473:
756:
as a hardware test, and then post-processes the random sequence with a shift register stream cipher. It is generally hard to use statistical tests to validate the generated random numbers. Wang and Nicol proposed a distance-based statistical testing technique that is used to identify the weaknesses of several random generators. Li and Wang proposed a method of testing random numbers based on laser chaotic entropy sources using
Brownian motion properties.
33:
823:. Again, a naive implementation may induce a modulo bias into the result, so more involved algorithms must be used. A method that nearly never performs division was described in 2018 by Daniel Lemire, with the current state-of-the-art being the arithmetic encoding-inspired 2021 "optimal algorithm" by Stephen Canon of
755:
Generated random numbers are sometimes subjected to statistical tests before use to ensure that the underlying source is still working, and then post-processed to improve their statistical properties. An example would be the TRNG9803 hardware random number generator, which uses an entropy measurement
503:
Most computer-generated random numbers use PRNGs which are algorithms that can automatically create long runs of numbers with good random properties but eventually the sequence repeats (or the memory usage grows without bound). These random numbers are fine in many situations but are not as random as
1041:
it would allow NSA to determine the state of the random number generator, and thereby eventually be able to read all data sent over the SSL connection. Even though it was apparent that Dual_EC_DRBG was a very poor and possibly backdoored pseudorandom number generator long before the NSA backdoor was
1053:
It has also been theorized that hardware RNGs could be secretly modified to have less entropy than stated, which would make encryption using the hardware RNG susceptible to attack. One such method that has been published works by modifying the dopant mask of the chip, which would be undetectable to
751:
Even given a source of plausible random numbers (perhaps from a quantum mechanically based hardware generator), obtaining numbers which are completely unbiased takes care. In addition, behavior of these generators often changes with temperature, power supply voltage, the age of the device, or other
654:. The recurrence relation can be extended to matrices to have much longer periods and better statistical properties . To avoid certain non-random properties of a single linear congruential generator, several such random number generators with slightly different values of the multiplier coefficient,
445:
noise, greatly aid the development of the physical random number generator. Among them, optical chaos has a high potential to physically produce high-speed random numbers due to its high bandwidth and large amplitude. A prototype of a high-speed, real-time physical random bit generator based on a
728:
and using them as a randomization source. However, most studies find that human subjects have some degree of non-randomness when attempting to produce a random sequence of e.g. digits or letters. They may alternate too much between choices when compared to a good random generator; thus, this
937:
Computational and hardware random number generators are sometimes combined to reflect the benefits of both kinds. Computational random number generators can typically generate pseudorandom numbers much faster than physical generators, while physical generators can generate "true randomness."
322:
While a pseudorandom number generator based solely on deterministic logic can never be regarded as a "true" random number source in the purest sense of the word, in practice they are generally sufficient even for demanding security-critical applications. Carefully designed and implemented
319:(CSPRNGs). The fallback occurs when the desired read rate of randomness exceeds the ability of the natural harvesting approach to keep up with the demand. This approach avoids the rate-limited blocking behavior of random number generators based on slower and purely environmental methods.
781:
Most random number generators natively work with integers or individual bits, so an extra step is required to arrive at the "canonical" uniform distribution between 0 and 1. The implementation is not as trivial as dividing the integer by its maximum possible value. Specifically:
312:. This type of generator typically does not rely on sources of naturally occurring entropy, though it may be periodically seeded by natural sources. This generator type is non-blocking, so they are not rate-limited by an external event, making large bulk reads a possibility.
696:
sufficient for cryptography purposes, as is explicitly stated in the language documentation. Such library functions often have poor statistical properties and some will repeat patterns after only tens of thousands of trials. They are often initialized using a computer's
117:, as well as countless other techniques. Because of the mechanical nature of these techniques, generating large quantities of sufficiently random numbers (important in statistics) required much work and time. Thus, results would sometimes be collected and distributed as
208:
randomness, many other operations only need a modest amount of unpredictability. Some simple examples might be presenting a user with a "random quote of the day", or determining which way a computer-controlled adversary might move in a computer game. Weaker forms of
72:(HRNGs), wherein each generation is a function of the current value of a physical environment's attribute that is constantly changing in a manner that is practically impossible to model. This would be in contrast to so-called "random number generations" done by
263:
There are two principal methods used to generate random numbers. The first method measures some physical phenomenon that is expected to be random and then compensates for possible biases in the measurement process. Example sources include measuring
812:'s STL. It does not use the extra precision and suffers from bias only in the last bit due to round-to-even. Other numeric concerns are warranted when shifting this "canonical" uniform distribution to a different range. A proposed method for the
687:
The quality i.e. randomness of such library functions varies widely from completely predictable output, to cryptographically secure. The default random number generator in many languages, including Python, Ruby, R, IDL and PHP is based on the
378:
The earliest methods for generating random numbers, such as dice, coin flipping and roulette wheels, are still used today, mainly in games and gambling as they tend to be too slow for most applications in statistics and cryptography.
