Linear Congruential Generator

Related Concepts

Pseudorandom Generator; Stream Cipher

Definition

A linear congruential generator is a pseudorandom generator that produces a sequence of numbers x ₁, x ₂, x ₃, … according to the following linear recurrence:

$${x}_{t} = a{x}_{t-1} + b\quad \mathrm{mod}\ n$$

for t ≥ 1 (modular arithmetic); integers a,  b, and n characterize entirely the generator, and the seed is x ₀.

Example

Considering for example a = 3,  b = 5,  n = 17, and x ₀ = 2, the sequence produced by the linear congruential generator will be 11, 4, 0, 5, 3, 14, 13, 10, 1, 8, 12, 7, 9, 15, 16, …

Background

Pseudorandom generators are very useful in cryptography, in protocols, but also in the generation of keystreams in stream ciphers. In this case, they have to present strong properties to face cryptanalysis.

Applications

Such generators are easy to implement and pass the following statistical tests: Golomb’s randomness postulates, frequency test, serial test, poker test, runs test, autocorrelation test, Maurer’s...