Editing Linear congruential generator (section)

=== ''m'' a power of 2, ''c'' = 0 ===
Choosing ''m'' to be a [[power of two]], most often ''m'' = 2<sup>32</sup> or ''m'' = 2<sup>64</sup>, produces a particularly efficient LCG, because this allows the modulus operation to be computed by simply truncating the binary representation. In fact, the most significant bits are usually not computed at all. There are, however, disadvantages.

This form has maximal period ''m''/4, achieved if ''a''&nbsp;≡&nbsp;±3 (mod 8) and the initial state ''X''<sub>0</sub> is odd.  Even in this best case, the low three bits of ''X'' alternate between two values and thus only contribute one bit to the state.  ''X'' is always odd (the lowest-order bit never changes), and only one of the next two bits ever changes.  If ''a''&nbsp;≡&nbsp;+3, ''X'' alternates ±1↔±3, while if ''a''&nbsp;≡&nbsp;−3, ''X'' alternates ±1↔∓3 (all modulo 8).

It can be shown that this form is equivalent to a generator with modulus ''m''/4 and ''c'' ≠ 0.{{r|KnuthV2}}

A more serious issue with the use of a power-of-two modulus is that the low bits have a shorter period than the high bits.  Its simplicity of implementation comes from the fact that bits are never affected by higher-order bits, so the low ''b'' bits of such a generator form a modulo-2<sup>''b''</sup> LCG by themselves, repeating with a period of 2<sup>''b''−2</sup>. Only the most significant bit of ''X'' achieves the full period.