669:. While simple to implement, its output is of poor quality. It has a very short period and severe weaknesses, such as the output sequence almost always converging to zero. A recent innovation is to combine the middle square with a
729:
approach is not widely used. However, for the very reason that humans perform poorly in this task, human random number generation can be used as a tool to gain insights into brain functions otherwise not accessible.
480:
Another common entropy source is the behavior of human users of the system. While people are not considered good randomness generators upon request, they generate random behavior quite well in the context of playing
865:, involves choosing an x and y value and testing whether the function of x is greater than the y value. If it is, the x value is accepted. Otherwise, the x value is rejected and the algorithm tries again.
705:. These functions may provide enough randomness for certain tasks (for example video games) but are unsuitable where high-quality randomness is required, such as in cryptography applications, or statistics.
268:, thermal noise, and other external electromagnetic and quantum phenomena. For example, cosmic background radiation or radioactive decay as measured over short timescales represent sources of natural
592:
275:
The speed at which entropy can be obtained from natural sources is dependent on the underlying physical phenomena being measured. Thus, sources of naturally occurring "true" entropy are said to be
994:
The Group at the
Taiyuan University of Technology generates random numbers sourced from a chaotic laser. Samples of random numbers are available at their physical random number generator service.
991:
harvests randomness from the quantum process of photonic emission in semiconductors. They supply a variety of ways of fetching the data, including libraries for several programming languages.
789:
The nature of floating-point math itself means there exists more precision the closer the number is to zero. This extra precision is usually not used due to the sheer number of bits required.
1058:
hardware RNG without mixing in the RDRAND output with other sources of entropy to counteract any backdoors in the hardware RNG, especially after the revelation of the NSA Bullrun program.
449:
Various imaginative ways of collecting this entropic information have been devised. One technique is to run a hash function against a frame of a video stream from an unpredictable source.
504:
numbers generated from electromagnetic atmospheric noise used as a source of entropy. The series of values generated by such algorithms is generally determined by a fixed number called a
172:, and other areas where producing an unpredictable result is desirable. Generally, in applications having unpredictability as the paramount feature, such as in security applications,
66:
chance is generated. This means that the particular outcome sequence will contain some patterns detectable in hindsight but impossible to foresee. True random number generators can be
485:
games. Some security-related computer software requires the user to make a lengthy series of mouse movements or keyboard inputs to create sufficient entropy needed to generate random
1375:
Li, Pu; Sun, Yuanyuan; Liu, Xianglian; Yi, Xiaogang; Zhang, Jianguo; Guo, Xiaomin; Guo, Yanqiang; Wang, Yuncai (2016-07-15). "Fully photonics-based physical random bit generator".
861:, involves integrating up to an area greater than or equal to the random number (which should be generated between 0 and 1 for proper distributions). A second method called the
124:
Several computational methods for pseudorandom number generation exist. All fall short of the goal of true randomness, although they may meet, with varying success, some of the
676:
Most computer programming languages include functions or library routines that provide random number generators. They are often designed to provide a random byte or word, or a
239:
random, and may even have ways to control the selection of music: a truly random system would have no restriction on the same item appearing two or three times in succession.
128:
intended to measure how unpredictable their results are (that is, to what degree their patterns are discernible). This generally makes them unusable for applications such as
946:
Some computations making use of a random number generator can be summarized as the computation of a total or average value, such as the computation of integrals by the
640:
958:
numbers. Such sequences have a definite pattern that fills in gaps evenly, qualitatively speaking; a truly random sequence may, and usually does, leave larger gaps.
315:
Some systems take a hybrid approach, providing randomness harvested from natural sources when available, and falling back to periodically re-seeded software-based
78:(PRNGs), which generate numbers that only look random but are in fact predetermined—these generations can be reproduced simply by knowing the state of the PRNG.
