The above equation gives a way to leap the generator ahead, similar in fashion to the leap ahead concept discussed for LCGs. But such an operation with a Fibonacci generator requires a matrix-vector multiply involving, at best, the precomputation of possibly many multiples of . This precomputation may be expensive, even for small values of (recall that is ), and may be prohibitive for large values of . Leaping a Fibonacci generator ahead is therefore not recommended. This does not mean that such a generator cannot be ``split,'' however. We now look at an efficient method for splitting a Fibonacci generator.

Before explaining the splitting technique, notice that the period, **P**, of a
properly constructed Fibonacci generator is . Consider a
given initial set of values of an **M**-bit, -long Fibonacci register.
This
state is a particular bit pattern in the rectangular
register.
If the register is advanced **P** times, the initial pattern will be replaced by
**P - 1** different patterns before the initial one reappears. But the number of
* possible* bit patterns in the register is ,
a number far
larger than **P**. This tells us that there are many **P**-long cycles that are
independent of one other and can be generated from the same
structure. The number of such full-period cycles is
[41].
For example, with the generator of our
previous example, there are separate full-period cycles
and a much smaller number of less than full-period cycles
(, to be precise).

The question now becomes, how do we initialize separate cycles?