线性同馀方法

线性同馀方法（LCG）是个产生伪随机数的方法。

它是根据以下的递回关系式：

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

其中${\displaystyle A,B,M}$是产生器设定的常数。

LCG的周期最大为${\displaystyle M}$，但大部分情况都会少于M。要令LCG达到最大周期，应符合以下条件：

1. ${\displaystyle B,M}$互质；
2. ${\displaystyle M}$的所有质因数都能整除${\displaystyle A-1}$；
3. 若${\displaystyle M}$是4的倍数，${\displaystyle A-1}$也是；
4. ${\displaystyle A,B,N_{0}}$都比${\displaystyle M}$小；
5. ${\displaystyle A,B}$是正整数。

随机性

因为通过线性同余方法构建的伪随机数生成器的内部状态可以轻易地由其输出演算得知，所以此种伪随机数生成器属于统计学伪随机数生成器。

设计密码学的应用必须至少使用密码学安全伪随机数生成器，故需要避免由线性同余方法获得的随机数在密码学中的应用。

参见

• Mersenne Twister

参考文献

• 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 [2012-12-30]. doi:10.1090/S0025-5718-99-00996-5. （原始内容存档于2005-05-16）.
• 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 [2012-12-30], ISBN 978-0-521-88068-8, （原始内容存档于2011-08-11）
• 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).

外部链接

• The simulation Linear Congruential Generator （页面存档备份，存于互联网档案馆） 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 （页面存档备份，存于互联网档案馆）
• 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. （页面存档备份，存于互联网档案馆）
• Additive Congruential Method : maths and logic behind its spread