1031:
316:
258:
133:
1919:"Uniform random floats: How to generate a double-precision floating-point number in [0, 1] uniformly at random given a uniform random source of bits"
2272:
2253:
304:
that can produce long sequences of apparently random results, which are in fact completely determined by a shorter initial value, known as a seed value or
308:. As a result, the entire seemingly random sequence can be reproduced if the seed value is known. This type of random number generator is often called a
759:
Statistical tests are also used to give confidence that the post-processed final output from a random number generator is truly unbiased, with numerous
792:
Rounding error in division may bias the result. At worst, a supposedly excluded bound may be drawn contrary to expectations based on real-number math.
981:
generates random numbers sourced from quantum vacuum. Samples of random numbers are available at their quantum random number generator research page.
2162:
2144:
2377:
M. Tomassini; M. Sipper; M. Perrenoud (October 2000). "On the generation of high-quality random numbers by two-dimensional cellular automata".
1073:
on the MUSL's secure RNG computer during routine maintenance. During the hacks the man won a total amount of $ 16,500,000 over multiple years.
716:
for
Microsoft Windows. Most programming languages, including those mentioned above, provide a means to access these higher-quality sources.
204:
is an important and common task in computer programming. While cryptography and certain numerical algorithms require a very high degree of
1046:. There have subsequently been accusations that RSA Security knowingly inserted a NSA backdoor into its products, possibly as part of the
279: – they are rate-limited until enough entropy is harvested to meet the demand. On some Unix-like systems, including most
39:
are an example of a mechanical hardware random number generator. When a cubical die is rolled, a random number from 1 to 6 is obtained.
974:
resource pages contain a number of hands-on interactive activities and demonstrations of random number generation using Java applets.
235:
are in fact not quite so simple. For instance, a system that "randomly" selects music tracks for a background music system must only
1646:
Matsumoto, M.; Nishimura, T. (1998). "MersenneTwister: A 623-dimensionally
Equidistributed Uniform Pseudo-Random Number Generator".
1525:
1139:
Lugrin, Thomas (2023), Mulder, Valentin; Mermoud, Alain; Lenders, Vincent; Tellenbach, Bernhard (eds.), "Random Number
Generator",
386:
can be based on an essentially random atomic or subatomic physical phenomenon whose unpredictability can be traced to the laws of
17:
2362:
2339:
2313:
2211:
1893:
1801:
1158:
658:, can be used in parallel, with a "master" random number generator that selects from among the several different generators.
2130:
853:
Given a source of uniform random numbers, there are a couple of methods to create a new random source that corresponds to a
524:
289:
will block until sufficient entropy is harvested from the environment. Due to this blocking behavior, large bulk reads from
1054:
optical reverse-engineering. For example, for random number generation in Linux, it is seen as unacceptable to use Intel's
434:, can be used to approach a uniform distribution of bits from a non-uniformly random source, though at a lower bit rate.
1042:
confirmed in 2013, it had seen significant usage in practice until 2013, for example by the prominent security company
1784:
Wang, Yongge (2014). "Statistical
Properties of Pseudo Random Sequences and Experiments with PHP and Debian OpenSSL".
323:
pseudorandom number generators can be certified for security-critical cryptographic purposes, as is the case with the
2494:
911:
681:
647:
2461:
bit-strings in blocks of 512 bits every 60 seconds. Designed to provide unpredictability, autonomy, and consistency.
2115:
1093:
844:
746:
383:
373:
252:
173:
68:
988:
2243:
1066:
1730:
W. A. Wagenaar (1972). "Generation of random sequences by human subjects: a critical survey of the literature".
1691:
W. A. Wagenaar (1972). "Generation of random sequences by human subjects: a critical survey of the literature".
1004:
498:
442:
2442:
a Java-based framework for the generation of simulation sequences, including pseudorandom sequences of numbers
2182:"I am so glad I resisted pressure from Intel engineers to let /dev/random rely only on the RDRAND instruction"
2066:
1038:
838:
515:
511:
309:
248:
74:
1985:"[stdlib] Floating-point random-number improvements by NevinBR · Pull Request #33560 · apple/swift"
854:
391:
269:
169:
2499:
862:
801:
431:
187:
is facilitated by the ability to run the same sequence of random numbers again by starting from the same
724:
Random number generation may also be performed by humans, in the form of collecting various inputs from
89:
data. Some of these have existed since ancient times, including well-known examples like the rolling of
2428:
2048:"An optimal algorithm for bounded random integers by stephentyrone · Pull Request #39143 · apple/swift"
1939:
813:
147:
82:
197:
is secret. The sender and receiver can generate the same set of numbers automatically to use as keys.
