Recently I have been doing some work testing the quality of random number generators (RNGs), so I thought I would record things that should be useful as a reference. I won’t provide too much background here since there are many good existing references to the theory and practice of RNGs, the ones of which I have encountered I have linked to.
Properties
More specifically, a pseudorandom number generator (PRNG) is a repeatable process for producing numbers that have good statistical random properties. A true RNG, in contrast, produces statistically random numbers in a nonrepeatable way, for example in electronics by using a physical source of entropy. True RNGs have an obvious importance in cryptographic applications.
A pseudorandom sequence can be repeated by starting with a particular seed number. The period of a PRNG is the longest unique sequence of numbers generated from any seed. The period is bounded by the size of the internal state of a generator ($n$ bits of state can encode $2^n$ numbers), however a generator may produce shorter repeated sequences, called cycles.
The properties of a ‘good’ PRNG are:

That the length of its period exceeds the number of values that taken from the generator by a program. As a rule of thumb, the period should be at least the square of the numbers used.

For independent uses of a generator concurrently, that the probability is low that any two sequences starting at different seeds overlap.

That successive values are uniformly distributed. In the literature this property is also described as equidistribution, which can be stated as the probability of finding a number in an interval of a sequence is proportional to the length of the interval.

That successive values are uncorrelated.

That each value can be computed efficiently.
The first two properties can typically be determined analytically and it is true that PRNGs are designed in order that they can be. Uniformity can be tested by sampling a large number of values and using statistical measures to analyse the difference from the expected distribution. The fifth characteristic is straightforward to determine, whether the generator is implemented in hardware or software. Much harder to determine, however, is the third property. If it were possible to prove whether a generator is free of correlation, no PRNG would be considered random since by definition there exists a welldefined relationship between successive numbers.
Empirical testing
Conventional approaches to testing RNGs subject them to a collection of tests, exploring different aspects of the generator’s statistics. They cannot be exhaustive, but are shown to be effective by their performance in detecting correlations in existing standard RNGs. This pragmatism is summed up well in this paper with the comment: “the different between good and bad RNGs, in a nutshell, is that the bad ones fail very simple tests whereas the good ones fail only very complicated tests that are hard to figure out or impractical to run.”
There are two popular empirical test suites:

TestU01, a comprehensive C library, containing example PRNGs, utilities and a collection of statistical tests drawn from the academic literature of RNGs. The statistical tests can be run individually, or as part of test batteries, which have various run times and levels of stringency.

PractRand, which provides similar functionality to TestU01 but implemented in C++, with more modern features such as multithreading, flexible interfaces and support for long sequence lengths (over 100 terabytes). According to the author, it’s tests are not drawn from the literature (presumably designed by the author instead) and are therefore a good complement to testing with TestU01 or similar. It also requires more random bits than TestU01 and therefore takes longer to run.
Also worth investigating are the Dieharder test suite (an updated version of the original Diehard) and the RaBiGeTe test suite. There is also some interesting discussion from 2010 between the authors of PractRand and RaBiGete here.
Testing with TestU01
TestU01 provides an interface to test external generators written in C. The
interface requires a method, GetU01
, to generate numbers in the unit interval
$[0, 1)$ as a double
and a method, GetBits
, to return 32 random bits as an
unsigned int
. Some tests will use random bits and some will use random floats.
Just one function can be provided, with the other defined by TestU01 with
the relationship $\texttt{GetU01}=\texttt{GetBits}/2^{32}$.
Converting 32 random bits to a doubleprecision float is lossless in that it
has be represented exactly in the 52 bits of mantissa. However, the conversion
biases the higher bits since the lowest bits will be most affected by numerical
errors. For this reason, it is considered good practice to also test the
reversed output of generator, to expose the lowest bits. To test generators
with a larger output, say 64 bits, it is important that all the bits are
exposed to the tests. One way to do this is to alternately use the high and low
bits of a 64bit value each call to GetBits
or GetU01
. A further reason to
run the reverse of a generator is that the Crush test battery are defined to
ignore the bottommost bit of the generator’s output, and most tests also
ignore the second bit (see the TestU01
documentation).
In his testing of the xorshift
family of generators, Sebastiano
Vigna takes the following approach to measuring
quality with TestU01: for a particular generator, run it with 100 different
seeds, which are spaced at regular intervals in the state space, i.e. for a
generator with $n$ bits of state, choose seeds at $1 + i\lfloor 2^n/100\rfloor$
for $0 \leq i < 100$. The quality of a generator is then measured by the total
number of failures over all seeds, with fewer failures meaning higher
quality. If a generator has 100 or more failures, the failure is called
systematic and the generator is disregarded. Quality is measured with the
BigCrush battery, but since it takes many hours to run (using approximately
$2^{38}$ random values in 106 tests), potential generators can be assessed by
running the smaller test batteries SmallCrush (10 tests) and Crush (96 tests),
continuing based on the number of failures.
Through my own experiments, I found that the reverse of a generator won’t
always catch weak lower bits. xoroshiro128+
is currently the highest
quality and fastest known generator,
as measured by the above process, with 31 failures and 27 failures when
reversed, but it has a known weak bit 0 that follows a linear
recurrence that is
not detected by BigCrush, even when reversed. However, the weak bit is
detectable with the matrix rank test with parameters $N=1$, $n=80$, $r=15$,
$s=15$, $L=k=5000$, or more simply by swapping the high and low 16bit portions
of each 32bit word to move the bottom bit into the middle. I didn’t discover
anything new about xoroshiro128+
here, but what this does highlight is that
comprehensive test sets like TestU01 are by no means exhaustive, and it
therefore worth testing some of their assumptions, particularly in this case if
you are interested in the quality of bit 0.
Incidentally, TestU01 includes a battery of nine tests called Alphabit, which
is allows specific bits or ranges of bits from a generator to be tested. It was
not stringent enough however to detect the correlation of xoroshirt128+
‘s bit 0.
An example
I’ve put together a simple example, available on
Github, of using TestU01 to
assess the quality of a PRNG, which replicates Vigna’s testing of
xoroshiro128+
. The code includes a C program to test xoroshiro128+
with
the Crush batteries and Python scripts to run the test over different seeds and
to summarise the output of TestU01.
Further reading and links
 The PRNG shootout.
 Sebastiano Vigna. An experimental exploration of Marsaglia’s xorshift generators, scrambled. ACM Trans. Math. Software, 42(4), 2016.
 For a comprehensive general introduction to PRNGs and testing them, see ‘The art of computer systems performance analysis’ by Raj Jain (1991) Chapters 26 and 27.