418:, whether there is any way to distinguish the output of a high-quality PRNG from a truly random sequence. In this setting, the distinguisher knows that either the known PRNG algorithm was used (but not the state with which it was initialized) or a truly random algorithm was used, and has to distinguish between the two. The security of most cryptographic algorithms and protocols using PRNGs is based on the assumption that it is infeasible to distinguish use of a suitable PRNG from use of a truly random sequence. The simplest examples of this dependency are
1896:. The algorithm is as follows: take any number, square it, remove the middle digits of the resulting number as the "random number", then use that number as the seed for the next iteration. For example, squaring the number "1111" yields "1234321", which can be written as "01234321", an 8-digit number being the square of a 4-digit number. This gives "2343" as the "random" number. Repeating this procedure gives "4896" as the next result, and so on. Von Neumann used 10 digit numbers, but the process was the same.
2441:
1515:
1324:
214:. The quality of LCGs was known to be inadequate, but better methods were unavailable. Press et al. (2007) described the result thus: "If all scientific papers whose results are in doubt because of were to disappear from library shelves, there would be a gap on each shelf about as big as your fist."
1899:
A problem with the "middle square" method is that all sequences eventually repeat themselves, some very quickly, such as "0000". Von
Neumann was aware of this, but he found the approach sufficient for his purposes and was worried that mathematical "fixes" would simply hide errors rather than remove
174:
The list of widely used generators that should be discarded is much longer . Do not trust blindly the software vendors. Check the default RNG of your favorite software and be ready to replace it if needed. This last recommendation has been made over and over again over the past 40 years. Perhaps
1903:
Von
Neumann judged hardware random number generators unsuitable, for, if they did not record the output generated, they could not later be tested for errors. If they did record their output, they would exhaust the limited computer memories then available, and so the computer's ability to read and
111:
Good statistical properties are a central requirement for the output of a PRNG. In general, careful mathematical analysis is required to have any confidence that a PRNG generates numbers that are sufficiently close to random to suit the intended use.
1510:{\displaystyle \forall E\in {\mathfrak {F}}\quad \forall \varepsilon >0\quad \exists N\in \mathbb {N} _{1}\quad \forall n\geq N,\quad \left|{\frac {\#\left\{i\in \left\{1,2,\dots ,n\right\}:f(i)\in E\right\}}{n}}-P(E)\right|<\varepsilon }
748:
434:. The design of cryptographically adequate PRNGs is extremely difficult because they must meet additional criteria. The size of its period is an important factor in the cryptographic suitability of a PRNG, but not the only one.
254:
family of generators was developed. The WELL generators in some ways improves on the quality of the
Mersenne Twister, which has a too-large state space and a very slow recovery from state spaces with a large number of zeros.
284:
in distinguishing the generator's output sequence from a random sequence. In other words, while a PRNG is only required to pass certain statistical tests, a CSPRNG must pass all statistical tests that are restricted to
1878:
859:
163:
simulations, or in other ways relied on PRNGs, were much less reliable than ideal as a result of using poor-quality PRNGs. Even today, caution is sometimes required, as illustrated by the following warning in the
507:
K3 – It should be impossible for an attacker (for all practical purposes) to calculate, or otherwise guess, from any given subsequence, any previous or future values in the sequence, nor any inner state of the
198:(discussed below), which was published in 1998. Other higher-quality PRNGs, both in terms of computational and statistical performance, were developed before and after this date; these can be identified in the
1774:
3041:
2088:
511:
K4 – It should be impossible, for all practical purposes, for an attacker to calculate, or guess from an inner state of the generator, any previous numbers in the sequence or any previous inner generator
1199:
1136:
1318:
587:
675:
2165:
116:
cautioned about the misinterpretation of a PRNG as a truly random generator, joking that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin."
2411:
2342:
448:
217:
A major advance in the construction of pseudorandom generators was the introduction of techniques based on linear recurrences on the two-element field; such generators are related to
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264:
105:
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1620:
1049:
993:
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796:
772:
641:
611:
1697:
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2628:
504:
zeros (or ones), the next bit a one (or zero) with probability one-half; and any selected subsequence contains no information about the next element(s) in the sequence.
