# Linear congruential generator Two modulo-9 LCGs show how different parameters lead to different cycle lengths. Each row shows the state evolving until it repeats. The top row shows a generator with m = 9, a = 2, c = 0, and a seed of 1, which produces a cycle of length 6. The second row is the same generator with a seed of 3, which produces a cycle of length 2. Using a = 4 and c = 1 (bottom row) gives a cycle length of 9 with any seed in [0, 8].

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 recurrence relation:

$X_{n+1}=\left(aX_{n}+c\right)~~{\bmod {~}}~m$ where $X$ is the sequence of pseudorandom values, and

$m,\,0 — the "modulus"
$a,\,0 — the "multiplier"
$c,\,0\leq c — the "increment"
$X_{0},\,0\leq X_{0} — 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.:4-

## Period length

A benefit of LCGs is that with appropriate choice of parameters, the period is known and long. Although not the only criterion, too short a period is a fatal flaw in a pseudorandom number generator.

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 m and a.[clarification needed] For example, a = 1 and c = 1 produces a simple modulo-m counter, which has a long period, but is obviously non-random.

Historically, poor choices for a have 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.

There are three common families of parameter choice:

### m prime, c = 0

This is the original Lehmer RNG formulation. The period is m−1 if the multiplier a is chosen to be a primitive element of the integers modulo m. The initial state must be chosen between 1 and m−1.

One disadvantage of a prime modulus is that the modular reduction requires a double-width product and an explicit reduction step. Often a prime just less than a power of 2 is used (the Mersenne primes 231−1 and 261−1 are popular), so that the reduction modulo 2e − d can be computed as (ax mod 2e) − d ⌊ax/2e⌋. This must be followed by a conditional addition of m if the result is negative, but the number of additions is limited to ad/m, which can be easily limited to one if d is small.

If a double-width product is unavailable, and the multiplier is chosen carefully, Schrage's method may be used. To do this, factor m = qa+r, i.e. q = ⌊m/a and r = m mod a. Then compute ax mod m = a(x mod q) − r ⌊x/q. Since x mod q < qm/a, the first term is strictly less than am/a = m. If a is chosen so that r ≤ q (and thus r/q ≤ 1), then the second term is also: r ⌊x/q⌋ ≤ rx/q = x(r/q) ≤ x(1) = x < m. Thus, both products can be computed with a single-witch product, and the difference between them lie in the range [1−mm−1], sp can be reduced to [0, m−1] with a single conditional add.

A second disadvantage is that it is awkward to convert the value 1 ≤ x < m to uniform random bits. If a prime just less than a power of 2 is used, sometimes the missing values are simply ignored.

### m a power of 2, c = 0

Choosing m to be a power of 2, most often m = 232 or m = 264, produces a particularly efficient LCG, because this allows the modulus operation to be computed by simply truncating the binary representation. In fact, the most significant bits are usually not computed at all. There are, however, disadvantages.

This form has maximal period m/4, achieved if a ≡  3 or a ≡ 5 (mod 8). The initial state X0 must be odd, and the low three bits of X alternate between two states and are not useful. It can be shown that this form is equivalent to a generator with a modulus a quarter the size and c ≠ 0.

A more serious issue with the use of a power-of-two modulus is that the low bits have a shorter period than the high bits. The lowest-order bit of X never changes (X is always odd), and the next two bits alternate between two states. (If a ≡ 5 (mod 8), then bit 1 never changes and bit 2 alternates. If a ≡ 3 (mod 8), then bit 2 never changes and bit 1 alternates.) Bit 3 repeats with a period of 4, bit 4 has a period of 8, and so on. Only the most significant bit of X achieves the full period.

### c ≠ 0

When c ≠ 0, correctly chosen parameters allow a period equal to m, for all seed values. This will occur if and only if::17—19

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.

This form may be used with any m. A power-of-2 modulus equal to a computer's word size is convenient and achieves the longest possible period (for any odd c and a ≡ 1 mod 4), but has the same problem mentioned above: only the most significant bit achieves the full period. Lower-order bit k repeats with a period of length 2k+1.

