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A pseudorandom number generator PRNG , also known as a deterministic random bit generator DRBG , [1] is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers.

The PRNG-generated sequence is not truly random , because it is completely determined by an initial value, called the PRNG's seed which may include truly random values. Although sequences that are closer to truly random can be generated using hardware random number generators , pseudorandom number generators are important in practice for their speed in number generation and their reproducibility. PRNGs are central in applications such as simulations e. Cryptographic applications require the output not to be predictable from earlier outputs, and more elaborate algorithms , which do not inherit the linearity of simpler PRNGs, are needed.

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.

John von Neumann cautioned about the misinterpretation of a PRNG as a truly random generator, and joked that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. A PRNG can be started from an arbitrary initial state using a seed state.

It will always produce the same sequence when initialized with that state. The period of a PRNG is defined thus: The period is bounded by the number of the states, usually measured in bits. However, since the length of the period potentially doubles with each bit of "state" added, it is easy to build PRNGs with periods long enough for many practical applications. If a PRNG's internal state contains n bits, its period can be no longer than 2 n results, and may be much shorter.

For some PRNGs, the period length can be calculated without walking through the whole period. Linear congruential generators have periods that can be calculated by factoring. In practice, the output from many common PRNGs exhibit artifacts that cause them to fail statistical pattern-detection tests. Defects exhibited by flawed PRNGs range from unnoticeable and unknown to very obvious.

It was seriously flawed, but its inadequacy went undetected for a very long time. In many fields, much research work prior to the 21st century that relied on random selection or on Monte Carlo simulations, or in other ways relied on PRNGs, is much less reliable than it might have been as a result of using poor-quality PRNGs.

The list of widely used generators that should be discarded is [long] Check the default [PRNG] 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. The first PRNG to avoid major problems and still run fairly quickly was the Mersenne Twister discussed below , which was published in Other high-quality PRNGs have since been developed.

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.

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 invention of the Mersenne Twister , [9] in particular, avoided many of the problems with earlier generators. In , 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.

In the WELL family of generators was developed. 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 the CSPRNG to a problem that is assumed to be hard , such as integer factorization. Most PRNG algorithms produce sequences which 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. 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.

We call a function f: Intuitively, an arbitrary distribution can be simulated from a simulation of the standard uniform distribution. The algorithm is as follows: For example, squaring the number "" yields "", which can be written as "", an 8-digit number being the square of a 4-digit number.

This gives "" as the "random" number. Repeating this procedure gives "" 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 "". 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. A recent innovation is to combine the middle square with a Weyl sequence. This method produces high quality output through a long period.

Numbers selected from a non-uniform probability distribution can be generated using a uniform distribution PRNG and a function that relates the two distributions. Using a random number c from a uniform distribution as the probability density to "pass by", we get.

Similar considerations apply to generating other non-uniform distributions such as Rayleigh and Poisson. From Wikipedia, the free encyclopedia. This page is about commonly encountered characteristics of pseudorandom number generators algorithms. For the formal concept in theoretical computer science, see Pseudorandom generator. Cryptographically secure pseudorandom number generator. Retrieved 19 August Journal of Statistical Software. Cryptanalytic Attacks on RSA. Design Principles and Practical Applications, Chapter 9.

Retrieved 20 July Introduction to modern cryptography. Anwendungshinweise und Interpretationen AIS. Archived from the original on May 27, Retrieved from " https: Views Read Edit View history. This page was last edited on 23 February , at By using this site, you agree to the Terms of Use and Privacy Policy.