166:
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2207:
340:
2290:
2006:
1977:
1541:
247:
generators, again based on a linear recurrence. Such generators are extremely fast and, combined with a nonlinear operation, they pass strong statistical tests.
30:
This page is about commonly encountered characteristics of pseudorandom number generator algorithms. For the formal concept in theoretical computer science, see
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1089:
1069:
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957:
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545:
380:
algorithm, which provide a strong security proof (such algorithms are rather slow compared to traditional constructions, and impractical for many applications)
680:
2592:
2567:
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2795:
236:
in (up to) 623 dimensions (for 32-bit values), and at the time of its introduction was running faster than other statistically reasonable generators.
3350:
3079:
489:
443:
251:
228:, in particular, avoided many of the problems with earlier generators. The Mersenne Twister has a period of 2 − 1 iterations (≈ 4.3
1799:
2955:
2455:
199:
801:
2891:
2637:
3193:
von
Neumann J., "Various techniques used in connection with random digits," in A.S. Householder, G.E. Forsythe, and H.H. Germond, eds.,
2706:
466:
K2 – A sequence of numbers is indistinguishable from "truly random" numbers according to specified statistical tests. The tests are the
373:
3337:
a simple online random number generator. Random numbers are generated by
Javascript pseudorandom number generators (PRNGs) algorithms
1908:
computer he was using, the "middle square" method generated numbers at a rate some hundred times faster than reading numbers in from
3214:
3188:
3173:
1941:
1722:
365:
combination PRNGs which attempt to combine several PRNG primitive algorithms with the goal of removing any detectable non-randomness
302:
1141:
500:
test. In essence, these requirements are a test of how well a bit sequence: has zeros and ones equally often; after a sequence of
3074:
1948:
1643:
72:
1098:
3197:, National Bureau of Standards Applied Mathematics Series, 12 (Washington, D.C.: U.S. Government Printing Office, 1951): 36–38.
3160:
2742:
1277:
550:
411:
290:
646:
2495:
2096:
218:
2664:
2368:
When using practical number representations, the infinite "tails" of the distribution have to be truncated to finite values.
2465:
1935:
211:
184:
3282:
2014:
2475:
459:, BSI) has established four criteria for quality of deterministic random number generators. They are summarized here:
383:
generic PRNGs: while it has been shown that a (cryptographically) secure PRNG can be constructed generically from any
180:
144:
Distances between where certain values occur are distributed differently from those in a random sequence distribution.
3008:
2527:
3328:
3271:
3257:
3230:
3114:
2460:
2293:
71:(which may include truly random values). Although sequences that are closer to truly random can be generated using
2374:
2305:
1880:. Intuitively, an arbitrary distribution can be simulated from a simulation of the standard uniform distribution.
463:
K1 – There should be a high probability that generated sequences of random numbers are different from each other.
1904:
write numbers. If the numbers were written to cards, they would take very much longer to write and read. On the
3275:
2490:
2470:
392:
281:
58:
866:
488:
test (checks whether there exists any run of length 34 or greater in 20 000 bits of the sequence)—both from
148:
Defects exhibited by flawed PRNGs range from unnoticeable (and unknown) to very obvious. An example was the
132:
Shorter-than-expected periods for some seed states (such seed states may be called "weak" in this context);
2947:
2505:
294:
31:
2976:
2422:
2300:
932:
306:
97:
2440:
1589:
1018:
962:
2426:
1923:
1893:
1226:
777:
753:
622:
592:
396:
298:
188:
104:. Cryptographic applications require the output not to be predictable from earlier outputs, and more
2916:"Cryptography Engineering: Design Principles and Practical Applications, Chapter 9.4: The Generator"
3118:
936:
289:
in the size of the seed. Though a proof of this property is beyond the current state of the art of
278:
2691:
2652:
1669:
210:
In the second half of the 20th century, the standard class of algorithms used for PRNGs comprised
3316:
3300:
2864:
2761:
2718:
2480:
2446:
2414:
2218:
160:
153:
89:
57:
for generating a sequence of numbers whose properties approximate the properties of sequences of
2839:
387:, this generic construction is extremely slow in practice, so is mainly of theoretical interest.
