Abstract
Recently, a reversible garbage-free 2k ±1 constant-multiplier circuit was presented by Axelsen and Thomsen. This was the first construction of a garbage-free, reversible circuit for multiplication with non-trivial 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 2k±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 2k ±1-multipliers, and for generating the minimal cascade of 2k ±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.
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Rotenberg, E., Cranch, J., Thomsen, M.K., Axelsen, H.B. (2013). Strength of the Reversible, Garbage-Free 2k ±1 Multiplier. In: Dueck, G.W., Miller, D.M. (eds) Reversible Computation. RC 2013. Lecture Notes in Computer Science, vol 7948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38986-3_5
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DOI: https://doi.org/10.1007/978-3-642-38986-3_5
Publisher Name: Springer, Berlin, Heidelberg
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