Random number generation

Random number generators have applications in gambling, statistical sampling, computer simulation, cryptography, completely randomized design, and other areas where producing an unpredictable result is desirable. Generally, in applications having unpredictability as the paramount feature, such as in security applications, hardware generators are generally preferred over pseudorandom algorithms, where feasible. Pseudorandom number generators are very useful in developing Monte Carlo-method simulations, as debugging is facilitated by the ability to run the same sequence of random numbers again by starting from the same random seed. They are also used in cryptography – so long as the seed is secret. The sender and receiver can generate the same set of numbers automatically to use as keys. The generation of pseudorandom numbers is an important and common task in computer programming. While cryptography and certain numerical algorithms require a very high degree of apparent 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 randomness are used in hash algorithms and in creating amortized searching and sorting algorithms. Some applications that appear at first sight to be suitable for randomization are in fact not quite so simple. For instance, a system that "randomly" selects music tracks for a background music system must only appear 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. == True vs. pseudo-random numbers ==

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 atmospheric noise, 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 entropy (as a measure of unpredictability or surprise of the number generation process). 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 blocking they are rate-limited until enough entropy is harvested to meet the demand. On some Unix-like systems, including most Linux distributions, the pseudo device file will block until sufficient entropy is harvested from the environment. Due to this blocking behavior, large bulk reads from , such as filling a hard disk drive with random bits, can often be slow on systems that use this type of entropy source. The second method uses computational algorithms 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 key. 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 pseudorandom number generator. 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. Standard cryptographic designs take a hybrid approach, using randomness harvested from natural sources to seed a cryptographically secure pseudorandom number generators (CSPRNGs). Hardware random number generators generally produce only a limited number of random bits per second. In order to increase the available output data rate, they are often used to generate the "seed" for a faster PRNG. PRNG also helps with the noise source "anonymization" (whitening out the noise source identifying characteristics) and entropy extraction. With a proper PRNG algorithm selected (cryptographically secure pseudorandom number generator, CSPRNG), the combination can satisfy the requirements of Federal Information Processing Standards and Common Criteria standards. == Generation methods ==

Physical methods 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. Computational methods For performance and security reasons, most random number generators use PRNGs in their construction (this architecture is actually mandated in the industrial standards like NIST SP 800-90A). By humans 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.g., digits or letters. They may alternate too much between choices when compared to a good random generator; thus, this 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. == Post-processing and statistical checks ==

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. 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 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. Statistical tests are also used to give confidence that the post-processed final output from a random number generator is truly unbiased, with numerous randomness test suites being developed. == Other considerations ==

Reshaping the distribution Uniform distributions 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: • The integer used in the transformation must provide enough bits for the intended precision. • 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. • 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. The mainstream algorithm, used by OpenJDK, Rust, and NumPy, is described in a proposal for C++'s STL. It does not use the extra precision and suffers from bias only in the last bit due to rounding half to even. Other numeric concerns are warranted when shifting this canonical uniform distribution to a different range. A proposed method for the Swift programming language claims to use the full precision everywhere. Uniformly distributed integers are commonly used in algorithms such as the Fisher–Yates shuffle. 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 Apple Inc. Most 0 to 1 RNGs include 0 but exclude 1, while others include or exclude both. Other distributions Given a source of uniform random numbers, there are a couple of methods to create a new random source that corresponds to a probability density function. 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. As an example for rejection sampling, to generate a pair of statistically independent standard normally distributed random numbers (x, y), one may first generate the polar coordinates (r, θ), where r2~χ22 and θ~UNIFORM(0,2π) (see Box–Muller transform). Whitening The outputs of multiple independent RNGs can be combined (for example, using a bit-wise XOR operation) to provide a combined RNG at least as good as the best RNG used. 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 pseudorandom numbers much faster than physical generators, while physical generators can generate true randomness. == Low-discrepancy sequences as an alternative ==

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. == Activities and demonstrations ==

The following sites make available random number samples: • The SOCR resource pages contain a number of hands-on interactive activities and demonstrations of random number generation using Java applets. • The Quantum Optics Group at the ANU generates random numbers sourced from quantum vacuum. Samples of random numbers are available at their quantum random number generator research page. • Random.org makes available random numbers that are sourced from the randomness of atmospheric noise. • The Quantum Random Bit Generator Service at the Ruđer Bošković Institute 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. • 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. == Backdoors ==

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. The NSA is reported to have inserted a backdoor into the NIST certified cryptographically secure pseudorandom number generator Dual EC DRBG. If for example an SSL connection is created using this random number generator, then according to Matthew Green 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 confirmed in 2013, it had seen significant usage in practice until 2013, for example by the prominent security company RSA Security. There have subsequently been accusations that RSA Security knowingly inserted a NSA backdoor into its products, possibly as part of the Bullrun program. RSA has denied knowingly inserting a backdoor into its products. 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 optical reverse-engineering. For example, for random number generation in Linux, it is seen as unacceptable to use Intel's RDRAND 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. In 2010, a U.S. lottery draw was rigged by the information security director of the Multi-State Lottery Association (MUSL), who surreptitiously installed backdoor malware 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. == See also ==