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.
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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)
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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
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DOI: https://doi.org/10.1007/978-3-642-36315-3_14
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