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prngs 4 even today caution is sometimes required as illustrated by the following warning in the international encyclopedia of statistical science 2010 5 the list of widely used generators that should be discarded is much longer than the list of good generators 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 amazingly it remains as relevant today as it was 40 years ago as an illustration consider the widely used programming language java up until 2020 java still relied on a linear congruential generator lcg for its prng 6 7 which is of low quality see further below java support was upgraded with java 17 one well known prng to avoid major problems and still run fairly quickly is the mersenne twister 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 list of pseudorandom number generators generators based on linear recurrences edit in the second half of the 20th century the standard class of algorithms used for prngs comprised linear congruential generators 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 lcgs and related were to disappear from library shelves there would be a gap on each shelf about as big as your fist 8 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 linear feedback shift registers the 1997 invention of the mersenne twister 9 in particular avoided many of the problems with earlier generators the mersenne twister has a period of 2 19 937 1 iterations 4 3 10 6001 is proven to be equidistributed in up to 623 dimensions for 32 bit values and at the time of its introduction was running faster than other statistically reasonable generators in 2003 george marsaglia introduced the family of xorshift generators 10 again based on a linear recurrence such generators are extremely fast and combined with a nonlinear operation they pass strong statistical tests 11 12 13 in 2006 the well family of generators was developed 14 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 counter based rngs edit main article counter based random number generator a counter based random number generation cbrng also known as a counter based pseudo random number generator or cbprng is a kind of prng that uses only an integer counter as its internal state output f n key displaystyle text output f n text key they are generally used for generating pseudorandom numbers for large parallel computations such as over gpu or cpu clusters 15 they have certain advantages the only state needed is the counter value and the key for a given counter and key the output is always the same this property makes cbrngs reproducible because each random number is computed independently of any previous outputs they can be generated in parallel for example in a massively parallel application each thread or gpu core can be assigned a range of counter values and compute random numbers without synchronization or shared state since the generator does not require stepping through every intermediate state it can jump to any point in the sequence in constant time this is particularly useful in applications like monte carlo simulations where independent streams are needed examples include 15 philox uses multiplication based mixing to combine the counter and key threefry based on a reduced strength version of the threefish block cipher cryptographic prngs edit main article cryptographically secure pseudorandom number generator a prng suitable for cryptographic applications is called a cryptographically secure prng csprng a requirement for a csprng is that an adversary not knowing the seed has only negligible advantage 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 polynomial time in the size of the seed though a proof of this property is beyond the current state of the art of computational complexity theory strong evidence may be provided by reducing to the csprng from a problem that is assumed to be hard such as integer factorization 16 in general years of review may be required before an algorithm can be certified as a csprng some classes of csprngs include the following stream ciphers block ciphers running in counter 17 or output feedback mode prngs that have been designed specifically to be cryptographically secure such as microsoft s cryptographic application programming interface function cryptgenrandom the yarrow algorithm incorporated in mac os x and freebsd and fortuna combination prngs which attempt to combine several prng primitive algorithms with the goal of removing any detectable non randomness special designs based on mathematical hardness assumptions examples include the micali schnorr generator 18 naor reingold pseudorandom function and the blum blum shub algorithm which provide a strong security proof such algorithms are rather slow compared to traditional constructions and impractical for many applications generic prngs while it has been shown that a cryptographically secure prng can be constructed generically from any one way function 19 this generic construction is extremely slow in practice so is mainly of theoretical interest it has been shown to be likely that the nsa has inserted an asymmetric backdoor into the nist certified pseudorandom number generator dual_ec_drbg 20 most prng algorithms produce sequences that are uniformly distributed by any of several tests it is an open question and one central to the theory and practice of cryptography 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 21 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 stream ciphers which most often work by exclusive or ing the plaintext of a message with the output of a prng producing ciphertext 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 bsi evaluation criteria edit the german federal office for information security german bundesamt für sicherheit in der informationstechnik bsi has established four criteria for quality of deterministic random number generators 22 they are summarized here k1 there should be a high probability that generated sequences of random numbers are different from each other