Quality Assurance

Core Testing

A values in the below are the maximum output size where a bit generator or sequence of bit generators has passed PractRand. A – indicates that configuration is not relevant. Failures are marked with FAIL. Most bit generators were only tested in their default configuration. Non-default configurations are indicated by listing the keyword arguments to the bit generator. Two sets of tests were performed. The first tested all configurations using 128GB of data using PractRand’s extended set of tests and additional bit folding. The second set of tests used 4TB of data wit the standard set of tests and folding.

All bit generators have been tested using the same SeedSequence initialized with the same 256-bits of entropy taken from random.org.

Method

Seed Sequence

Jumped

Streams

1

4

8196

4

8196

AESCounter

4TB

4TB

4TB

4TB

4TB

ChaCha(rounds=20)

4TB

4TB

4TB

4TB

4TB

ChaCha(rounds=8)

4TB

4TB

4TB

4TB

4TB

DSFMT⁴

4TB

FAIL at 64 GB¹

4TB

FAIL at 64 GB¹

FAIL at 64 GB¹

EFIIX64

4TB

4TB

4TB

HC128

4TB

4TB

4TB

JSF

4TB

4TB

4TB

JSF(seed_size=3)

4TB

4TB

4TB

LCG128Mix(output=upper)

4TB

4TB

4TB

4TB

4TB

LXM

4TB

4TB

4TB

4TB

4TB

MT19937⁴,⁵

4TB

FAIL at 64 GB¹

4TB

FAIL at 64 GB¹

4TB

PCG64DXSM²

4TB

4TB

4TB

4TB

4TB

PCG64(variant=dxsm-128)

4TB

4TB

4TB

4TB

4TB

PCG64⁵

4TB

4TB

4TB

4TB

4TB

Philox⁵

4TB

4TB

4TB

4TB

4TB

Romu

4TB

4TB

4TB

Romu(variant=trio)

4TB

4TB

4TB

SFC64⁵

4TB

4TB

4TB

SFC64(k=3394385948627484371)

4TB

4TB

4TB

SFC64(k=Weyl)³

4TB

4TB

4TB

SFMT⁴

4TB

FAIL at 64 GB¹

4TB

FAIL at 64 GB¹

FAIL at 4 TB¹

SPECK128

4TB

4TB

4TB

4TB

4TB

ThreeFry

4TB

4TB

4TB

4TB

4TB

Xoshiro256

4TB

4TB

4TB

4TB

4TB

Xoshiro512

4TB

4TB

4TB

4TB

4TB

Notes

¹ Failures at or before 128GB were generated by tests that used the expanded set of tests and extra bt folds (-te 1 and -tf 2). Failures at sample sizes above 128GB were produces using the default configuration (-te 0 and -tf 1).

² PCG64DXSM and PCG64(variant=dxsm) are identical and so the latter not separately reported.

³ SFC64(k=weyl) uses distinct Weyl increments that have 50% or fewer non-zero bits.

⁴ The Mersenne Twisters begin to fail at 64GB. This is a known limitation of MT-family generators. These should not be used in large studies except when backward compatibility is required.

⁵ Identical output to the version included in NumPy 1.19.

Example Configuration

All configurations are constructed using the same template. The code below tests a configuration using 8,196 streams of AESCounter. The other configurations simply make changes to either JUMPED or STREAMS.

import numpy as np

import randomgen as rg

ENTROPY = 86316980830225721106033794313786972513572058861498566720023788662568817403978
JUMPED = False
STREAMS = 8196
BIT_GENERATOR_KWARGS = {}

SEED_SEQ = np.random.SeedSequence(ENTROPY)


BASE_GEN = rg.AESCounter(SEED_SEQ, **BIT_GENERATOR_KWARGS)
if STREAMS == 1:
   bit_gens = [BASE_GEN]
elif JUMPED:
   bit_gens = [BASE_GEN]
   for _ in range(STREAMS - 1):
      bit_gens.append(bit_gens[-1].jumped())
else:
   bit_gens = []
   for child in SEED_SEQ.spawn(STREAMS):
      bit_gens.append(rg.AESCounter(child, **BIT_GENERATOR_KWARGS))
output = 64

Additional Experiments

The best practice for using any of the bit generators is to initialize a single SeedSequence with a reasonably random seed, and then to use this seed sequence to initialize all bit generators. Some additional experiments were used to check that the quality of output streams is not excessively sensitive to use that deviates from this best practice.

Correlated Seeds

While the recommended practice is to use a SeedSequence, it is natural to worry about bad seeds. A common sequence of bad seeds are those which set a single bit to be non-zero: 1, 2, 4, 8, 16, and so on. By default, bit generators use a SeedSequence to transform seed values into an initial state for the bit generator. SeedSequence is itself a random number generator that always escapes low-entropy states – that is, those with many 0s or 1s – immediately. All bit generators were tested with 8 streams using seeds of the form \(2^i\) for i in 0, 1, …, 7. Only three bit generators failed this experiment: DSFMT, MT19937, and SFMT. These are all members of the Mersenne Twister family which commonly fail BRank tests.

Sequential Seeds

The recommended practice for constructing multiple SeedSequence’s spawn() method.

from numpy.random import default_rng, Generator, SeedSequence
from randomgen import Romu

NUM_STREAMS = 2**15
seed_seq = SeedSequence(5897100938578919857511)
# To use the default bit generator, which is not guaranteed to be stable
generators = [default_rng(child) for child in seed_seq.spawn(NUM_STREAMS)]

# To use a specific bit generator
generators = [Generator(Romu(child)) for child in seed_seq.spawn(NUM_STREAMS)]

It is common to see examples that use sequential seed that resemble:

generators = [default_rng(i) for i in range(NUM_STREAMS)]

This practice was examined with all bit generators using 8,196 streams seeded using 0, 1, 2, …, 8,195 by intertwining the output of the generators. None of the generators failed these tests.

Zero (0) Seeding

Bit generators use a SeedSequence that always escapes low-entropy states immediately to transform seed values into an initial state for the bit generator. To ensure that this is not an issue, all bit generators were tested using 4, 32 or 8196 streams using 128GB in PractRand with expanded tests and extra folding. The table below reports only the configurations that failed. These were all Mersenne Twister-class generators and so failure is attributable to the bit generator and not the seeding. All other generators passed these tests.

Streams

4

32

8196

DSFMT

FAIL at 64 GB

FAIL at 64 GB

MT19937

FAIL at 64 GB

FAIL at 64 GB

SFMT

FAIL at 64 GB

FAIL at 64 GB

The non-failures at 8196 are due to the relatively short length of each sequence tested since 128GB shared across 8196 streams only samples \(2^{37}/(2^{13}\times2^{3})=2^{21}\) values from each stream since each value is 8-bytes.