線性同餘方法

線性同餘方法（LCG）是個產生偽隨機數的方法。

它是根據遞歸公式：

$N_{j+1}\equiv (A\times N_{j}+B){\pmod {M}}$

其中$A,B,M$是產生器設定的常數。

LCG的週期最大為$M$，但大部分情況都會少於M。要令LCG達到最大週期，應符合以下條件：

1. $B,M$互質；
2. $M$的所有質因數都能整除$A-1$；
3. 若$M$是4的倍數，$A-1$也是；
4. $A,B,N_{0}$都比$M$小；
5. $A,B$是正整數。

參見

• Mersenne Twister

參考文獻

• S.K. Park and K.W. Miller. Random Number Generators: Good Ones Are Hard To Find. Communications of the ACM. 1988, 31 (10): 1192–1201. doi:10.1145/63039.63042. 
• D. E. Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 3.2.1: The Linear Congruential Method, pp. 10–26.
• P. L'Ecuyer. Tables of Linear Congruential Generators of Different Sizes and Good Lattice Structure. Mathematics of Computation. 1999, 68 (225): 249–260. doi:10.1090/S0025-5718-99-00996-5. 
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP, Section 7.1.1. Some History, Numerical Recipes: The Art of Scientific Computing 3rd, New York: Cambridge University Press, 2007, ISBN 978-0-521-88068-8 
• Gentle, James E., (2003). Random Number Generation and Monte Carlo Methods, 2nd edition, Springer, ISBN 0-387-00178-6.
• Joan Boyar. Inferring sequences produced by pseudo-random number generators. Journal of the ACM. 1989, 36 (1): 129–141. doi:10.1145/58562.59305.  (in this paper, efficient algorithms are given for inferring sequences produced by certain pseudo-random number generators).

外部鏈接

• The simulation Linear Congruential Generator visualizes the correlations between the pseudo-random numbers when manipulating the parameters.
• Security of Random Number Generation: An Annotated Bibliography
• Linear Congruential Generators post to sci.math
• The "Death of Art" computer art project at Goldstein Technologies LLC, uses an LCG to generate 33,554,432 images
• P. L'Ecuyer and R. Simard, ``TestU01: A C Library for Empirical Testing of Random Number Generators, May 2006, Revised November 2006, ACM Transactions on Mathematical Software, 33, 4, Article 22, August 2007.
• Additive Congruential Method : maths and logic behind its spread