## 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 |
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Title of host publication | Reversible Computation - 5th International Conference, RC 2013, Proceedings |

Volume | 7948 LNCS |

Publication date | 2013 |

Pages | 46-57 |

ISBN (Print) | 9783642389856 |

DOIs | |

Publication status | Published - 2013 |

Event | 5th International Conference on Reversible Computation, RC 2013 - Victoria, BC, Canada Duration: 4 Jul 2013 → 5 Jul 2013 |

### Conference

Conference | 5th International Conference on Reversible Computation, RC 2013 |
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Country/Territory | Canada |

City | Victoria, BC |

Period | 04/07/2013 → 05/07/2013 |

Sponsor | Pacific Institute for the Mathematical Sciences, University of Victoria BC |

Series | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7948 LNCS |

ISSN | 0302-9743 |

## Keywords

- constant multiplication
- Mersenne numbers
- Number theory
- reversible circuit design

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