309:. In general, years of review may be required before an algorithm can be certified as a CSPRNG.
187:(LCG) for its PRNG, which is of low quality (see further below). Java support was upgraded with
2692:"Mersenne twister: a 623-dimensionally equi-distributed uniform pseudo-random number generator"
516:
For cryptographic applications, only generators meeting the K3 or K4 standards are acceptable.
414:
by any of several tests. It is an open question, and one central to the theory and practice of
3225:
3210:
3184:
3169:
2926:
2887:
2633:
159:
In many fields, research work prior to the 21st century that relied on random selection or on
3042:"Functionality Classes and Evaluation Methodology for Deterministic Random Number Generators"
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233:
93:
81:
are important in practice for their speed in number generation and their reproducibility.
2364:
would produce a sequence of (positive only) values with a
Gaussian distribution; however
1523:
2526:
Barker, Elaine; Barker, William; Burr, William; Polk, William; Smid, Miles (July 2012).
743:{\displaystyle {\mathfrak {F}}=\left\{\left(-\infty ,t\right]:t\in \mathbb {R} \right\}}
277:(CSPRNG). A requirement for a CSPRNG is that an adversary not knowing the seed has only
3135:
2911:
2907:
2626:
L'Ecuyer, Pierre (2010). "Uniform random number generators". In Lovric, Miodrag (ed.).
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1940:
Numbers selected from a non-uniform probability distribution can be generated using a
3344:
3308:
3304:
3267:
3203:
3120:
Recommendation for Random Number
Generation Using Deterministic Random Bit Generators
2765:
1919:
419:
316:
156:. It was seriously flawed, but its inadequacy went undetected for a very long time.
3312:
3156:
2868:
2793:
Vigna S. (2016), "An experimental exploration of
Marsaglia’s xorshift generators",
2722:
2421:
Similar considerations apply to generating other non-uniform distributions such as
1909:
423:
415:
404:
335:
PRNGs that have been designed specifically to be cryptographically secure, such as
325:
321:
270:
101:
3253:
1915:
The middle-square method has since been supplanted by more elaborate generators.
368:
special designs based on mathematical hardness assumptions: examples include the
194:
One well-known PRNG to avoid major problems and still run fairly quickly is the
67:
2546:
3322:
3236:
2825:
2500:
2436:
431:
85:
3243:", in Proc. 19th Annual Computer Security Applications Conference, Dec. 2003.
135:
Lack of uniformity of distribution for large quantities of generated numbers;
65:, because it is completely determined by an initial value, called the PRNG's
2860:
2171:
from a uniform distribution as the probability density to "pass by", we get
614:
427:
336:
128:
that cause them to fail statistical pattern-detection tests. These include:
54:
3334:
2816:
Vigna S. (2017), "Further scramblings of
Marsaglia’s xorshift generators",
1071:
is not specified, it is assumed to be some set contained in the support of
3240:
2756:
2714:
1873:{\displaystyle F^{*}(x):=\inf \left\{t\in \mathbb {R} :x\leq F(t)\right\}}
2737:
352:
244:
854:{\displaystyle \left\{\left(-\infty ,t\right]:t\in \mathbb {R} \right\}}
2915:
2668:
356:
2840:"Improved long-period generators based on linear recurrences modulo 2"
1922:. This method produces high-quality output through a long period (see
17:
3133:, "Some long-period random number generators using shifts and xors",
62:
27:
Algorithm that generates an approximation of a random number sequence
2804:
1586:
is a pseudo-random number generator for the uniform distribution on
3261:
3223:
Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P. (2007),
108:, which do not inherit the linearity of simpler PRNGs, are needed.
3293:
1905:
179:
As an illustration, consider the widely used programming language
149:
3124:
3084:
2838:
Panneton, François; L'Ecuyer, Pierre; Matsumoto, Makoto (2006).
2538:
493:
400:
175:
amazingly, it remains as relevant today as it was 40 years ago.
