Besides cryptographic random number generators (RNGs), the following are examples of highquality pseudorandom number generators (PRNGs). The “Fails PractRand Starting At” column in this and other tables in this page means the number of bytes (rounded up to the nearest power of two) at which PractRand detects a failure in the PRNG. (Note that highquality PRNGs, as I define them, are not necessarily appropriate for information security.)
PRNG  Seeds Allowed  Cycle Length  Fails PractRand Starting At  Notes 

xoshiro256**  2^256  1  2^256  1  ??? TiB  
xoshiro256+  2^256  1  2^256  1  ??? TiB  Lowest bits have low linear complexity (see (Blackman and Vigna 2019)^{1} and see also “Testing low bits in isolation”); if the application or library cares, it can discard those bits before using this PRNG’s output. 
xoshiro256++  2^256  1  2^256  1  ??? TiB  
xoshiro512**  2^512  1  2^512  1  ??? TiB  
xoshiro512+  2^512  1  2^512  1  ??? TiB  Lowest bits have low linear complexity 
xoshiro512++  2^512  1  2^512  1  ??? TiB  
xoroshiro128++  2^128  1  2^128  1  ??? TiB  
xoroshiro128**  2^128  1  2^128  1  ??? TiB  
SFC64 (C. DotyHumphrey)  2^192  At least 2^64 per seed  ??? TiB  256bit state 
Philox4×647  2^128  At least 2^256 per seed  ??? TiB  384bit state 
Velox3b  2^64  At least 2^128 per seed  ??? TiB  256bit state 
gjrand named after Geronimo Jones 
2^128  At least 2^64 per seed  ??? TiB  256bit state 
MRG32k3a (L’Ecuyer 1999; L’Ecuyer et al. 2002)^{2}  Near 2^192  2 cycles with length near 2^191  ??? TiB  192bit state 
MRG31k3p (L’Ecuyer and Touzin 2000)^{3}  Near 2^186  2 cycles with length near 2^185  ??? TiB  192bit state 
JLKISS (Jones 2007/2010)^{4}  2^64 * (2^64  1)^2  At least (2^128  2^64)  ??? TiB  192bit state 
JLKISS64 (Jones 2007/2010)^{4}  2^64 * (2^64  1)^3  At least (2^128  2^64)  ??? TiB  256bit state 
A multiplicative linear congruential generator (LCG) with prime modulus greater than 2^{63} described in Table 2 of (L’Ecuyer 1999)^{5}  Modulus  1  Modulus  1  ??? TiB  Memory used depends on modulus size 
XorShift* 128/64  2^128  1  2^128  1  ??? TiB  128bit state. Described by M. O’Neill in “You don’t have to use PCG!”, 2017.^{6} 
XorShift* 64/32  2^64  1  2^64  1  ??? TiB  64bit state. Described by M. O’Neill in “You don’t have to use PCG!”, 2017. 
Some PRNGs support multiple “streams” that behave like independent uniform random bit sequences. The test for independence involves interleaving two “streams”’ outputs and sending the interleaved outputs to the PractRand tests.
The following lists highquality PRNGs that support streams and their PractRand results for different strategies of forming pseudorandom number “streams”.
PRNG  Fails PractRand Starting At  Notes 

xoshiro256**  Jumpahead by 2^64: ??? TiB Jumpahead by 2^128: ??? TiB Jumpahead by 2^256/φ: ??? TiB Consecutive seeds: ??? TiB 

xoshiro256++  Jumpahead by 2^64: ??? TiB Jumpahead by 2^128: ??? TiB Jumpahead by 2^256/φ: ??? TiB Consecutive seeds: ??? TiB 

xoroshiro128**  Jumpahead by 2^64: ??? TiB Jumpahead by 2^128/φ: ??? TiB Consecutive seeds: ??? TiB 

xoroshiro128++  Jumpahead by 2^64: ??? TiB Jumpahead by 2^128/φ: ??? TiB Consecutive seeds: ??? TiB 

SFC64  Consecutive seeds: ??? TiB Seed increment by 2^64: ??? TiB 

Philox4×647  Consecutive seeds: ??? TiB Seed increment by 2^64: ??? TiB 

PCG64  Jumpahead by period/φ: ??? TiB  What PCG calls “streams” does not produce independent sequences. 
???  Jumpahead by period/φ: ??? TiB 
Constructions for counterbased PRNGs (using the definition from (Salmon et al. 2011, section 2)^{7} include:
More specifically, let C and S each be 64 or greater and divisible by 8. Then:
H(seed  0x5F  counter)
(where H
is a hash function, 
means concatenation, 0x5F
is the 8bit block 0x5F, and seed
and counter
are littleendian encodings of the seed or counter, respectively), and adds 1 to the counter by wraparound addition. Or…The following lists hash functions and block ciphers that form highquality counterbased PRNGs. It’s possible that reducedround versions of these and other functions will also produce highquality counterbased PRNGs.
Function  Fails PractRand Starting At  Notes 

