## Casting Lots in the Digital AgeJune 23, 2016 Random numbers are an important part of computer programming. Aside from their obvious use as a method of throwing "digital dice" for computer versions of board games and generating random play in other computer games, they're used extensively in scientific simulations. One of the earliest uses of random numbers in computer simulation was for the Monte Carlo method, developed principally by Nicholas Metropolis in the early 1950s. John von Neumann (1903-1957), a founding figure in computing, was involved also with the development of the Monte Carlo method. Von Neumann was most concerned that the numbers generated by a computer were as close to random as possible. Since these "random" numbers are actually produced by an algorithm, they aren't really random, just pseudorandom. Von Neumann cautioned against taking the pseudorandom character of these numbers too lightly when he said, "Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin."[1] Von Neumann advanced pseudorandom number generation in two ways. First, he invented the middle-square method, a rapid generator suitable for simple applications.[1] An example of this method for generating four digit random numbers follows:• Select a four-digit "seed" number: 5678In spreadsheet programs, such as Gnumeric, you can select the middle digits of a number n using the modulus and integer functions; i.e., =mod(int(n/100),10000). Spreadsheets have their own random function, which is better to use. The first few subsequent numbers generated in the above example are 7408, 8784, 1586, 5153, 5534, 6251, 750, 5625, 6406, and 368. There are two problems with von Neumann's middle-square method. If the middle digits become all zeros, subsequent output is always zero. The generator can also enter a mode that outputs the same short sequence, again, and again.
Von Neumann further created a process to extract a greater degree of randomness from a biased coin toss. Since there are two states (head/tail) in a coin toss, this Von Neumann Whitening is simply applied to a bitstream of ones and zeros. The method takes two successive bits at a time, and maps them to a random stream as follows: (0,1) → 0, (1,0) → 1, (0,0) → NULL, (1,1) → NULL, the NULL meaning that the result is ignored, and you move on to the next two bits of input data. As can be seen in the example below of a bitstream slightly biased to 1, this method discards a lot of bits.
^{19937} - 1. It passes the stringent Diehard suite of statistical tests for randomness.[3]
Since algorithmic sources of randomness are always problematic, you can look to nature to give you some physical sources of randomness. In 1926, John B. Johnson of Bell Labs discovered that resistors produce a noise voltage.[4] His Bell Labs colleague, Harry Nyquist, was able to explain this phenomenon, which is now known as Johnson-Nyquist noise.[5]
This random voltage noise exists across any resistor at any temperature above absolute zero. Its value is,
v is the root-mean-square (rms) noise voltage, _{n}k is the Boltzmann constant (1.3806 x 10_{B}^{−23} joule/kelvin), R is the resistance (ohms), T is the absolute temperature (kelvin), and Δf is the frequency interval over which the noise voltage is measured (hertz). Since the noise voltage doesn't depend on where the frequency interval is taken, it's frequency-independent noise; that is, "white noise."
The amplitude of Johnson-Nyquist noise is small, so electrical engineers have found better noise sources. Although the primary purpose of a Zener diode is to act as a voltage reference or voltage limiter, a reverse-biased Zener diode will produce a noise voltage. A basic circuit is shown in the figure, and two amplifiers are usually cascaded to produce volt-level signals. Zener diodes with higher breakdown voltage produce stronger noise signals.
## References:- John von Neumann, “Various techniques used in connection with random digits,” in A.S. Householder, G.E. Forsythe, and H.H. Germond, eds., Monte Carlo Method, National Bureau of Standards Applied Mathematics Series, vol. 12 (Washington, D.C.: U.S. Government Printing Office, 1951): pp. 36-38 (PDF File).
- Geoff Kuenning, "Mersenne Twist Pseudorandom Number Generator Package, 2013.
- Dieharder is a GPL version of Diehard, Robert G. Brown, Dirk Eddelbuettel, and David Bauer, "Dieharder: A Random Number Test Suite," Duke University Web Site.
- J. B. Johnson, "Thermal Agitation of Electricity in Conductors," Phys. Rev., vol. 32 (July 1, 1928), pp. 97ff..
- H. Nyquis, "Thermal Agitation of Electric Charge in Conductors," Phys. Rev., vol. 32 (July 1, 1928), pp. 110ff..
- Willie D. Jones, "Intel Makes a Digital Coin Tosser for Future Processors," IEEE Spectrum Online, June 29, 2010.
- One method is described on the Hot Bits Web Site.
- Eshan Chattopadhyay and David Zuckerman, "Explicit Two-Source Extractors and Resilient Functions," The Electronic Colloquium on Computational Complexity, March 20, 2016.
- New Method of Producing Random Numbers Could Improve Cybersecurity, University of Texas Press Release, May 16, 2016.
Linked Keywords: Random number generation; random number; computer programming; dice; board game; randomness; random; computer game; science; scientific; computer simulation; Monte Carlo method; Nicholas Metropolis; 1950s; John von Neumann (1903-1957); computing; algorithm; pseudorandom number generator; Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin; invention; invented; middle-square method; numerical digit; random seed; exponentiation; square; data structure alignment; Pad; spreadsheet; computer program; Gnumeric; modulo operation; modulus; floor function; integer; function; mode; sequence; fair coin; biased coin toss; Von Neumann extractor; Von Neumann Whitening; bitstream; Null string; NULL; Inkscape; computer hardware; software; linear feedback shift register; shift register; exclusive-or (XOR) gate; feedback; linear congruential generator; recurrence relation; Derrick Henry Lehmer; cryptography; Mersenne Twister; Mersenne prime; Diehard tests; statistical hypothesis testing; statistical test; nature; physics; physical; John Bertrand Johnson; Bell Labs; colleague; Harry Nyquist; phenomenon; Johnson-Nyquist noise; voltage; resistor; temperature; absolute zero; root-mean-square; Boltzmann constant; joule; kelvin; resistance; ohm; frequency; hertz; independent variable; frequency-independent; white noise; amplitude; electrical engineering; electrical engineer; Zener diode; voltage reference; limiter; reverse-bias; electronic circuit; amplifier; breakdown voltage; analog signal; digital; bitstream; electronics; God does not play dice; quantum mechanics; radioactive decay; dice; physicist; M-theory dimensions; Wikimedia Commons; Clément Bucco-Lechat; computer scientist; University of Texas at Austin (Austin, Texas); scientific literature; publish; research; professor; David Zuckerman; postgraduate education; graduate student; Eshan Chattopadhyay; draft document; Electronic Colloquium on Computational Complexity; arXiv; author; journal. |
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