1769:{\displaystyle F^{*}:\left(0,1\right)\rightarrow \mathbb {R} }
1194:{\displaystyle \mathbb {N} _{1}=\left\{1,2,3,\dots \right\}}
3296:" by Zvi Gutterman, Benny Pinkas, and Tzachy Reinman (2006)
3087:. 1994-01-11. p. 4.11.1 Power-Up Tests. Archived from
1918:
A recent innovation is to combine the middle square with a
2780:
2593:"Various techniques used in connection with random digits"
1131:{\displaystyle f:\mathbb {N} _{1}\rightarrow \mathbb {R} }
1313:{\displaystyle f\left(\mathbb {N} _{1}\right)\subseteq A}
582:{\displaystyle \left(\mathbb {R} ,{\mathfrak {B}}\right)}
1944:
PRNG and a function that relates the two distributions.
670:{\displaystyle {\mathfrak {F}}\subseteq {\mathfrak {B}}}
484:
test (counts the frequency of runs of various lengths),
472:
test (equal numbers of ones and zeros in the sequence),
2600:
National Bureau of
Standards Applied Mathematics Series
2160:{\displaystyle 0=F(-\infty )\leq F(b)\leq F(\infty )=1}
891:– a non-empty set (not necessarily a Borel set). Often
124:
In practice, the output from many common PRNGs exhibit
3205:
The Jungles of Randomness : a mathematical safari
2781:"xorshift*/xorshift+ generators and the PRNG shootout"
2344:
with an ideal uniform PRNG with range (0, 1) as input
265:
Cryptographically secure pseudorandom number generator
2377:
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2180:
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2017:
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141:
Poor dimensional distribution of the output sequence;
3149:
Hörmann W., Leydold J., Derflinger G. (2004, 2011),
2699:
ACM Transactions on Modeling and Computer Simulation
3053:
Bundesamt für Sicherheit in der Informationstechnik
2083:{\displaystyle F(b)=\int _{-\infty }^{b}f(b')\,db'}
457:
Bundesamt für Sicherheit in der Informationstechnik
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1091:and containing its interior, depending on context.
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790:
766:
742:
669:
635:
605:
581:
539:
430:of a message with the output of a PRNG, producing
3075:"Security requirements for cryptographic modules"
2655:, Java Platform Standard Edition 8 Documentation.
2629:International Encyclopedia of Statistical Science
1543:denotes the number of elements in the finite set
167:International Encyclopedia of Statistical Science
3144:Random Number Generation and Monte Carlo Methods
2818:Journal of Computational and Applied Mathematics
2263:is a number randomly selected from distribution
1825:
410:Most PRNG algorithms produce sequences that are
3181:Pseudorandomness and Cryptographic Applications
341:Cryptographic Application Programming Interface
312:Some classes of CSPRNGs include the following:
172:
3241:Practical Random Number Generation in Software
3151:Automatic Nonuniform Random Variate Generation
3294:Analysis of the Linux Random Number Generator
3049:Anwendungshinweise und Interpretationen (AIS)
2690:Matsumoto, Makoto; Nishimura, Takuji (1998).
8:
3035:
3033:
2931:: CS1 maint: multiple names: authors list (
2406:{\displaystyle \operatorname {erf} ^{-1}(x)}
2337:{\displaystyle \operatorname {erf} ^{-1}(x)}
959:is the uniform distribution on the interval
152:random number algorithm used for decades on
1888:An early computer-based PRNG, suggested by
61:. The PRNG-generated sequence is not truly
84:PRNGs are central in applications such as
2977:"Lecture 11: The Goldreich-Levin Theorem"
2848:ACM Transactions on Mathematical Software
2796:ACM Transactions on Mathematical Software
2755:
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560:
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532:
403:-certified pseudorandom number generator
3022:Katz, Jonathan; Yehuda, Lindell (2014).
1699:is a pseudo-random number generator for
391:It has been shown to be likely that the
183:. Up until 2020, Java still relied on a
3168:, Third Edition. Addison-Wesley, 1997.