2445:
1113:
869:
1874:
Goualard, F. (2020). "Generating Random
Floating-Point Numbers by Dividing Integers: A Case Study".
1746:
1707:
1660:
1220:
Herrero-Collantes, Miguel; Garcia-Escartin, Juan Carlos (2016). "Quantum random number generators".
1037:. If for example an SSL connection is created using this random number generator, then according to
984:
Random.org makes available random numbers that are sourced from the randomness of atmospheric noise.
2474:
2087:
951:
930:
operation) to provide a combined RNG at least as good as the best RNG used. This is referred to as
915:
900:
820:
2181:
1788:. Lecture Notes in Computer Science. Vol. 8712. Heidelberg: Springer LNCS. pp. 454–471.
1015:
Since much cryptography depends on a cryptographically secure random number generator for key and
646:
as a series of pseudorandom numbers. The maximum number of numbers the formula can produce is the
411:
2257:
1062:
950:. For such problems, it may be possible to find a more accurate solution by the use of so-called
708:
Much higher quality random number sources are available on most operating systems; for example
422:. However, physical phenomena and tools used to measure them generally feature asymmetries and
1741:
1732:
1702:
1693:
1655:
742:
702:
701:
as the seed, since such a clock is 64 bit and measures in nanoseconds, far beyond the person's
462:
125:
1918:
2475:
Statistical
Properties of Pseudo Random Sequences and Experiments with PHP and Debian OpenSSL
2439:
1103:
612:
476:
Demonstration of a simple random number generator based on where and when a button is clicked
2276:
1434:
Wang, Anbang; Li, Pu; Zhang, Jianguo; Zhang, Jianzhong; Li, Lei; Wang, Yuncai (2013-08-26).
1830:
1447:
1384:
1331:
1108:
1020:
1010:
786:
The integer used in the transformation must provide enough bits for the intended precision.
662:
427:
276:
157:
138:(CSPRNGS) also exist, with special features specifically designed for use in cryptography.
8:
1819:"Brownian motion properties of optoelectronic random bit generators based on laser chaos"
1016:
872:
201:
161:
118:
1834:
1536:
1451:
1388:
1335:
926:
The outputs of multiple independent RNGs can be combined (for example, using a bit-wise
2394:
2029:
2011:
1899:
1673:
1626:
1481:
1416:
1295:
1271:
1266:
1247:
1229:
947:
931:
486:
305:
280:
180:
472:
2358:
2335:
2309:
1903:
1889:
1856:
1848:
1797:
1473:
1465:
1408:
1400:
1357:
1349:
1300:
1251:
1154:
1088:
884:
848:
395:
387:
297:
with random bits, can often be slow on systems that use this type of entropy source.
265:
225:
2398:
2033:
2002:
Lemire, Daniel (23 February 2019). "Fast Random
Integer Generation in an Interval".
1485:
1019:
generation, if a random number generator can be made predictable, it can be used as
469:
uses variations in the amplitude of atmospheric noise recorded with a normal radio.
2386:
2350:
2299:
2291:
2021:
1879:
1838:
1789:
1751:
1712:
1677:
1665:
1455:
1420:
1392:
1339:
1290:
1280:
1239:
1144:
858:
738:
689:
666:
324:
221:
2450:
2371:
1793:
1243:
1118:
1098:
1047:
760:
698:
423:
294:
272:(as a measure of unpredictability or surprise of the number generation process).
2295:
2197:"Re: [PATCH] /dev/random: Insufficient of entropy on many architectures"
1958:
1884:
1605:
1149:
829:
Most 0 to 1 RNGs include 0 but exclude 1, while others include or exclude both.
1817:
Li, Pu; Yi, Xiaogang; Liu, Xianglian; Wang, Yuncai; Wang, Yongge (2016-07-11).
