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The most significant b−2 msbits are used to select a shift amount between b−2 and 2 b−2 +b−3. M: A multiply by a fixed constant. Each of these operations is either invertible (and thus one-to-one ) or a truncation (and thus 2 k -to-one for some fixed k ), so their composition maps the same fixed number of input states to each output value.
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A modification of Lagged-Fibonacci generators. A SWB generator is the basis for the RANLUX generator, [19] widely used e.g. for particle physics simulations. Maximally periodic reciprocals: 1992 R. A. J. Matthews [20] A method with roots in number theory, although never used in practical applications. KISS: 1993 G. Marsaglia [21]
Blum Blum Shub takes the form + =, where M = pq is the product of two large primes p and q.At each step of the algorithm, some output is derived from x n+1; the output is commonly either the bit parity of x n+1 or one or more of the least significant bits of x n+1.
Note that a power-of-2 modulus shares the problem as described above for c = 0: the low k bits form a generator with modulus 2 k and thus repeat with a period of 2 k; only the most significant bit achieves the full period. If a pseudorandom number less than r is desired, ⌊ rX/m ⌋ is a much higher-quality result than X mod r.
In some cases, data reveals an obvious non-random pattern, as with so-called "runs in the data" (such as expecting random 0–9 but finding "4 3 2 1 0 4 3 2 1..." and rarely going above 4). If a selected set of data fails the tests, then parameters can be changed or other randomized data can be used which does pass the tests for randomness.
Babbage went on to design his much more general analytical engine, but later designed an improved "Difference Engine No. 2" design (31-digit numbers and seventh-order differences), [9] between 1846 and 1849. Babbage was able to take advantage of ideas developed for the analytical engine to make the new difference engine calculate more quickly ...
For example, the built-in pseudorandom number generator provided by many programming languages and/or libraries may often have only 32 bits of internal state, which means it can only produce 2 32 different sequences of numbers. If such a generator is used to shuffle a deck of 52 playing cards, it can only ever produce a very small fraction of ...