2518:
2299:For example, the inverse of cumulative
1646:of some given probability distribution
884:{\displaystyle A\subseteq \mathbb {R} }
643:– a non-empty collection of Borel sets
444:Federal Office for Information Security
2984:COM S 687 Introduction to Cryptography
2924:
2456:List of pseudorandom number generators
206:Generators based on linear recurrences
200:List of pseudorandom number generators
3040:Schindler, Werner (2 December 1999).
293:, strong evidence may be provided by
7:
3331:by Stephan Lavavej (Microsoft, 2013)
3303:" by Parikshit Gopalan, Raghu Meka,
3209:. New York: John Wiley & Sons.
3024:Introduction to modern cryptography
2528:"Recommendation for Key Management"
2413:should be reduced by means such as
1339:
1232:
1203:pseudo-random number generator for
1201:is the set of positive integers) a
783:
774:is not specified, it may be either
759:
686:
662:
652:
628:
598:
569:
374:Naor-Reingold pseudorandom function
2145:
2115:
2041:
1527:
1404:
1380:
1358:
1345:
1328:
818:
707:
47:deterministic random bit generator
25:
138:Correlation of successive values;
73:hardware random number generators
3009:"The Many Flaws of Dual_EC_DRBG"
2948:"IV.4 Perfect Random Generators"
2568:"Pseudorandom number generators"
2439:
1949:cumulative distribution function
1615:{\displaystyle \left(0,1\right)}
1044:{\displaystyle \left(0,1\right]}
988:{\displaystyle \left(0,1\right]}
547:– a probability distribution on
476:test (a special instance of the
3161:The Art of Computer Programming
2882:Song Y. Yan (7 December 2007).
2743:Journal of Statistical Software
2736:Marsaglia, George (July 2003).
2535:NIST Special Publication 800-57
1395:
1379:
1357:
1344:
1240:{\displaystyle {\mathfrak {F}}}
791:{\displaystyle {\mathfrak {B}}}
767:{\displaystyle {\mathfrak {F}}}
636:{\displaystyle {\mathfrak {F}}}
606:{\displaystyle {\mathfrak {B}}}
291:computational complexity theory
219:linear-feedback shift registers
3351:Pseudorandom number generators
3301:Better pseudorandom generators
3183:, Princeton Univ Press, 1996.
2886:. Springer, 2007. p. 73.
2496:Random number generator attack
2400:
2394:
2331:
2325:
2279:
2273:
2247:
2241:
2190:
2184:
2148:
2142:
2133:
2127:
2118:
2109:
2065:
2054:
2027:
2021:
1995:
1989:
1966:
1960:
1862:
1856:
1819:
1813:
1758:
1493:
1487:
1461:
1455:
1120:
212:linear congruential generators
78:pseudorandom number generators
1:
2680:Press et al. (2007) §7.1
2466:Linear congruential generator
1936:Pseudo-random number sampling
422:, which (most often) work by
275:cryptographically-secure PRNG
185:linear congruential generator
39:pseudorandom number generator
3256:: A free, state-of-the-art (
2884:Cryptanalytic Attacks on RSA
2476:Pseudorandom binary sequence
2371:Repetitive recalculation of
1692:{\displaystyle F^{*}\circ f}
395:has inserted an asymmetric
2653:Random (Java Platform SE 8)
2616:Press et al. (2007), chap.7
2253:{\displaystyle b=F^{-1}(c)}
1979:of the target distribution
3367:
3231:Cambridge University Press
2946:Klaus Pommerening (2016).
2914:, Tadayoshi Kohno (2010).
2632:. Springer. p. 1629.
2591:Von Neumann, John (1951).
2547:10.6028/NIST.SP.800-57p1r3
2461:Applications of randomness
2294:inverse transform sampling
1933:
262:
224:The 1997 invention of the
29:
3324:rand() Considered Harmful
3283:Generating random numbers
3278:Random Number Test Suite.
3264:Random Number Test Suite.
2826:10.1016/j.cam.2016.11.006
2167:. Using a random number
1892:in 1946, is known as the
273:applications is called a
243:introduced the family of
3201:Peterson, Ivars (1997).
