Period
Named by 6 essays across one field — each of them below, with the objects they name alongside it.
The planes a recurrence cannot leave
One multiplication and one addition, taken modulo a fixed number, produce a sequence that passes for random one value at a time. Taken two or three at a time it does not, and the reason is a whole-number relation that pins every point onto one of a small family of parallel lines.
Nineteen thousand bits of state
The generator most simulations actually use is not clever. It is a linear recurrence over the two-element field with an enormous state, and its virtues are a proved period, a proved equidistribution and speed — none of which is unpredictability, which it does not have and does not claim.
A filter that changes only the spread
The generator most simulations use passes every output through a last scrambling step before anyone sees it. The step is reversible, it changes nothing about the period, and it cannot make the generator any less predictable. What it changes is which patterns of consecutive outputs can occur at all — on a small twisted generator, from half of them to every one.
A rotation that hides the lattice
A congruential generator modulo a power of two has a lowest bit that alternates and pairs of outputs that lie on a few lines. Keep the generator exactly as it is, and show only eight bits of each state, rotated by an amount the state's own top bits choose: every output bit now runs the full cycle, the pairs fill the square as a random sequence would, and triples pass a test the state's own bits fail by a factor of six. Nothing about the state has changed, and four outputs still give it away.
A seed is a deck
A shuffle that is perfectly correct, driven by a generator that passes every statistical test, can still deal almost none of the possible decks. A shuffle program is a function from the generator's starting state to an ordering of the cards, so it can never produce more orderings than the generator has states: a 32-bit seed reaches one deck in about 10⁵⁸. On nine cards and a 16-bit generator every case can be checked — 62,616 of the 362,880 orders are ever dealt — and on a full deck, six cards on the table are enough to name the seed and every card still to come.
Birthdays spaced too evenly
Scatter n birthdays through a year of k days, sort them, and count how many of the gaps between neighbours repeat an earlier gap. For random birthdays the count is close to a Poisson variable with mean n³/4k. A linear congruential generator with m states, read in pairs, matches that law until the grid has as many cells as the generator has states, and then leaves it at once: at sixteen times as many cells, every run of the spacings count rejects it. The sample that catches it is about five times the cube root of the period — 8,192 outputs for a 32-bit generator, thirteen million for a 64-bit one — and the same states, scrambled before they are read, pass.
Named alongside it
The objects these essays reach for when they reach for this one.
PseudorandomnessEquidistributionLinear recurrenceBirthday problemExhaustive searchFinite fieldLatticeModular arithmeticPigeonhole principleUniform distributionBasisCollision