Hash Functions: BEBB4185; BLAKE2S256; BLAKE3; CityHash64; Falkhash; FarmHash128; FarmHash32; FarmHash64; Farsh32; Farsh64; Floppsyhash; GoodOAAT; HalfSipHash; Hasshe2; MD5 (low 32 bits); Metrohash128; Mirhash; Mirhashstrict (low 32 bits); MUM; MurmurHash64A for x64; MurmurHash3 (128bit) for x64; Seahash; Seahash (low 32 bits); SHA256; SHA256 (low 64 bits); SHA3256; SipHash; Spooky64; Fast Positive Hash (32bit bigendian); TSip; xxHash v3 64bit (both full and low 32 bits)  > 1 TiB  S = 64, C = 128. Failure figure applies to regular sequence; 2, 4, and 11 interleaved streams from consecutive seeds; 2, 4, and 11 interleaved streams from counters incremented by 2^{64}; 2, 4, and 11 interleaved streams from counters incremented by 2^{96}. 
???  ??? TiB (Consecutive seeds: ??? TiB)  
???  ??? TiB (Consecutive seeds: ??? TiB)  
???  ??? TiB (Consecutive seeds: ??? TiB) 
The following lists highquality combined PRNGs. See “Testing PRNGs for HighQuality Randomness” for more information on combining PRNGs.
Function  Fails PractRand Starting At  Notes 

??? combined with Weyl sequence  ??? TiB  
??? combined with 128bit LCG  ??? TiB  
JSF64 combined with ???  ??? TiB  
JSF64 combined with ???  ??? TiB  
Tyche combined with ???  ??? TiB  
Tychei combined with ???  ??? TiB  
??? combined with ???  ??? TiB 
The following lists highquality splittable PRNGs. See “Testing PRNGs for HighQuality Randomness” for more information on testing splittable PRNGs, and see the appendix for splittable PRNG constructions.
Function  Fails PractRand Starting At  Notes 

???  ??? TiB  
???  ??? TiB  
???  ??? TiB 
Although the following are technically highquality PRNGs, they are not preferred:
PRNG  Notes 

C++’s std::ranlux48 engine 
Usually takes about 192 8bit bytes of memory. Admits up to 2^577  2 seeds; seed’s bits cannot be all zeros or all ones (Lüscher 1994)^{8}. The maximum cycle length for ranlux48 ’s underlying generator is very close to 2^576. 
A highquality PRNG that is an LCG with nonprime modulus (or a PRNG based on one, such as PCG)  If the modulus is a power of 2, this PRNG can produce highly correlated pseudorandom number sequences from seeds that differ only in their high bits (see S. Vigna, “The wrapup on PCG generators”) and lowest bits have short cycles. (What PCG calls “streams” does not produce independent sequences.) 
The following are not considered highquality PRNGs:
Algorithm  Notes 

Sequential counter  Doesn’t behave like independent random sequence 
A linear congruential generator with modulus less than 2^{63} (such as java.util.Random and C++’s std::minstd_rand and std::minstd_rand0 engines) 
Admits fewer than 2^{63} seeds 
Mersenne Twister (MT19937)  Shows a systematic failure in BigCrush’s LinearComp test (part of L’Ecuyer and Simard’s “TestU01”). (See also (Vigna 2019)^{9}.) Moreover, it usually takes about 2500 8bit bytes of memory. 
Marsaglia’s xorshift family (“Xorshift RNGs”, 2003) 
Shows systematic failures in SmallCrush’s MatrixRank test (Vigna 2016)^{10} 
System.Random , as implemented in the .NET Framework 4.7 
Admits fewer than 2^{63} seeds 
Ran2 (Numerical Recipes)  Minimum cycle length less than 2^{63} 
msws (Widynski 2017)^{11} 
Admits fewer than 2^{63} seeds (about 2^{54.1} valid seeds) 
JSF32 (B. Jenkins’s “A small noncryptographic PRNG”)  Admits fewer than 2^{63} seeds; proven minimum cycle length is only 2^{20} or more 
JSF64 (B. Jenkins’s “A small noncryptographic PRNG”)  No proven minimum cycle of at least 2^{63} values 
Middle square  No proven minimum cycle of at least 2^{63} values 
Many cellularautomaton PRNGs (especially if they are neither reversible nor maximallength^{12})  No proven minimum cycle of at least 2^{63} values 
Tyche/Tychei (Neves and Araujo 2011)^{13}  No proven minimum cycle of at least 2^{63} values 
ISAAC (“ISAAC and RC4” by B. Jenkins)  Proven minimum cycle length is only 2^{40} or more 
Here are implementation notes on splittable PRNGs. The pseudocode conventions apply to this section. In addition, the following notation is used:

symbol means concatenation.TOBYTES(x, n)
converts an integer to a sequence of n
8bit bytes, in “littleendian” encoding: the first byte is the 8 least significant bits, the second byte is the next 8 bits, and so on. No more than n
times 8 bits are encoded, and unused bytes become zeros.BLOCKLEN
is the hash function’s block size in bits. For noncryptographic hash functions, this can be the function’s output size in bits instead. BLOCKLEN
is rounded up to the closest multiple of 8.TOBLOCK(x)
is the same as TOBYTES(x, BLOCKLEN / 8)
.The splittable PRNG designs described here use keyed hash functions, which hash a message with a given key and output a hash code. An unkeyed hash function can become a keyed hash function by hashing the following data: key  TOBYTES(0x5F, 1)  message
.
The Claessen–Pałka splittable PRNG (Claessen and Pałka 2013)^{14} can be described as follows:
TOBLOCK(seed)
and its path is an empty bit vector.split
creates two new states from the old one; the first (or second) is a copy of the old state, except a 0 (or 1, respectively) is appended to the path. If a new state’s path reaches BLOCKLEN
bits this way, the state’s seed is set to the result of hashing BitsToBytes(path)
with the seed as the key, and the state’s path is set to an empty bit vector.generate
creates a random number by hashing BitsToBytes(path)
with the seed as the key.(The Claessen paper, section 5, also shows how a sequence of numbers can be generated from a state, essentially by hashing the path with the seed as the key, and in turn hashing a counter with that hash code as the key, but uses a rather complicated encoding to achieve this.)
The following helper method, in pseudocode, is used in the description above:
METHOD BitsToBytes(bits)
// Unfortunately, the Claessen paper sec. 3.3 pads
// blocks with zeros, creating a risk that different paths
// encode to the same byte sequence (for example, <1100> vs.
// <11000> or <0011> vs. <00011>). Even without this padding,
// this risk exists unless the underlying hash function hashes
// bit sequences (not just byte sequences), which is rare.
// Therefore, encode the bits to a sequence of bytes
// rather than using the encoding given in sec. 3.3.
v = []
for i in 0...size(bits): v = v  TOBYTES(bits[i], 1)
return v
END METHOD
The splittable PRNG described in “JAX PRNG Design” can be described as follows:
TOBLOCK(seed)
.split
creates two new states from the old one; the first (or second) is a hash of TOBLOCK(0)
(or TOBLOCK(1)
, respectively) with the old state as the key.generate
creates one or more random numbers by hashing TOBLOCK(n)
with the state as the key, where n
starts at 1 and increases by 1 for each new random number.Blackman, D., Vigna, S. “Scrambled Linear Pseudorandom Number Generators”, 2019 (xoroshiro and xoshiro families); S. Vigna, “An experimental exploration of Marsaglia’s xorshift
generators, scrambled”, 2016 (scrambled xorshift family). ↩
L’Ecuyer, P., “Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators”, Operations Research 47(1), 1999; L’Ecuyer, P., Simard, R., et al., “An ObjectOriented Random Number Package with Many Long Streams and Substreams”, Operations Research 50(6), 2002. ↩
L’Ecuyer, P., Touzin, R., “Fast Combined Multiple Recursive Generators with Multipliers of the Form a = ±2^{q} ± 2^{r}”, Proceedings of the 2000 Winter Simulation Conference, 2000. ↩
Jones, D., “Good Practice in (Pseudo) Random Number Generation for Bioinformatics Applications”, 2007/2010. ↩ ↩^{2}
P. L’Ecuyer, “Tables of Linear Congruential Generators of Different Sizes and Good Lattice Structure”, Mathematics of Computation 68(225), January 1999, with errata. ↩
This XorShift* generator is not to be confused with S. Vigna’s *scrambled PRNGs, which multiply the PRNG state differently than this one does. ↩
Salmon, John K., Mark A. Moraes, Ron O. Dror, and David E. Shaw. “Parallel random numbers: as easy as 1, 2, 3.” In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis, pp. 112. 2011. ↩
Lüscher, M., “A Portable HighQuality Random Number Generator for Lattice Field Theory Simulations”, arXiv:heplat/9309020 (1994). See also Conrads, C., “Faster RANLUX PseudoRandom Number Generators”. ↩
S. Vigna, “It Is High Time We Let Go of the Mersenne Twister”, arXiv:1910.06437 [cs.DS], 2019. ↩
S. Vigna, “An experimental exploration of Marsaglia’s xorshift
generators, scrambled”, 2016. ↩
Widynski, B., “Middle Square Weyl Sequence RNG”, arXiv:1704.00358 [cs.CR], 2017. ↩
Bhattacharjee, K., “Cellular Automata: Reversibility, Semireversibility and Randomness”, arXiv:1911.03609 [cs.FL], 2019. ↩
Neves, S., and Araujo, F., “Fast and Small Nonlinear Pseudorandom Number Generators for Computer Simulation”, 2011. ↩
Claessen, K, Pałka, M., “Splittable Pseudorandom Number Generators using Cryptographic Hashing”, ACM SIGPLAN Notices 48(12), December 2013. ↩