3166:Seminumerical Algorithms
3026:. CRC press. p. 70.
2491:Random number generation
2471:Low-discrepancy sequence
1566:It can be shown that if
370:Micali–Schnorr generator
3127:SP800-90A, January 2012
2861:10.1145/1132973.1132974
2292:. This is based on the
861:, depending on context.
520:Mathematical definition
438:BSI evaluation criteria
2506:Statistical randomness
2417:for faster generation.
2407:
2358:
2338:
2286:
2254:
2203:
2202:{\displaystyle F(b)=c}
2161:
2084:
2002:
1973:
1930:Non-uniform generators
1874:
1790:
1770:
1713:
1693:
1660:
1636:
1616:
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1557:
1537:
1511:
1314:
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1241:
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1195:
1132:
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1065:
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989:
953:
925:
905:
885:
855:
792:
768:
744:
671:
637:
607:
583:
541:
456:
301:that is assumed to be
177:
32:Pseudorandom generator
3289:) by Eric Uner (2004)
3007:(18 September 2013).
2757:10.18637/jss.v008.i14
2715:10.1145/272991.272995
2408:
2359:
2339:
2301:Gaussian distribution
2287:
2255:
2204:
2162:
2085:
2003:
1974:
1947:First, one needs the
1875:
1791:
1776:is the percentile of
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1714:
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1133:
1086:
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954:
926:
906:
886:
856:
793:
769:
745:
672:
638:
608:
584:
542:
412:uniformly distributed
307:integer factorization
297:to the CSPRNG from a
232:10), is proven to be
98:procedural generation
3142:Gentle J.E. (2003),
3139:, 2007; 48:C188–C202
2375:
2348:
2306:
2285:{\displaystyle f(b)}
2267:
2219:
2178:
2097:
2015:
2001:{\displaystyle f(b)}
1983:
1972:{\displaystyle F(b)}
1954:
1942:uniform distribution
1924:middle-square method
1894:middle-square method
1800:
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681:
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593:
551:
531:
269:A PRNG suitable for
106:elaborate algorithms
2050:
1536:{\displaystyle \#S}
1095:We call a function
939:; for instance, if
259:Cryptographic PRNGs
154:mainframe computers
45:), also known as a
3317:Microsoft Research
3195:Monte Carlo Method
3153:, Springer-Verlag.
2481:Pseudorandom noise
2447:Mathematics portal
2415:ziggurat algorithm
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90:Monte Carlo method
3226:Numerical Recipes
2893:978-0-387-48741-0
2639:978-3-642-04897-5
2357:{\displaystyle x}
1789:{\displaystyle P}
1712:{\displaystyle P}
1659:{\displaystyle P}
1635:{\displaystyle F}
1579:{\displaystyle f}
1556:{\displaystyle S}
1479:
1260:{\displaystyle A}
1247:taking values in
1216:{\displaystyle P}
1084:{\displaystyle P}
1064:{\displaystyle A}
1008:{\displaystyle A}
952:{\displaystyle P}
924:{\displaystyle P}
911:is a set between
904:{\displaystyle A}
617:on the real line)
540:{\displaystyle P}
351:(incorporated in
16:(Redirected from
3358:
3325:
3287:embedded systems
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1890:John von Neumann
1884:Early approaches
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613:is the standard
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546:
544:
543:
538:
478:chi-squared test
451:
385:one-way function
349:Yarrow algorithm
241:George Marsaglia
231:
226:Mersenne Twister
196:Mersenne Twister
120:Potential issues
114:John von Neumann
94:electronic games
21:
3366:
3365:
3361:
3360:
3359:
3357:
3356:
3355:
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3105:
3104:
3094:
3092:
3091:on May 27, 2013
3073:
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3055:. pp. 5–11
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2805:10.1145/2845077
2792:
2788:
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2777:
2773:
2738:"Xorshift RNGs"
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558:
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549:
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529:
528:
522:
498:autocorrelation
447:
440:
330:output feedback
287:polynomial time
267:
261:
234:equidistributed
229:
208:
122:
35:
28:
23:
22:
15:
12:
11:
5:
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3320:
3297:
3290:
3279:
3265:
3249:
3248:External links
3246:
3245:
3244:
3234:
3221:
3215:
3198:
3191:
3177:
3154:
3147:
3140:
3136:ANZIAM Journal
3128:
3109:
3106:
3103:
3102:
3066:
3029:
3014:
2996:
2975:Pass, Rafael.
