Linear congruential generator

File:Linear congruential generator visualisation.svg

Visualisation of generation of pseudo-random numbers in [0, 8] using a linear congruential generator. The top two rows show a generator with m = 9, a = 2 and c = 0 outputting numbers from left to right until the output equals the seed, when the sequence repeats. A seed of 1 gives a cycle length of 6 but a seed of 3 gives a cycle length of only 2. Using a = 4 and c = 1 (bottom row) gives a full cycle length of 9 with any seed.

A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation. The method represents one of the oldest and best-known pseudorandom number generator algorithms. The theory behind them is relatively easy to understand, and they are easily implemented and fast, especially on computer hardware which can provide modulo arithmetic by storage-bit truncation.

The generator is defined by the recurrence relation:

$X_{n+1} = \left( a X_n + c \right)~~\bmod~~m$

where $X$ is the sequence of pseudorandom values, and

$m,\, 0<m$ – the "modulus"

$a,\,0 < a < m$ – the "multiplier"

$c,\,0 \le c < m$ – the "increment"

$X_0,\,0 \le X_0 < m$ – the "seed" or "start value"

are integer constants that specify the generator. If c = 0, the generator is often called a multiplicative congruential generator (MCG), or Lehmer RNG. If c ≠ 0, the method is called a mixed congruential generator.

Period length

The period of a general LCG is at most m, and for some choices of factor a much less than that. The LCG will have a full period for all seed values if and only if:

1. $\,m$ and the offset $\,c$ are relatively prime,
2. $\,a - 1$ is divisible by all prime factors of $\,m$,
3. $\,a - 1$ is divisible by 4 if $\,m$ is divisible by 4.

These three requirements are referred to as the Hull-Dobell Theorem. While LCGs are capable of producing pseudorandom numbers which can pass formal tests for randomness, this is extremely sensitive to the choice of the parameters c, m, and a.

Historically, poor choices had led to ineffective implementations of LCGs. A particularly illustrative example of this is RANDU, which was widely used in the early 1970s and led to many results which are currently being questioned because of the use of this poor LCG.

Parameters in common use

The most efficient LCGs have an m equal to a power of 2, most often m = 2³² or m = 2⁶⁴, because this allows the modulus operation to be computed by merely truncating all but the rightmost 32 or 64 bits. The following table lists the parameters of LCGs in common use, including built-in rand() functions in runtime libraries of various compilers.

Source	m	(multiplier) a	(increment) c	output bits of seed in rand() / Random(L)
Numerical Recipes	232	1664525	1013904223
Borland C/C++	232	22695477	1	bits 30..16 in rand(), 30..0 in lrand()
glibc (used by GCC)	231	1103515245	12345	bits 30..0
ANSI C: Watcom, Digital Mars, CodeWarrior, IBM VisualAge C/C++	231	1103515245	12345	bits 30..16
C99, C11: Suggestion in the ISO/IEC 9899	231	1103515245	12345	bits 30..16
Borland Delphi, Virtual Pascal	232	134775813	1	bits 63..32 of (seed * L)
Microsoft Visual/Quick C/C++	232	214013 (343FD16)	2531011 (269EC316)	bits 30..16
Microsoft Visual Basic (6 and earlier)	224	1140671485 (43FD43FD16)	12820163 (C39EC316)
RtlUniform from Native API	231 − 1	2147483629 (7FFFFFED16)	2147483587 (7FFFFFC316)
Apple CarbonLib, C++11's minstd_rand0	231 − 1	16807	0	see MINSTD
C++11's minstd_rand	231 − 1	48271	0	see MINSTD
MMIX by Donald Knuth	264	6364136223846793005	1442695040888963407
Newlib, Musl	264	6364136223846793005	1	bits 63...32
VMS's MTH$RANDOM, old versions of glibc	232	69069	1
Java's java.util.Random, POSIX [ln]rand48, glibc [ln]rand48[_r]	248	25214903917 (5DEECE66D16)	11	bits 47...16
POSIX [jm]rand48, glibc [mj]rand48[_r]	248	25214903917 (5DEECE66D16)	11	bits 47...15
POSIX [de]rand48, glibc [de]rand48[_r]	248	25214903917 (5DEECE66D16)	11	bits 47...0
Formerly common: RANDU	231	65539	0

As shown above, LCGs do not always use all of the bits in the values they produce. For example, the Java implementation operates with 48-bit values at each iteration but returns only their 32 most significant bits. This is because the higher-order bits have longer periods than the lower-order bits (see below). LCGs that use this truncation technique produce statistically better values than those that do not.