An odd (usually prime) modulus avoids this problem, but as mentioned previously cannot produce certain values.

Although the above requirements provide maximum period, they are not sufficient to guarantee a good generator. For example, it is desirable for a−1 to not be any more divisible by prime factors of m than necessary. Thus, if m is a power of 2, then a−1 should be divisible by 4 but not divisible by 8, i.e. a ≡ 5 (mod 8).:§3.2.1.3 Indeed, most multipliers a produce a sequence which fails one test for non-randomness or another, and finding a multiplier which is satisfactory to all applicable criteria:§3.3.3 is quite challenging.

The generator is not sensitive to the choice of c, as long as it is relatively prime to the modulus (e.g. if m is a power of 2, then c must be odd), so the value c=1 is commonly chosen.

The series produced by other choices of c can be written as a simple function of the series when c=1.:11 Specifically, if Y is the prototypical series defined by Y0 = 0 and Yn+1aYn+1 mod m, then a general series Xn+1aXn+c mod m can be written as an affine function of Y:

$X_{n}=(X_{0}(a-1)+c)Y_{n}+X_{0}=(X_{1}-X_{0})Y_{n}+X_{0}{\pmod {m}}.$

More generally, any two series X and Z with the same multiplier and modulus are related by

${X_{n}-X_{0} \over X_{1}-X_{0}}=Y_{n}={a^{n}-1 \over a-1}={Z_{n}-Z_{0} \over Z_{1}-Z_{0}}{\pmod {m}}.$

## Parameters in common use

The following table lists the parameters of LCGs in common use, including built-in rand() functions in runtime libraries of various compilers.

Source modulus
m
multiplier
a
increment
c
output bits of seed in rand() or 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++ 
C90, C99, C11: Suggestion in the ISO/IEC 9899 , C18
2³¹ 1103515245 12345 bits 30..16
Borland Delphi, Virtual Pascal 2³² 134775813 1 bits 63..32 of (seed * L)
Turbo Pascal 2³² 134775813 (0x8088405₁₆) 1
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 (10DCD₁₆) 1 Java's java.util.Random, POSIX [ln]rand48, glibc [ln]rand48[_r] 2⁴⁸ 25214903917 (5DEECE66D₁₆) 11 bits 47...16 random0 If Xₙ is even then Xₙ₊₁ will be odd, and vice versa—the lowest bit oscillates at each step. 134456 = 2³7⁵ 8121 28411 ${\frac {X_{n}}{134456}}$ 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 cc65 2²³ 65793 (10101₁₆) 4282663 (415927₁₆) bits 22...8 cc65 2³² 16843009 (1010101₁₆) 826366247 (31415927₁₆) bits 31...16 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. This is especially noticeable in scripts that use the mod operation to reduce range; modifying the random number mod 2 will lead to alternating 0 and 1 without truncation. ## Advantages and disadvantages LCGs are fast and require minimal memory (one modulo-m number, often 32 or 64 bits) to retain state. This makes them valuable for simulating multiple independent streams. Hyperplanes of a linear congruential generator in three dimensions. This structure is what the spectral test measures. Although LCGs have a few specific weaknesses, many of their flaws come from having too small a state. The fact that people have been lulled for so many years into using them with such small moduli can be seen as a testament to strength of the technique. A LCG with large enough state can pass even stringent statistical tests; a modulo-2 LCG which returns the high 32 bits passes TestU01's SmallCrush suite,[citation needed] and a 96-bit LCG passes the most stringent BigCrush suite. For a specific example, an ideal random number generator with 32 bits of output is expected (by the Birthday theorem) to begin duplicating earlier outputs after m ≈ 216 results. Any PRNG whose output is its full, untruncated state will not produce duplicates until its full period elapses, an easily detectable statistical flaw. For related reasons, any PRNG should have a period longer than the square of the number of outputs required. Given modern computer speeds, this means a period of 264 for all but the least demanding applications, and longer for demanding simulations. One flaw specific to LCGs is that, if used to choose points in an n-dimensional space, the points will lie on, at most, nn!⋅m hyperplanes (Marsaglia's Theorem, developed by George Marsaglia). This is due to serial correlation between successive values of the sequence Xn. Carelessly chosen multipliers will usually have far fewer, widely spaced planes, which can lead to problems. The spectral test, which is a simple test of an LCG's quality, measures this spacing and allows a good multiplier to be chosen. The plane spacing depends both on the modulus and the multiplier, a large enough modulus can reduce this distance below the resolution of double precision numbers. The choice of the multiplier becomes less important when the modulus is large. It is still necessary to calculate the spectral index and make sure that the multiplier is not a bad one, but purely probabilistically it becomes extremely unlikely to encounter a bad multiplier when the modulus is larger than about 264. Another flaw specific to LCGs is the short period of the low-order bits when m is chosen to be a power of 2. This can be mitigated by using a modulus larger than the required output, and using the most significant bits of the state. LCGs should be chosen very carefully for applications where high-quality randomness is critical. Any Monte Carlo simulation should use an LCG with a modulus greater and preferably much greater than the cube of the number of random samples which are required. This means, for example, that a (good) 32-bit LCG can be used to obtain about a thousand random numbers; a 64-bit LCG is good for about 221 random samples which is a little over two million, etc. For this reason, LCGs are in practice not suitable for large scale Monte Carlo simulations. LCGs are not intended, and must not be used, for cryptographic applications; use a cryptographically secure pseudorandom number generator for such applications. 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. ## Sample Python code The following is an implementation of an LCG in Python: def lcg(modulus, a, c, seed): while True: seed = (a * seed + c) % modulus yield seed  ## Sample Free Pascal code Free Pascal uses a Mersenne Twister as its default pseudo random number generator whereas Delphi uses a LCG. Here is a Delphi compatible example in Free Pascal based on the information in the table above. Given the same RandSeed value it generates the same sequence of random numbers as Delphi. unit lcg_random; {$ifdef fpc}{$mode delphi}{$endif}
interface

