Abstract
Multiplication of integers is non-injective and, thus, requires garbage lines for any reversible logic implementation. However, multiplying with a fixed constant is injective, and should therefore be implementable in reversible logic without garbage. Despite this, the only reported circuits for constant multiplication without garbage are restricted to powers of 2, i.e., the multiplication is a simple bit-shift.
Here, we show how to generate a garbage-free linear-depth reversible logic circuit for multiplying an input integer with a constant of the form 2k±1 or 2k ±2l ±1, by building on a simple strength reduction to addition. Using several such circuits in sequence allows us to support a greater variety of constants. This enables wider use of constant multiplication in garbage-free reversible circuits than was previously possible.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Axelsen, H.B., Glück, R.: What Do Reversible Programs Compute? In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 42–56. Springer, Heidelberg (2011)
Burignat, S., Vermeirsch, K., De Vos, A., Thomsen, M.K.: Garbageless Reversible Implementation of Integer Linear Transformations. In: Glück, R., Yokoyama, T. (eds.) RC 2012. LNCS, vol. 7581, pp. 160–170. Springer, Heidelberg (2013)
Cuccaro, S.A., Draper, T.G., Kutin, S.A., Moulton, D.P.: A new quantum ripple-carry addition circuit. arXiv:quant-ph/0410184v1 (2005)
Daubechies, I., Sweldens, W.: Factoring wavelet transforms into lifting steps. Journal of Fourier Analysis and Applications 4(3), 247–269 (1998)
De Vos, A., Burignat, S., Thomsen, M.K.: Reversible implementation of a discrete integer linear transform. J. Mult.-Val. Log. S. 18(1), 25–35 (2012)
Draper, T.G., Kutin, S.A., Rains, E.M., Svore, K.M.: A logarithmic-depth quantum carry-lookahead adder. arXiv:quant-ph/0406142v1 (2004)
Kowada, L.A.B., Portugal, R., de Figueiredo, C.M.H.: Reversible Karatsuba’s algorithm. J. Universal Computer Science 12(5), 499–511 (2006)
Mogensen, T.Æ.: Private communication (2012)
Offermann, S., Wille, R., Dueck, G.W., Drechsler, R.: Synthesizing multiplier in reversible logic. In: 13th IEEE Symposium on Design and Diagnostics of Electronic Circuits and Systems, pp. 335–340. IEEE (2010)
Rotenberg, E.: Mersennary numbers, University of Copenhagen (report in preparation, 2012)
Thomsen, M.K., Axelsen, H.B.: Parallelization of reversible ripple-carry adders. Parallel Processing Letters 19(1), 205–222 (2009)
Thomsen, M.K., Glück, R., Axelsen, H.B.: Reversible arithmetic logic unit for quantum arithmetic. J. Phys. A: Math. Theor. 43(38), 382002 (2010)
Van Rentergem, Y., De Vos, A.: Optimal design of a reversible full adder. International Journal of Unconventional Computing 1(4), 339–355 (2005)
Vedral, V., Barenco, A., Ekert, A.: Quantum networks for elementary arithmetic operations. Physical Review A 54(1), 147–153 (1996)
Wille, R., Drechsler, R.: Towards a Design Flow for Reversible Logic. Springer Science (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Axelsen, H.B., Thomsen, M.K. (2013). Garbage-Free Reversible Integer Multiplication with Constants of the Form 2k±2l±1. In: Glück, R., Yokoyama, T. (eds) Reversible Computation. RC 2012. Lecture Notes in Computer Science, vol 7581. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36315-3_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-36315-3_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36314-6
Online ISBN: 978-3-642-36315-3
eBook Packages: Computer ScienceComputer Science (R0)