k2 a sequence of numbers is indistinguishable from truly random numbers according to specified statistical tests the tests are the monobit test equal numbers of ones and zeros in the sequence poker test a special instance of the chi squared test runs test counts the frequency of runs of various lengths longruns test checks whether there exists any run of length 34 or greater in 20 000 bits of the sequence both from bsi 22 and nist 23 and the autocorrelation test in essence these requirements are a test of how well a bit sequence has zeros and ones equally often after a sequence of n 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 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 generator 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 states for cryptographic applications only generators meeting the k3 or k4 standards are acceptable mathematical definition edit given p displaystyle p a probability distribution on r b displaystyle left mathbb r mathfrak b right where b displaystyle mathfrak b is the sigma algebra of all borel subsets of the real line f displaystyle mathfrak f a non empty collection of borel sets f b displaystyle mathfrak f subseteq mathfrak b e g f t t r displaystyle mathfrak f left left infty t right t in mathbb r right if f displaystyle mathfrak f is not specified it may be either b displaystyle mathfrak b or t t r displaystyle left left infty t right t in mathbb r right depending on context a r displaystyle a subseteq mathbb r a non empty set not necessarily a borel set often a displaystyle a is a set between p displaystyle p s support and its interior for instance if p displaystyle p is the uniform distribution on the interval 0 1 displaystyle left 0 1 right a displaystyle a might be 0 1 displaystyle left 0 1 right if a displaystyle a is not specified it is assumed to be some set contained in the support of p displaystyle p and containing its interior depending on context we call a function f n 1 r displaystyle f mathbb n _ 1 rightarrow mathbb r where n 1 1 2 3 displaystyle mathbb n _ 1 left 1 2 3 dots right is the set of positive integers a pseudo random number generator for p displaystyle p given f displaystyle mathfrak f taking values in a displaystyle a if and only if f n 1 a displaystyle f left mathbb n _ 1 right subseteq a e f ε 0 n n 1 n n i 1 2 n f i e n p e ε 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 s displaystyle s denotes the number of elements in the finite set s displaystyle s it can be shown that if f displaystyle f is a pseudo random number generator for the uniform distribution on 0 1 displaystyle left 0 1 right and if f displaystyle f is the cdf of some given probability distribution p displaystyle p then f f displaystyle f circ f is a pseudo random number generator for p displaystyle p where f 0 1 r displaystyle f left 0 1 right rightarrow mathbb r is the quantile function of p displaystyle p i e f x inf t r x f t displaystyle f x inf left t in mathbb r x leq f t right intuitively an arbitrary distribution can be simulated from a simulation of the standard uniform distribution early approaches edit an early computer based prng suggested by john von neumann in 1946 is known as the middle square method 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 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 them 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 write numbers if the numbers were written to cards they would take very much longer to write and read on the eniac computer he was using the middle square method generated numbers at a rate some hundred times faster than reading numbers in from punched cards the middle square method has since been supplanted by more elaborate generators a recent innovation is to combine the middle square with a weyl sequence this method produces high quality output through a long period see middle square method non uniform generators edit main article pseudo random number sampling numbers selected from a non uniform probability distribution can be generated using a uniform distribution prng and a function that relates the two distributions first one needs the cumulative distribution function f b displaystyle f b of the target distribution f b displaystyle f b f b b f b d b displaystyle f b int _ infty b f b db note that 0 f f b f 1 displaystyle 0 f infty leq f b leq f infty 1 using a random number c from a uniform distribution as the probability density to pass by we get f b c displaystyle f b c so that b f 1 c displaystyle b f 1 c is a number randomly selected from distribution f b displaystyle f b this is based on the inverse transform sampling for example the inverse of cumulative gaussian distribution erf 1 x displaystyle operatorname erf 1 x with an ideal uniform prng with range 0 1 as input x displaystyle x would produce a sequence of positive only values with a gaussian distribution however when using practical number representations the infinite tails of the distribution have to be truncated to finite values repetitive recalculation of erf 1 x displaystyle operatorname erf 1 x should be reduced by means such as ziggurat algorithm for faster generation similar considerations apply to generating other non uniform distributions such as rayleigh and poisson see also edit mathematics portal list of pseudorandom number generators applications of randomness linear congruential generator low discrepancy sequence pseudorandom binary sequence pseudorandom noise pseudorandomness random number generation random number generator attack randomness statistical randomness references edit barker elaine barker william burr william polk william smid miles july 2012 recommendation for key management pdf nist special publication 800 57 nist doi 10 6028 nist sp 800 57p1r3 retrieved 19 august 2013 pseudorandom number generators khan academy retrieved 2016 01 11 von neumann john 1951 various techniques used in connection with random digits pdf national bureau of standards applied mathematics series 12 36 38 archived from the original pdf on 28 november ...