  A targeted exponentiation algorithm computes a group exponentiation operation
$a^k$ with a reversible circuit in such a way that the initial state of the
circuit consists of only the base $a$ and fixed values, and the final state
consists of only the exponential $a^k$ and fixed values. Three targeted
exponentiation algorithms based on Fibonacci addition chains are considered,
offering tradeoffs in terms of the number of working registers and the number
of iterations. The approaches also motivate related results on the Fibonacci
Zeckendorf array, including a new \emph{modular Hofstadter G problem} and an
improvement to Anderson's recent algorithm for locating pairs of adjacent
integers in the extended Fibonacci Zeckendorf array. The algorithms have
applications in quantum computing.

Number Theory