1285:
713:
677:
482:
328:
214:
2488:
1852:
1770:
1561:
1469:
1404:
1353:
670:
661:
A simple pen-and-paper method for generating random numbers is the so-called
438:
399:
232:
94:
2424:
2047:
1984:
1878:. ICCS. Lecture Notes in Computer Science. Vol. 12138. pp. 15–28.
1265:
Jacak, Marcin M.; Jóźwiak, Piotr; Niemczuk, Jakub; Jacak, Janusz E. (2021).
2479:
2458:
2239:
1860:
1477:
1412:
1361:
1304:
1043:
1034:
819:
Uniformly distributed integers are commonly used in algorithms such as the
165:
129:
102:
2163:"Researchers can slip an undetectable trojan into Intel's Ivy Bridge CPUs"
1669:
27:
Producing a sequence that cannot be predicted better than by random chance
2323:
1843:
1818:
1460:
1435:
1396:
1344:
1319:
955:
506:
419:
407:
189:
2304:
2145:"We don't enable backdoors in our crypto products, RSA tells customers"
1198:
824:
466:
403:
356:
110:
2390:
176:
are generally preferred over pseudorandom algorithms, where feasible.
1755:
1716:
1521:
454:
415:
301:
218:
184:
98:
2327:
2025:
1625:
Widynski, Bernard (19 May 2020). "Middle-Square Weyl
Sequence RNG".
2464:
2196:
2016:
1631:
1500:
1234:
1208:
1184:
1050:. RSA has denied knowingly inserting a backdoor into its products.
725:
458:
450:
153:
673:. This method produces high-quality output through a long period.
32:
2468:
2376:
1204:
1083:
1070:
797:
352:
336:
114:
2348:
1219:
231:
Some applications that appear at first sight to be suitable for
85:
have led to the development of different methods for generating
2349:
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007).
1055:
712:
on various BSD flavors, Linux, Mac OS X, IRIX, and Solaris, or
348:
106:
86:
63:
59:
55:
1174:
941:
179:
Pseudorandom number generators are very useful in developing
1436:"4.5 Gbps high-speed real-time physical random bit generator"
1180:
809:
805:
437:
The appearance of wideband photonic entropy sources, such as
344:
868:
As an example for rejection sampling, to generate a pair of
2454:
2322:
1027:
971:
90:
36:
2418:
2284:
Handbook of Computational Statistics: Concepts and Methods
2085:
1581:
966:
The following sites make available random number samples:
2434:
1264:
1026:
The NSA is reported to have inserted a backdoor into the
978:
927:
732:
340:
1959:"Drawing random floating-point numbers from an interval"
2372:
NIST SP800-90A, B, C series on random number generation
1318:
Li, Pu; Wang, Yun-Cai; Zhang, Jian-Zhong (2010-09-13).
317:
cryptographically secure pseudorandom number generators
135:
cryptographically secure pseudorandom number generators
2357:(3rd ed.). New York: Cambridge University Press.
1032:
cryptographically secure pseudorandom number generator
587:{\displaystyle X_{n+1}=(aX_{n}+b)\,{\textrm {mod}}\,m}
259:
Cryptographically secure pseudorandom number generator
193:. They are also used in cryptography – so long as the
2412:
2248:. Vol. 2: Seminumerical algorithms (3 ed.).
1606:"High Dimensionality Pseudo Random Number Generators"
1143:, Cham: Springer Nature Switzerland, pp. 31–34,
1141:
Trends in Data Protection and Encryption Technologies
615:
527:
2290:(second ed.). Springer-Verlag. pp. 35–71.
2265:
Proceedings of the 2017 Winter Simulation Conference
2004:
ACM Transactions on Modeling and Computer Simulation
1648:
ACM Transactions on Modeling and Computer Simulation
709:
285:
2203:
62:that cannot be reasonably predicted better than by
2480:Random Sequence Generator based on Avalanche Noise
2355:Numerical Recipes: The Art of Scientific Computing
1645:
1519:
634:
586:
426:that make their outcomes not uniformly random. A
141:
1940:"A new specification for std::generate_canonical"
489:or to initialize pseudorandom number generators.
242:
2486:
2471:article describing a dedicated Linux system call
2334:. New York: John Wiley & Sons. p. 772.
1559:
1433:
987:The Quantum Random Bit Generator Service at the
961:
2446:Random number generators in NAG Fortran Library
2282:. In J. E. Gentle; W. Haerdle; Y. Mori (eds.).