The Knuth representation for 3 variables is as below: X_n+1 = (8121 X_n + 28411) mod 134456

Because there are only 134456 distinct possible values, according to the parameter definition, it tends to make it a bit more predictable. If X_n is even then X_n+1 will be odd, and vice versa, so the lowest order of bit oscillates at each step. This makes the generator to produce bits in each number that are usually not equally random.

Advantages and disadvantages of LCGs

LCGs are fast and require minimal memory (typically 32 or 64 bits) to retain state. This makes them valuable for simulating multiple independent streams.

File:Lcg 3d.gif

Hyperplanes of a linear congruential generator in three dimensions

LCGs should not be used for applications where high-quality randomness is critical. For example, it is not suitable for a Monte Carlo simulation because of the serial correlation (among other things). They also must not be used for cryptographic applications; see cryptographically secure pseudo-random number generator for more suitable generators. If a linear congruential generator is seeded with a character and then iterated once, the result is a simple classical cipher called an affine cipher; this cipher is easily broken by standard frequency analysis.

LCGs tend to exhibit some severe defects. For instance, if an LCG is used to choose points in an n-dimensional space, the points will lie on, at most, (n!m)^1/n hyperplanes (Marsaglia's Theorem, developed by George Marsaglia). This is due to serial correlation between successive values of the sequence X_n. The spectral test, which is a simple test of an LCG's quality, is based on this fact.

A further problem of LCGs is that the lower-order bits of the generated sequence have a far shorter period than the sequence as a whole if m is set to a power of 2. In general, the nth least significant digit in the base b representation of the output sequence, where b^k = m for some integer k, repeats with at most period bⁿ.

Yet another problem is that LCGs are not suitable for parallel programming. Multiple threads may access the currently stored state simultaneously causing a race condition. In implementations which use same initialization for different threads, equal sequences of random numbers may occur on simultaneously executing threads. Random number generators, particularly for parallel computers, should not be trusted. It is strongly recommended to check the results of simulation with more than one RNG to check if bias is introduced. Among the recommended generators for use on a parallel computer include combined linear congruential generators using sequence splitting and lagged Fibonacci generators using independent sequences.

Nevertheless, for some applications LCGs may be a good option. For instance, in an embedded system, the amount of memory available is often severely limited. Similarly, in an environment such as a video game console taking a small number of high-order bits of an LCG may well suffice. The low-order bits of LCGs when m is a power of 2 should never be relied on for any degree of randomness whatsoever. Indeed, simply substituting 2ⁿ for the modulus term reveals that the low order bits go through very short cycles. In particular, any full-cycle LCG when m is a power of 2 will produce alternately odd and even results.

Comparison with other PRNGs

Generators based on linear recurrences (such as xorshift*) or on good avalanching functions (such as SplitMix64[1]) outperform linear congruential generators even at small state sizes. Moreover, linear generators can generate very long sequences, and when suitably perturbed at the output, they pass strong statistical tests. Several examples can be found in the Xorshift family. The Mersenne twister algorithm provides a very long period (2¹⁹⁹³⁷ − 1) and variate uniformity, but it fails some statistical tests. A common Mersenne twister implementation, interestingly enough, uses an LCG to generate seed data.

Linear congruential generators have the problem that all of the bits in each number are usually not equally random. A linear feedback shift register PRNG produces a stream of pseudo-random bits, each of which are truly pseudo-random, and can be implemented with essentially the same amount of memory as a linear congruential generator, albeit with a bit more computation.

The linear feedback shift register has a strong relationship to linear congruential generators. Given a few values in the sequence, some techniques can predict the following values in the sequence for not only linear congruent generators but any other polynomial congruent generator.

See also

• Full cycle
• Inversive congruential generator
• Multiply-with-carry
• Lehmer RNG (sometimes called the Park-Miller RNG)
• Combined Linear Congruential Generator

References

• 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.
• Gentle, James E., (2003). Random Number Generation and Monte Carlo Methods, 2nd edition, Springer, ISBN 0-387-00178-6.
• (in this paper, efficient algorithms are given for inferring sequences produced by certain pseudo-random number generators).

External links

• 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.
• Article about another way of cracking LCG