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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 , the quality of the output is extremely sensitive to the choice of the parameters m and a.

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 s and led to many results which are currently being questioned because of the use of this poor LCG.

This is the original Lehmer RNG construction. One disadvantage of a prime modulus is that the modular reduction requires a double-width product and an explicit reduction step.

If a double-width product is unavailable, and the multiplier is chosen carefully, Schrage's method [10] may be used. If a prime just less than a power of 2 is used, sometimes the missing values are simply ignored.

In fact, the most significant bits are usually not computed at all. There are, however, disadvantages.

The initial state X 0 must be odd, and the low three bits of X alternate between two states and are not useful. 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. 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. This will occur if and only if : [2] : 17— These three requirements are referred to as the Hull—Dobell Theorem.

This form may be used with any m , but only works well for m with many repeated prime factors, such as a power of 2; using a computer's word size is the most common choice.

Although the Hull—Dobell theorem provides maximum period, it is not sufficient to guarantee a good generator.

The spectral test is one of the most important tests. The generator is not sensitive to the choice of c , as long as it is relatively prime to the modulus e.

More generally, any two series X and Z with the same multiplier and modulus are related by. The following table lists the parameters of LCGs in common use, including built-in rand functions in runtime libraries of various compilers.

This table is to show popularity, not examples to emulate; many of these parameters are poor.

Tables of good parameters are available. As shown above, LCGs do not always use all of the bits in the values they produce.

For example, the Java implementation operates with 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.

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.

LCGs are not intended, and must not be used, for cryptographic applications; use a cryptographically secure pseudorandom number generator for such applications.

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.

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 2 64 for all but the least demanding applications, and longer for demanding simulations.

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 2 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.

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.

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.

LCGs should be evaluated very carefully for suitability in non-cryptographic applications where high-quality randomness is critical.

For Monte Carlo simulations, an LCG must use 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 bit LCG can be used to obtain about a thousand random numbers; a bit LCG is good for about 2 21 random samples a little over two million , etc.

The following is an implementation of an LCG in Python :. 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. 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.

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.

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