1069:(MUSL), who surreptitiously installed backdoor
453:used this technique with images of a number of
2328:"Chapter 1 – Uniform Random Number Generation"
2131:"RSA warns developers not to use RSA products"
1729:
1690:
355:uses a pseudorandom number algorithm known as
152:Random number generators have applications in
2465:A system call for random numbers: getrandom()
2258:"History of Uniform Random Number Generation"
2194:
2179:
2128:
2113:
816:claims to use the full precision everywhere.
771:
2271:
2252:
2238:
1065:by the information security director of the
1816:
1317:
942:Low-discrepancy sequences as an alternative
46:is a process by which, often by means of a
1374:
1320:"All-optical fast random number generator"
2303:
2064:
2015:
1883:
1842:
1745:
1706:
1659:
1630:
1582:"RANDOM.ORG – True Random Number Service"
1459:
1343:
1294:
1284:
1233:
1148:
580:
572:
1956:
1916:
1873:
1768:
1624:
1023:by an attacker to break the encryption.
776:
492:
471:
31:
1771:"TRNG9803 True Random Number Generator"
1562:"TrueCrypt Beginner's Tutorial, Part 3"
766:
14:
2487:
2242:(1997). "Chapter 3 – Random Numbers".
2212:"Inside the Biggest Lottery Scam Ever"
2209:
2001:
1267:"Quantum generators of random numbers"
1138:
832:
733:Post-processing and statistical checks
414:, the timing of actual movements of a
2122:
362:
300:The second method uses computational
2288:Handbook of Computational Statistics
1783:
1982:
1121:, contains a chance-dependent value
367:
24:
2326:; Taimre, T.; Botev, Z.I. (2011).
2232:
1498:
796:The mainstream algorithm, used by
25:
2511:
2406:
1876:Computational Science – ICCS 2020
1526:"Games for Extracting Randomness"
1501:"HotBits: Genuine Random Numbers"
808:, is described in a proposal for
446:chaotic laser was built in 2013.
331:. The former is the basis of the
69:hardware random-number generators
2116:"The Many Flaws of Dual_EC_DRBG"
2088:"G05 – Random Number Generators"
2086:The Numerical Algorithms Group.
1957:Goualard, Frédéric (July 2021).
1786:Computer Security - ESORICS 2014
1094:List of random number generators
977:The Quantum Optics Group at the
845:cumulative distribution function
747:List of random number generators
461:measured radioactive decay with
384:hardware random number generator
374:Hardware random number generator
253:Hardware random number generator
126:statistical tests for randomness
2419:Quantum random number generator
2332:Handbook of Monte Carlo Methods
2267:. IEEE Press. pp. 202–230.
2245:The Art of Computer Programming
2188:
2173:
2155:
2137:
2107:
2079:
2058:
2040:
1995:
1976:
1950:
1932:
1910:
1867:
1810:
1777:
1762:
1723:
1684:
1639:
1618:
1598:
1574:
1553:
1513:
1067:Multi-State Lottery Association
142:Practical applications and uses
2379:IEEE Transactions on Computers
1773:. Manufacturer: www.TRNG98.se.
1492:
1427:
1368:
1311:
1258:
1213:
1190:
1166:
1132:
1063:a U.S. lottery draw was rigged
1005:Random number generator attack
883:), one may first generate the
569:
547:
499:Pseudorandom number generation
443:amplified spontaneous emission
243:True vs. pseudo-random numbers
132:. However, carefully designed
75:pseudorandom number generators
13:
1:
2210:Nestel, M.L. (July 7, 2015).
1533:Weizmann Institute of Science
1126:
962:Activities and demonstrations
873:standard normally distributed
839:Pseudo-random number sampling
516:linear congruential generator
310:pseudorandom number generator
249:Pseudorandom number generator
2129:Matthew Green (2013-09-20).
2114:matthew Green (2013-09-18).
1917:Campbell, Taylor R. (2014).
