Suppose that we need to compute the value of for some reasonably large n. Such problems occur in primality testing for cryptography, as discussed in Section .
The simplest algorithm performs n-1 multiplications, by computing . However, we can do better by observing that . If n is even, then . If n is odd, then . In either case, we have halved the size of our exponent at the cost of at most two multiplications, so multiplications suffice to compute the final value.
if (n = 0) return(1)
x = power( )
if (n is even) then return( )
else return( )
This simple algorithm illustrates an important principle of divide and conquer. It always pays to divide a job as evenly as possible. This principle applies to real life as well. When n is not a power of two, the problem cannot always be divided perfectly evenly, but a difference of one element between the two sides cannot cause any serious imbalance.