How does java.util.Random work and how good is it?

The java.util.Random class was once the standard Java random nubmer generator. It still provides the blueprint for various different random number generation methods such as nextInt(), nextDouble(). But in terms of the algorithm it uses, it has now been largely surpassed by ThreadLocalRandom for reasons that we will explore below and on the pages referred to below.

The algorithm: a 48-bit LCG generator

The java.util.Random class implements what is generally called a linear congruential generator (LCG). An LCG is essentially a formula of the following form:

In other words, we begin with some start or "seed" number which ideally is "genuinely unpredictable", and which in practice is "unpredictable enough". For example, the number of milliseconds— or even nanoseconds— since the computer was switched on is available on most systems. Then, each time we want a random number, we multiply the current seed by some fixed number, a, add another fixed number, c, then take the result modulo another fixed number, m. The number a is generally large. This method of random number generation goes back pretty much to the dawn of computing¹. Pretty much every "casual" random number generator you can think of— from those of scientific calculators to 1980s home computers to currentday C and Visual Basic library functions— uses some variant of the above formula to generate its random numbers.

Depending on the values chosen for a, c and and m, the quality of random numbers produced by this method varies between "unbelievably disastrous" and "OK for casual applications". For practical reasons, it is generally common to do one of the following:

• make a close to a power of 2 (so that the multiplication can be performed by shifting and adding/subtracting²), or;
• make m a power of 2, often the register size of the machine (such as 2³² for a 32-bit machine) so that the modulo is carried out either "for free", or at worst via an AND operation rather than an expensive division³.

With or without these constraints, values for the parameters are then generally sought so that:

• every possibly value between 0 and m-1 inclusive is generated before the pattern repeats itself;
• the numbers generated are as "statistically random" as we can get them (see below).

Since for a given "current seed" value, the "next seed" will always be completely predictable based on that value, the series of numbers must repeat after at most m generations. This is called the period of the random number generator. In the case of java.util.Random, m is 2⁴⁸ and the other values have indeed been chosen so that the generator has its maximum period. Therefore:

LCG parameters used by java.util.Random

The actual parameters used by java.util.Random are essentially taken from the UNIX rand48 generator (though with a slightly different seeding function). For reasons discussed later, only the top 32 bits of each 48 bits generated are used. With these parameters, the resulting random number generator appears to be about as "good as it gets" for an LCG.

In decimal, 2⁴⁸ is a few hundred million million. That might sound like enough— and it is for certain applications— but it does mean some quite severe limitations in other cases. For example, consider an application where you pull out a number of 2-integer pairs (and where you use the full range of the integer). One integer has 2³² possible values. So the number of possible combinations of 2-integer pairs is 2³² * 2³², or 2⁶⁴. In other words, java.util.Random will not be able to produce every possible combination. Of course, even a generator that produced "perfect" random numbers with a 2⁴⁸ period would have this limitation.

For some testing or scientific applications, that would be bad enough. But it turns out that with LCGs, things are actually worse:

• When taking combinations of values, e.g. for coordinates, the resulting pairs, triples etc always have a particular mathematical relationship, sometimes described as "falling in the planes".
• Not all bits are produced with equal randomness: the lower the bit in the number generated, the less "random" it actually is. This can introduce artefects in some cases. See the section on the randomness of bits with LCGs for an illustration and guidelines on minimising this problem.

Improving on java.util.Random

Because of the limitations of the java.util.Random algorithm, various random number generators have been added to the Java platform. The standard random number generators included in the Java platform include:

• ThreadLocalRandom, a faster, higher quality random number generator that is often the default choice nowadays;
• SplittableRandom, which allows multiple instances to work together to achieve the effect of a single generator with a larger period;
• SecureRandom, a cryptographically secure random nubmer generator.

The Random class is also designed to be pluggable: in other words, is also possible to subclass java.util.Random with a desired implementation and gain the benefits of the various Random methos such as nextDouble() etc without having to implement these from scratch unless there is a reason to do so.

What to read next

The following pages contain more information about java.util.Random and other random nubmer generators in Java:

• random number generators in Java
• ThreadLocalRandom, the newer default generator

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1. It is generally attributed to Dick Lehmer, who appears to have intoduced it formally in a 1948 conference paper.
2. For example, 65539 is 2¹⁶+3. So 65539x can be calculated as x<<16+x+x+x. Shift and addition/subtraction instructions are generally much faster operations than mutliplication.
3. For example, to calculate a number modulo 2⁴⁸— the modulus chosen in java.util.Random, we AND it with (2⁴⁸-1).

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