Strength of the Reversible, Garbage-Free 2~k±1 Multiplier

摘要

Recently, a reversible garbage-free 2~k±1 constant-multiplier circuit was presented by Axelsen and Thomsen. This was the first construction of a garbage-free, reversible circuit for multiplication with nontrivial constants. At the time, the strength, that is, the range of constants obtainable by cascading these circuits, was unknown. In this paper, we show that there exist infinitely many constants we cannot multiply by using cascades of 2~k±1-multipliers; in fact, there exist infinitely many primes we cannot multiply by. Using these results, we further provide an algorithm for determining whether one can multiply by a given constant using a cascade of 2~k±1-multipliers, and for generating the minimal cascade of 2~k±1-multipliers for an obtainable constant, giving a complete characterization of the problem. A table of minimal cascades for multiplying by small constants is provided for convenience.