Strength of the reversible, garbage-free 2k ± 1 multiplier

Eva Rotenberg; James Cranch; Michael Kirkedal Thomsen; Holger Bock Axelsen

doi:10.1007/978-3-642-38986-3_5

Strength of the reversible, garbage-free 2^k ± 1 multiplier

Eva Rotenberg, James Cranch, Michael Kirkedal Thomsen, Holger Bock Axelsen

Department of Computer Science

Abstract

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 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 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.

Original language	English
Title of host publication	Reversible Computation : 5th International Conference, RC 2013, Victoria, BC, Canada, July 4-5, 2013. Proceedings
Editors	Gerhard W. Dueck, D. Michael Miller
Number of pages	12
Publisher	Springer
Publication date	2013
Pages	46-57
ISBN (Print)	978-3-642-38985-6
ISBN (Electronic)	978-3-642-38986-3
DOIs	https://doi.org/10.1007/978-3-642-38986-3_5
Publication status	Published - 2013
Event	5th International Conference on Reversible Computation - Victoria, Canada Duration: 4 Jul 2013 → 5 Jul 2013 Conference number: 5

Conference

Conference	5th International Conference on Reversible Computation
Number	5
Country/Territory	Canada
City	Victoria
Period	04/07/2013 → 05/07/2013

Series	Lecture notes in computer science
Volume	7948
ISSN	0302-9743

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10.1007/978-3-642-38986-3_5

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Strength of the reversible, garbage-free 2^k ± 1 multiplier. / Rotenberg, Eva; Cranch, James; Thomsen, Michael Kirkedal et al.
Reversible Computation: 5th International Conference, RC 2013, Victoria, BC, Canada, July 4-5, 2013. Proceedings. ed. / Gerhard W. Dueck; D. Michael Miller. Springer, 2013. p. 46-57 (Lecture notes in computer science, Vol. 7948).

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

Rotenberg, E, Cranch, J, Thomsen, MK & Axelsen, HB 2013, Strength of the reversible, garbage-free 2^k ± 1 multiplier. in GW Dueck & DM Miller (eds), Reversible Computation: 5th International Conference, RC 2013, Victoria, BC, Canada, July 4-5, 2013. Proceedings. Springer, Lecture notes in computer science, vol. 7948, pp. 46-57, 5th International Conference on Reversible Computation, Victoria, Canada, 04/07/2013. https://doi.org/10.1007/978-3-642-38986-3_5

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AB - 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 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 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.

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Strength of the reversible, garbage-free 2k ± 1 multiplier

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Strength of the reversible, garbage-free 2^k ± 1 multiplier