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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.
Indeed, carefully designed and implemented pseudo-random number generators can be certified for security-critical cryptographic purposes, as is the case with the yarrow algorithm and fortuna.
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.
A physical random number generator can be based on an essentially random atomic or subatomic physical phenomenon whose unpredictability can be traced to the laws of quantum mechanics.
However, physical phenomena and tools used to measure them generally feature asymmetries and systematic biases that make their outcomes not uniformly random.
A randomness extractor , such as a cryptographic hash function , can be used to approach a uniform distribution of bits from a non-uniformly random source, though at a lower bit rate.
The appearance of wideband photonic entropy sources, such as optical chaos and amplified spontaneous emission 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 chaotic laser was built in 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. Lavarand used this technique with images of a number of lava lamps.
HotBits measures radioactive decay with Geiger—Muller tubes ,  while Random. 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 mixed strategy games.
Most computer generated random numbers use pseudorandom number generators 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 numbers generated from electromagnetic atmospheric noise used as a source of entropy.
One of the most common PRNG is the linear congruential generator , which uses the recurrence. The maximum number of numbers the formula can produce is one less than the modulus , m The recurrence relation can be extended to matrices to have much longer periods and better statistical properties.
A simple pen-and-paper method for generating random numbers is the so-called middle square method suggested by John von Neumann. 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 Weyl sequence. This method produces high quality output through a long period.
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 floating point number uniformly distributed between 0 and 1.
The default random number generator in many languages, including Python, Ruby, R, IDL and PHP is based on the Mersenne Twister algorithm and is not 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.
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, statistics or numerical analysis.
Most programming languages, including those mentioned above, provide a means to access these higher quality sources. There are a couple of methods to generate a random number based on a probability density function.
These methods involve transforming a uniform random number in some way. Because of this, these methods work equally well in generating both pseudo-random and true random numbers.
One method, called the inversion method , 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 acceptance-rejection method , 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. Random number generation may also be performed by humans, in the form of collecting various inputs from end users 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.
They may alternate too much between choices when compared to a good random generator;  thus, this approach is not widely used. 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 outside interference.
And a software bug in a pseudo-random number routine, or a hardware bug in the hardware it runs on, may be similarly difficult to detect. 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 TRNG  hardware random number generator, which uses an entropy measurement 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. Random numbers uniformly distributed between 0 and 1 can be used to generate random numbers of any desired distribution by passing them through the inverse cumulative distribution function CDF of the desired distribution see Inverse transform sampling.
Inverse CDFs are also called quantile functions. This is referred to as software whitening. Computational and hardware random number generators are sometimes combined to reflect the benefits of both kinds.
Computational random number generators can typically generate pseudo-random numbers much faster than physical generators, while physical generators can generate "true randomness.
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 Monte Carlo method.
For such problems, it may be possible to find a more accurate solution by the use of so-called low-discrepancy sequences , also called quasirandom 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.
Since much cryptography depends on a cryptographically secure random number generator for key and cryptographic nonce generation, if a random number generator can be made predictable, it can be used as backdoor by an attacker to break the encryption.
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Generate using characterset X This generates codes of a given length consisting of the selected charactersets. Generation options Codes to generate is the number of codes that will be generated.