2967:
2938:
2912:Bruce Schneier
2908:Niels Ferguson
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1934:Main article:
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1269:if and only if
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420:stream ciphers
389:
388:
381:
378:Blum Blum Shub
366:
363:
345:CryptGenRandom
333:
319:
317:stream ciphers
263:Main article:
260:
257:
207:
204:
146:
145:
142:
139:
136:
133:
121:
118:
88:(e.g. for the
59:random numbers
26:
24:
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13:
10:
9:
6:
4:
3:
2:
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3349:
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3309:Luca Trevisan
3306:
3305:Omer Reingold
3302:
3298:
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3216:0-471-16449-6
3212:
3207:
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3189:9780691025469
3186:
3182:
3178:
3176:. Chapter 3.
3175:
3174:0-201-89684-2
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3006:
3005:Matthew Green
3000:
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1920:Weyl sequence
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1910:punched cards
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322:block ciphers
320:
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304:
300:
296:
292:
288:
283:
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271:cryptographic
266:
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250:In 2006, the
248:
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3224:
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3180:
3165:
3164:, Volume 2:
3159:
3150:
3143:
3134:
3119:
3108:Bibliography
3093:. Retrieved
3089:the original
3078:
3069:
3057:. Retrieved
3048:
3023:
3017:
2999:
2987:. Retrieved
2983:
2970:
2959:. Retrieved
2956:uni-mainz.de
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2586:
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2572:Khan Academy
2571:
2562:
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2534:
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1939:
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424:exclusive or
416:cryptography
409:
405:Dual_EC_DRBG
390:
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102:cryptography
83:
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3146:, Springer.
3113:Barker E.,
2855:(1): 1–16.
2665:Random.java
442:The German
324:running in
161:Monte Carlo
86:simulations
3270:: A free (
3157:Knuth D.E.
3131:Brent R.P.
2961:2017-11-12
2952:Cryptology
2577:2016-01-11
2513:References
2501:Randomness
2093:Note that
508:generator.
496:, and the
432:ciphertext
305:, such as
279:negligible
96:(e.g. for
53:), is an
3268:DieHarder
3179:Luby M.,
3115:Kelsey J.
3095:19 August
3059:19 August
2779:S.Vigna.
2766:250501391
2552:19 August
2392:
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1851:≤
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1732:∗
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1121:→
1015:might be
874:⊆
839:∈
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816:−
728:∈
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658:⊆
615:Borel set
449:‹See Tfd›
428:plaintext
426:-ing the
399:into the
343:function
337:Microsoft
282:advantage
239:In 2003,
126:artifacts
55:algorithm
3345:Category
3237:Viega J.
2927:cite web
2709:: 3–30.
2606:: 36–38.
2433:See also
2423:Rayleigh
2212:so that
2077:′
2062:′
1719:, where
937:interior
935:and its
486:longruns
397:backdoor
376:and the
353:Mac OS X
295:reducing
245:xorshift
170:(2010).
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3329:YouTube
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3254:TestU01
2989:20 July
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2669:OpenJDK
2427:Poisson
1796:, i.e.
1666:, then
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1138:(where
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677:, e.g.
589:(where
524:Given:
512:states.
469:monobit
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359:), and
357:FreeBSD
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299:problem
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453:German
347:, the
63:random
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2762:S2CID
2719:S2CID
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2695:(PDF)
2596:(PDF)
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1906:ENIAC
1051:. If
750:. If
474:poker
150:RANDU
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3185:ISBN
3170:ISBN
3125:NIST
3097:2013
3085:NIST
3080:FIPS
3061:2013
2991:2016
2933:link
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492:and
482:runs
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355:and
332:mode
303:hard
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3272:GPL
3262:C++
3258:GPL
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