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

  • Eva Rotenberg
  • James Cranch
  • Michael Kirkedal Thomsen
  • Holger Bock Axelsen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7948)


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.


Number theory constant multiplication reversible circuit design Mersenne numbers 


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  1. 1.
    Axelsen, H.B., Thomsen, M.K.: Garbage-free reversible integer multiplication with constants of the form 2k±2l±1. In: Glück, R., Yokoyama, T. (eds.) RC 2012. LNCS, vol. 7581, pp. 171–182. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  2. 2.
    Cuccaro, S.A., Draper, T.G., Kutin, S.A., Moulton, D.P.: A New Quantum Ripple-carry Addition Circuit arXiv:quant-ph/0410184 (2005)Google Scholar
  3. 3.
    Draper, T.G., Kutin, S.A., Rains, E.M., Svore, K.M.: A Logarithmic-Depth Quantum Carry-Lookahead Adder arXiv:quant-ph/0406142 (2008)Google Scholar
  4. 4.
    Fredkin, E., Toffoli, T.: Conservative Logic. International Journal of Theoretical Physics 21(3-4), 219–253 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Mogensen, T.Æ.: Garbage-Free Reversible Constant Multipliers for Arbitrary Integers. In: Dueck, G.W., Miller, D.M. (eds.) RC 2013. LNCS, vol. 7948, pp. 70–83. Springer, Heidelberg (2013)Google Scholar
  6. 6.
    Parvardi, A.H.: Lifting The Exponent, LTE (2011),
  7. 7.
    Thomsen, M.K., Axelsen, H.B.: Parallelization of Reversible Ripple-carry Adders. Parallel Processing Letters 19(1), 205–222 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Toffoli, T.: Reversible Computing. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 632–644. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  9. 9.
    Van Rentergem, Y., De Vos, A.: Optimal Design of a Reversible Full Adder. International Journal of Unconventional Computing 1(4), 339–355 (2005)Google Scholar
  10. 10.
    Vedral, V., Barenco, A., Ekert, A.: Quantum Networks for Elementary Arithmetic Operations. Physical Review A 54(1), 147–153 (1996)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zsigmondy, K.: Zur Theorie der Potenzreste. Monatshefte für Mathematik und Physik 3(1), 265–284 (1892)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Eva Rotenberg
    • 1
  • James Cranch
    • 2
  • Michael Kirkedal Thomsen
    • 1
  • Holger Bock Axelsen
    • 1
  1. 1.DIKU, Dept. of Computer ScienceUniversity of CopenhagenDenmark
  2. 2.Dept. of Computer ScienceUniversity of SheffieldUnited Kingdom

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