true random stream to a computationally difficult problem These approaches are theoretically important but are too slow to be practical in most applications They include Blum Micali algorithm 1984 Blum Blum Shub 1986 Naor Reingold pseudorandom function 1997 Random number generators that use external entropy EditThese approaches combine a pseudo random number generator often in the form of a block or stream cipher with an external source of randomness e g mouse movements delay between keyboard presses etc a href Dev random html class mw redirect title Dev random dev random a Unix like systems CryptGenRandom Microsoft Windows Fortuna RDRAND instructions called Intel Secure Key by Intel available in Intel x86 CPUs since 2012 They use the AES generator built into the CPU reseeding it periodically True Random Number Generator using Corona Discharge 39 YarrowSee also EditDiceware Diehard tests statistical test suite for random number generators Non uniform random variate generation Hardware random number generator Random number generator attack Randomness TestU01 statistical test suite for random number generatorsReferences Edit Some of von Neumann s 1949 papers were printed only in 1951 John von Neumann Various techniques used in connection with random digits in A S Householder G E Forsythe and H H Germond eds Monte Carlo Method National Bureau of Standards Applied Mathematics Series vol 12 Washington D C U S Government Printing Office 1951 pp 36 38 Lehmer Derrick H 1951 Mathematical methods in large scale computing units Proceedings of 2nd Symposium on Large Scale Digital Calculating Machinery 141 146 Thomson W E 1958 A Modified Congruence Method of Generating Pseudo random Numbers The Computer Journal 1 2 83 doi 10 1093 comjnl 1 2 83 Rotenberg A 1960 A New Pseudo Random Number Generator Journal of the ACM 7 1 75 77 doi 10 1145 321008 321019 S2CID 16770825 D E Knuth The Art of Computer Programming Vol 2 Seminumerical Algorithms 3rd ed Addison Wesley Longman 1998 See pag 27 Tausworthe R C 1965 Random Numbers Generated by Linear Recurrence Modulo Two PDF Mathematics of Computation 19 90 201 209 doi 10 1090 S0025 5718 1965 0184406 1 Wichmann Brian A Hill David I 1982 Algorithm AS 183 An Efficient and Portable Pseudo Random Number Generator Journal of the Royal Statistical Society Series C Applied Statistics 31 2 188 190 doi 10 2307 2347988 JSTOR 2347988 Microsoft Support Description of the RAND function in Excel Apr 17 2018 Documentation The Python Standard Library 9 Numeric and Mathematical Modules 9 6 random Generate pseudo random numbers Wolfram S 1983 Statistical mechanics of cellular automata Rev Mod Phys 55 3 601 644 Bibcode 1983RvMP 55 601W doi 10 1103 RevModPhys 55 601 Eichenauer Jurgen Lehn Jurgen 1986 A nonlinear congruential pseudorandom number generator Statistische Hefte 27 315 326 doi 10 1007 BF02932576 S2CID 122052399 Blum L Blum M Shub M 1986 05 01 A Simple Unpredictable Pseudo Random Number Generator SIAM Journal on Computing 15 2 364 383 doi 10 1137 0215025 ISSN 0097 5397 Park Stephen K Miller Keith W 1988 Random Number Generators Good Ones Are Hard To Find PDF Communications of the ACM 31 10 1192 1201 doi 10 1145 63039 63042 S2CID 207575300 Pseudo random number generation cppreference com Retrieved 14 November 2021 Wikramaratna R S 1989 ACORN A new method for generating sequences of uniformly distributed Pseudo random Numbers Journal of Computational Physics 83 1 16 31 Bibcode 1989JCoPh 83 16W doi 10 1016 0021 9991 89 90221 0 Wikramaratna R S Theoretical and empirical convergence results for additive congruential random number generators Journal of Computational and Applied Mathematics 2009 doi 10 1016 j cam 2009 10 015 Savvidy G K Ter Arutyunyan Savvidy N G 1991 On the Monte Carlo simulation of physical systems Journal of Computational Physics 97 2 566 Bibcode 1991JCoPh 97 566S doi 10 1016 0021 9991 91 90015 D a b George Marsaglia