1794:10.1007/978-3-319-11203-9_26
1244:10.1103/RevModPhys.89.015004
998:
921:
855:probability density function
719:
518:, which uses the recurrence
170:completely randomized design
7:
2351:"Chapter 7. Random Numbers"
2296:10.1007/978-3-642-21551-3_3
2095:NAG Library Manual, Mark 23
2067:"Common generation methods"
1885:10.1007/978-3-030-50417-5_2
1150:10.1007/978-3-031-33386-6_7
1076:
863:acceptance-rejection method
597:to generate numbers, where
432:cryptographic hash function
10:
2516:
2415:True Random Number Service
2277:"Random Number Generation"
1286:10.1038/s41598-021-95388-7
1008:
1002:
842:
836:
814:Swift programming language
772:Reshaping the distribution
736:
496:
371:
256:
246:
148:Applications of randomness
145:
83:applications of randomness
1222:Reviews of Modern Physics
1114:Random password generator
952:low-discrepancy sequences
870:statistically independent
510:. One of the most common
283:, the pseudo device file
2495:Random number generation
989:Ruđer Bošković Institute
857:. One method called the
763:suites being developed.
609:are large integers, and
44:Random number generation
2425:Random and Pseudorandom
635:{\displaystyle X_{n+1}}
48:random number generator
18:Random Number Generator
1733:Psychological Bulletin
1694:Psychological Bulletin
1560:TrueCrypt Foundation.
752:outside interference.
743:Statistical randomness
636:
588:
477:
40:
1769:Dömstedt, B. (2009).
1670:10.1145/272991.272995
1104:Procedural generation
1009:Further information:
777:Uniform distributions
682:uniformly distributed
637:
589:
493:Computational methods
475:
418:read-write head, and
406:, avalanche noise in
335:source of entropy on
35:
1844:10.1364/OE.24.015822
1461:10.1364/OE.21.020452
1397:10.1364/OL.41.003347
1345:10.1364/OE.18.020360
1109:Randomized algorithm
1011:Backdoor (computing)
916:Box–Muller transform
821:Fisher–Yates shuffle
767:Other considerations
663:middle-square method
613:
525:
428:randomness extractor
293:, such as filling a
202:pseudorandom numbers
158:statistical sampling
119:random number tables
1835:2016OExpr..2415822L
1829:(14): 15822–15833.
1452:2013OExpr..2120452W
1446:(17): 20452–20462.
1389:2016OptL...41.3347L
1336:2010OExpr..1820360L
1330:(19): 20360–20369.
1017:cryptographic nonce
833:Other distributions
463:Geiger–Muller tubes
281:Linux distributions
174:hardware generators
162:computer simulation
2500:Information theory
1272:Scientific Reports
1207:Library Functions
948:Monte Carlo method
932:software whitening
632:
584:
478:
363:Generation methods
226:sorting algorithms
200:The generation of
181:Monte Carlo-method
41:
2451:Randomness Beacon
2391:10.1109/12.888056
2385:(10): 1146–1151.
2364:978-0-521-88068-8
2341:978-0-470-17793-8
2315:978-3-642-21550-6
1895:978-3-030-50416-8
1803:978-3-319-11202-2
1383:(14): 3347–3350.
1160:978-3-031-33386-6
1089:League of entropy
885:polar coordinates
849:quantile function
692:algorithm and is
684:between 0 and 1.
577:
424:systematic biases
396:radioactive decay
388:quantum mechanics
266:atmospheric noise
54:), a sequence of
16:(Redirected from
2507:
2402:
2368:
2345:
2319:
2307:
2281:
2273:L'Ecuyer, Pierre
2268:
2262:
2254:L'Ecuyer, Pierre
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1541:
1535:. Archived from
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875:random numbers (
859:inversion method
739:Randomness tests
690:Mersenne Twister
667:John von Neumann
657:
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368:Physical methods
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325:yarrow algorithm
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217:and in creating
183:simulations, as
21:
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2485:
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2457:, broadcasting
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2279:
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2235:
2233:Further reading
2230:
2220:
2218:
2216:The Daily Beast
2208:
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2195:Theodore Ts'o.
2193:
2189:
2180:Theodore Ts'o.
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2065:The MathWorks.
2063:
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2026:10.1145/3230636
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1747:10.1.1.211.9085
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1708:10.1.1.211.9085
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1187:– Special Files
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1119:Random variable
1099:PP (complexity)
1079:
1048:Bullrun program
1013:
1007:
1001:
964:
944:
924:
904:
851:
841:
835:
779:
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761:randomness test
749:
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699:real-time clock
655:
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642:is the next in
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295:hard disk drive
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247:Main articles:
245:
215:hash algorithms
150:
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28:
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2407:External links
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2184:. Google Plus.
