A permuted congruential generator (PCG) is a pseudorandom number generation algorithm developed in 2014 by Dr. M.E. O'Neill which applies an output permutation function to improve the statistical properties of a modulo-2n linear congruential generator. It achieves excellent statistical performance[1] [2] [3] [4] with small and fast code, and small state size.[5]
A PCG differs from a classical linear congruential generator (LCG) in three ways:
It is the variable rotation which eliminates the problem of a short period in the low-order bits that power-of-2 LCGs suffer from.
The PCG family includes a number of variants. The core LCG is defined for widths from 8 to 128 bits, although only 64 and 128 bits are recommended for practical use; smaller sizes are for statistical tests of the technique.
The additive constant in the LCG can be varied to produce different streams. The constant is an arbitrary odd integer,[6] so it does not need to be stored explicitly; the address of the state variable itself (with the low bit set) can be used.
There are several different output transformations defined. All perform well, but some have a larger margin than others. They are built from the following components:
x ^= x >> constant. The constant is chosen to be half of the bits not discarded by the next operation (rounded down).These are combined into the following recommended output transformations, illustrated here in their most common sizes:
(64→32 bits) count = (int)(x >> 59); x ^= x >> 18; return rotr32((uint32_t)(x >> 27), count);.
(64→32 bits) count = (int)(x >> 61); x ^= x >> 22; return (uint32_t)(x >> (29 - count));.
(128→64 bits) count = (int)(x >> 122); x64 = (uint64_t)(x ^ (x >> 64)); return rotr64(x64, count);
(32→32 bits) count=(int)(x >> 28); x ^= x >> (4 + count); x *= 277803737u; return x ^ (x >> 22);
(64→64 bits) count=(int)(x >> 59); x ^= x >> (5 + count); x *= 12605985483714917081u; return x ^ (x >> 43);
(128→128 bits) count = (int)(x >> 122); low64 = rotr64((uint64_t)(x ^ (x >> 64)), count); high64 = rotr64((uint64_t)(x >> 64), low64 & 63); return (uint128_t)high64 << 64 | low64;
Each step of these output transformations is either invertible (and thus one-to-one) or a truncation, so their composition maps the same fixed number of input states to each output value. This preserves the equidistribution of the underlying LCG.
Finally, if a cycle length longer than 2128 is required, the generator can be extended with an array of sub-generators. One is chosen (in rotation) to be added to the main generator's output, and every time the main generator's state reaches zero, the sub-generators are cycled in a pattern which provides a period exponential in the total state size.
The generator recommended for most users is PCG-XSH-RR with 64-bit state and 32-bit output. It can be implemented as:
static uint64_t state = 0x4d595df4d0f33173; // Or something seed-dependentstatic uint64_t const multiplier = 6364136223846793005u;static uint64_t const increment = 1442695040888963407u; // Or an arbitrary odd constant
static uint32_t rotr32(uint32_t x, unsigned r)
uint32_t pcg32(void)
void pcg32_init(uint64_t seed)
The generator applies the output transformation to the initial state rather than the final state in order to increase the available instruction-level parallelism to maximize performance on modern superscalar processors.
A slightly faster version eliminates the increment, reducing the LCG to a multiplicative (Lehmer-style) generator with a period of only 262, and uses the weaker XSH-RS output function:
uint32_t pcg32_fast(void)
void pcg32_fast_init(uint64_t seed)
The time saving is minimal, as the most expensive operation (the 64×64-bit multiply) remains, so the normal version is preferred except in extremis. Still, this faster version also passes statistical tests.
When executing on a 32-bit processor, the 64×64-bit multiply must be implemented using three 32×32→64-bit multiply operations. To reduce that to two, there are 32-bit multipliers which perform almost as well as the 64-bit one, such as 0xf13283ad, or 0xf2fc5985.
Melissa O'Neill proposes testing PRNGs by applying statistical tests to their reduced-size variants and determining the minimum number of internal state bits required to pass.[7] TestU01's BigCrush examines enough data to detect a period of 235, so even an ideal generator requires 36 bits of state to pass it. Some very poor generators can pass if given a large enough state;[8] passing despite a small state is a measure of an algorithm's quality, and shows how large a safety margin exists between that lower limit and the state size used in practical applications.
PCG-RXS-M-XS (with 32-bit output) passes BigCrush with 36 bits of state (the minimum possible), PCG-XSH-RR (pcg32 above) requires 39, and PCG-XSH-RS (pcg32_fast above) requires 49 bits of state. For comparison, xorshift*, one of the best of the alternatives, requires 40 bits of state, and Mersenne twister fails despite 19937 bits of state.[9]
It has been shown that it is practically possible (with a large computation) to recover the seed of the pseudo-random generator given 512 consecutive output bytes.[10] This implies that it is practically possible to predict the rest of the pseudo-random stream given 512 bytes.