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The Complexity of Nonuniform Random Number Generation Donald E. Knuth and Andrew C. Yao Titles and Abstracts of Contributed Papers. LIST OF INVITED AUTHORS. Cent Results in Algorithms and Complexity held by the Computer Science Department, Carnegie-Mellon University, April 7-9, 1976. A Hardware Efficient Random Number Generator for Nonuniform Distributions with Arbitrary Precision Article (PDF Available) in International Journal of Reconfigurable Computing 2012(6) January.
Are an example of a mechanical hardware random number generator. When a cubical die is rolled, a random number between 1 and 6 is obtained. A random number generator (RNG) is a device that generates a sequence of or symbols that cannot be reasonably predicted better than by a chance. Random number generators can be true (HRNG), which generate genuinely random numbers, or (PRNG) which generate numbers which look random, but are actually deterministic, and can be reproduced if the state of the PRNG is known. Various have led to the development of several different methods for generating data, of which some have existed since ancient times, among whose ranks are well-known 'classic' examples, including the rolling of, the of, the use of stalks (for ) in the, as well as countless other techniques. Because of the mechanical nature of these techniques, generating large numbers of sufficiently random numbers (important in statistics) required a lot of work and/or time. Thus, results would sometimes be collected and distributed as.
Several computational methods for pseudo-random number generation exist. All fall short of the goal of true randomness, although they may meet, with varying success, some of the intended to measure how unpredictable their results are (that is, to what degree their patterns are discernible). This generally makes them unusable for applications such as.
However, carefully designed (CSPRNG) also exist, with special features specifically designed for use in cryptography. Main article: Random number generators have applications in, and other areas where producing an unpredictable result is desirable.
Generally, in applications having unpredictability as the paramount, such as in security applications, are generally preferred over pseudo-random algorithms, where feasible. Random number generators are very useful in developing simulations, as is facilitated by the ability to run the same sequence of random numbers again by starting from the same. They are also used in – so long as the seed is secret. Sender and receiver can generate the same set of numbers automatically to use as keys.
The generation of 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 and in creating. Some applications which appear at first sight to be suitable for randomization are in fact not quite so simple.
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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 true random system would have no restriction on the same item appearing two or three times in succession. Pseudo-random numbers. See also: 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, 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. The speed at which entropy can be harvested from natural sources is dependent on the underlying physical phenomena being measured. Thus, sources of naturally occurring 'true' entropy are said to be – they are rate-limited until enough entropy is harvested to meet the demand. On some Unix-like systems, including most, 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 with random bits, can often be slow on systems that use this type of entropy source. The second method uses computational 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. 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. 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. Some systems take a hybrid approach, providing randomness harvested from natural sources when available, and falling back to periodically re-seeded software-based (CSPRNGs). The fallback occurs when the desired read rate of randomness exceeds the ability of the natural harvesting approach to keep up with the demand. This approach avoids the rate-limited blocking behavior of random number generators based on slower and purely environmental methods. 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. The former is the basis of the /dev/random source of entropy on, and others. Also uses a pseudo-random number algorithm based on known as. Generation methods Physical methods.
See also: and 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 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.
Other considerations 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 (CDF) of the desired distribution (see ). Inverse CDFs are also called. To generate a pair of random numbers ( x, y), one may first generate the ( r, θ), where r and θ (see ).
Some 0 to 1 RNGs include 0 but exclude 1, while others include or exclude both. The outputs of multiple independent RNGs can be combined (for example, using a bit-wise operation) to provide a combined RNG at least as good as the best RNG used.
This is referred to as. 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.' 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. For such problems, it may be possible to find a more accurate solution by the use of so-called, also called 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 resource pages contain a number of of random number generation using Java applets. The Quantum Optics Group at the generates random numbers sourced from quantum vacuum. You can download a sample of random numbers by visiting their research page. makes available random numbers that are sourced from the randomness of atmospheric noise.
The at the 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 chaotic laser. You can obtain a sample of random number by visiting their. Backdoors. Main article: Since much cryptography depends on a cryptographically secure random number generator for key and generation, if a random number generator can be made predictable, it can be used as by an attacker to break the encryption.
The NSA is reported to have inserted a backdoor into the certified. If for example an SSL connection is created using this random number generator, then according to 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 DualECDRBG 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. There have subsequently been accusations that RSA Security knowingly inserted a NSA backdoor into its products, possibly as part of the. 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 which 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 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, by the information security director of the (MUSL), who surreptitiously installed backdoor on the MUSL's secure RNG computer during routine maintenance. During the hacks the man won a total amount of $16,500,000 by predicting the numbers correct a few times in year. ASLR or Address Space Layout Randomization, a mitigation against rowhammer and related attacks on the physical hardware of memory chips has been found to be inadequate as of early 2017 by VUSec.
The random number algorithm if based on a shift register implemented in hardware is predictable at sufficiently large values of p and can be reverse engineered with enough processing power. This also indirectly means that malware using this method can run on both GPUs and CPUs if coded to do so, even using GPU to break ASLR on the CPU itself. See also.