implementation
function IM:cardinal;inline;
begin
RandSeed := RandSeed * 134775813 + 1;
Result := RandSeed;
end;

begin
Result := IM * 2.32830643653870e-10;
end;

begin
Result := IM * range shr 32;
end;


Like all pseudorandom number generators, a LCG needs to store state and alter it each time it generates a new number. Multiple threads may access this state simultaneously causing a race condition. Implementations should use different state each with unique initialization for different threads to avoid equal sequences of random numbers on simultaneously executing threads.

## LCG derivatives

There are several generators which are linear congruential generators in a different form, and thus the techniques used to analyze LCGs can be applied to them.

One method of producing a longer period is to sum the outputs of several LCGs of different periods having a large least common multiple; the Wichmann–Hill generator is an example of this form. (We would prefer them to be completely coprime, but a prime modulus implies an even period, so there must be a common factor of 2, at least.) This can be shown to be equivalent to a single LCG with a modulus equal to the product of the component LCG moduli.

Marsaglia's add-with-carry and subtract-with-borrow PRNGs with a word size of b=2w and lags r and s (r > s) are equivalent to LCGs with a modulus of br ± bs ± 1.

Multiply-with-carry PRNGs with a multiplier of a are equivalent to LCGs with a large prime modulus of abr−1 and a power-of-2 multiplier b.

A permuted congruential generator begins with a power-of-2-modulus LCG and applies an output transformation to eliminate the short period problem in the low-order bits.

## Comparison with other PRNGs

Generators based on linear recurrences (such as xorshift*) or on good avalanching functions (such as SplitMix64 ) 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 uses an LCG to generate seed data.[citation needed]

Linear congruential generators have the problem that all of the bits in each number are usually not equally random. Either the modulus is a power of 2 and the least-significant bits are less random than the most-significant bits, or the modulus is not a power of 2 and the outputs are biased. A linear feedback shift register PRNG produces a stream of pseudo-random bits, each of which is 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.