Zaman Arif 1991 A new class of random number generators Annals of Applied Probability 1 3 462 480 doi 10 1214 aoap 1177005878 Martin Luscher 1994 A portable high quality random number generator for lattice field theory simulations Computer Physics Communications 79 1 100 110 arXiv hep lat 9309020 Bibcode 1994CoPhC 79 100L doi 10 1016 0010 4655 94 90232 1 S2CID 17608961 Matthews Robert A J 1992 Maximally periodic reciprocals Bull Inst Math Appl 28 147 148 Marsaglia George Zaman Arif 1993 The KISS generator Technical Report Department of Statistics Florida State University Tallahassee FL USA Post by George Marsaglia on the newsgroup sci stat math dated 1 August 2018 with title Yet another RNG Koc Cemal 1995 Recurring with Carry Sequences Journal of Applied Probability 32 4 966 971 doi 10 2307 3215210 JSTOR 3215210 S2CID 123798320 Couture Raymond L Ecuyer Pierre 1997 Distribution properties of multiply with carry random number generators PDF Mathematics of Computation 66 Number 218 218 591 607 Bibcode 1997MaCom 66 591C doi 10 1090 S0025 5718 97 00827 2 Matsumoto M Nishimura T 1998 MersenneTwister A623 dimensionally Equidistributed Uniform Pseudo Random Number Generator ACM Transactions on Modeling and Computer Simulation 8 1 3 30 CiteSeerX 10 1 1 215 1141 doi 10 1145 272991 272995 S2CID 3332028 Marsaglia George July 2003 Xorshift RNGs Journal of Statistical Software 8 14 doi 10 18637 jss v008 i14 Panneton Francois O l Ecuyer Pierre Matsumoto Pierre March 2006 Improved long period generators based on linear recurrences modulo 2 PDF ACM Transactions on Mathematical Software 32 1 1 16 CiteSeerX 10 1 1 73 5499 doi 10 1145 1132973 1132974 S2CID 7368302 Jenkins Bob 2009 A small noncryptographic PRNG a b c Salmon John Moraes Mark Dror Ron Shaw David 2011 Parallel random numbers as easy as 1 2 3 Proceedings of 2011 International Conference for High Performance Computing Networking Storage and Analysis Article No 16 doi 10 1145 2063384 2063405 Balkova Lubomira Bucci Michelangelo De Luca Alessandro Hladky Jiri Puzynina Svetlana September 2016 Aperiodic pseudorandom number generators based on infinite words Theoretical Computer Science 647 85 100 arXiv 1311 6002 doi 10 1016 j tcs 2016 07 042 S2CID 2175443 Steele Guy L Jr Lea Doug Flood Christine H 2014 Fast splittable pseudorandom number generators PDF OOPSLA 14 Proceedings of the 2014 ACM International Conference on Object Oriented Programming Systems Languages amp Applications O Neill Melissa E 2014 PCG A Family of Simple Fast Space Efficient Statistically Good Algorithms for Random Number Generation PDF Technical Report Cookman Richard 2016 random cycle bit generator rcb generator Technical Report Widynski Bernard 2017 Middle Square Weyl Sequence RNG arXiv 1704 00358 cs CR Kneusel Ron 2018 Random Numbers and Computers 1 ed Springer pp 13 14 ISBN 9783319776972 Blackman David Vigna Sebastiano 2018 Scrambled Linear Pseudorandom Generators arXiv 1805 01407 cs DS Harase S Kimoto T 2018 Implementing 64 bit Maximally Equidistributed F2 Linear Generators with Mersenne Prime Period ACM Transactions on Mathematical Software 44 3 30 1 30 11 arXiv 1505 06582 doi 10 1145 3159444 S2CID 14923086 Widynski Bernard 2020 Squares A Fast Counter Based RNG arXiv 2004 06278 cs DS True Random Number Generator using Corona Discharge Indian Patent Office Patent Application Number 201821026766External links EditSP800 90 series on Random Number Generation NIST Random Number Generation in the GNU Scientific Library Reference Manual Random Number Generation Routines in the NAG Numerical Library Chris Lomont s overview of PRNGs including a good implementation of the WELL512 algorithm Source code to read data from a TrueRNG V2 hardware TRNG Retrieved from https en wikipedia org w index php title List of random number generators amp oldid 1178338141, wikipedia, wiki, book, books, library,

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