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1586:www.random.org
1573:
1552:
1520:Halprin, Ran;
1512:
1499:Walker, John.
1491:
1440:Optics Express
1426:
1377:Optics Letters
1367:
1324:Optics Express
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1003:Main article:
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497:Main article:
494:
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483:mixed strategy
372:Main article:
369:
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364:
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351:, and others.
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146:Main article:
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2324:Kroese, D. P.
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2169:. 2013-09-18.
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1542:on 2011-08-07
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1203: –
1200:
1199:arc4random(3)
1193:
1186:
1183:Programmer's
1182:
1179: –
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912:UNIFORM(0,2Ď€)
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671:Weyl sequence
668:
665:suggested by
664:
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439:optical chaos
435:
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400:thermal noise
397:
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390:. Sources of
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105:, the use of
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103:playing cards
100:
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95:coin flipping
92:
88:
84:
79:
77:
76:
71:
70:
65:
61:
57:
53:
49:
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38:
34:
30:
19:
2459:full entropy
2429:
2382:
2378:
2354:
2331:
2287:
2283:
2264:
2244:
2240:Donald Knuth
2219:. Retrieved
2215:
2205:
2190:
2175:
2167:Ars Technica
2166:
2157:
2149:Ars Technica
2148:
2139:
2124:
2109:
2098:. Retrieved
2094:
2081:
2070:. Retrieved
2060:
2051:
2042:
2007:
2003:
1997:
1988:
1978:
1966:. Retrieved
1962:
1952:
1943:
1934:
1922:. Retrieved
1912:
1875:
1869:
1826:
1822:
1812:
1785:
1779:
1764:
1740:(1): 65–72.
1737:
1731:
1725:
1701:(1): 65–72.
1698:
1692:
1686:
1651:
1647:
1641:
1620:
1609:. Retrieved
1600:
1589:. Retrieved
1585:
1576:
1565:. Retrieved
1555:
1544:. Retrieved
1537:the original
1532:
1515:
1504:. Retrieved
1494:
1443:
1439:
1429:
1380:
1376:
1370:
1327:
1323:
1313:
1279:(1): 16108.
1276:
1270:
1260:
1225:
1221:
1215:
1192:
1168:
1140:
1134:
1060:
1052:
1044:RSA Security
1035:Dual EC DRBG
1025:
1014:
965:
945:
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723:
707:
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502:
479:
448:
436:
430:, such as a
408:Zener diodes
381:
377:
321:
314:
299:
274:
262:
236:
230:
213:are used in
210:
205:
199:
194:
188:
178:
166:cryptography
151:
134:
130:cryptography
123:
109:stalks (for
80:
73:
67:
51:
47:
43:
42:
29:
2430:In Our Time
2305:10419/22195
2010:(1): 1–12.
1968:4 September
1924:4 September
1654:(1): 3–30.
956:quasirandom
710:/dev/random
420:radio noise
412:clock drift
333:/dev/random
291:/dev/random
286:/dev/random
190:random seed
2489:Categories
2413:RANDOM.ORG
2100:2012-02-09
2072:2024-09-08
2017:1805.10941
1632:1704.00358
1611:2018-11-21
1591:2016-01-14
1567:2009-06-27
1546:2009-06-27
1522:Naor, Moni
1506:2009-06-27
1235:1604.03304
1228:: 015004.
1127:References
1030:certified
843:See also:
825:Apple Inc.
737:See also:
467:Random.org
455:lava lamps
404:shot noise
357:arc4random
302:algorithms
257:See also:
211:randomness
111:divination
1983:NevinBR.
1904:219889587
1853:1094-4087
1742:CiteSeerX
1703:CiteSeerX
1656:CiteSeerX
1470:1094-4087
1405:1539-4794
1354:1094-4087
1252:118592321
1175:random(4)
1061:In 2010,
999:Backdoors
922:Whitening
895:), where
726:end users
720:By humans
703:precision
416:hard disk
222:searching
219:amortized
185:debugging
113:) in the
99:shuffling
2399:10139169
2275:(2012).
2256:(2017).
2221:July 10,
2034:44061046
1861:27410852
1486:10397141
1478:24105589
1413:27420532
1362:20940928
1305:34373502
1077:See also
1021:backdoor
465:, while
451:Lavarand
394:include
277:blocking
206:apparent
154:gambling
81:Various
2469:LWN.net
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