Reversible Karatsuba's Algorithm

Karatsuba discovered the first algorithm that accomplishes multiprecision integer multiplication with complexity below that of the grade-school method. This al- gorithm is implemented nowadays in computer algebra systems using irreversible logic. In this paper we describe reversible circuits for the Karatsuba's algorithm and analyze their computational complexity. We discuss garbage disposal methods and compare with the well known Bennett's schemes. These circuits can be used in reversible com- puters which have the advantage of being very efficient in terms of energy consumption. The algorithm can also be used in quantum computers and is an improvement of pre- vious circuits for the same purpose described in the literature.

[1]  Arnold Schönhage,et al.  Schnelle Multiplikation großer Zahlen , 1971, Computing.

[2]  Christof Zalka.  Fast versions of Shor's quantum factoring algorithm , 1998 .

[3]  Thomas G. Draper.  Addition on a Quantum Computer , 2000, quant-ph/0008033.

[4]  Alan T. Sherman,et al.  A Note on Bennett's Time-Space Tradeoff for Reversible Computation , 1990, SIAM J. Comput..

[5]  D. J. Bernstein.  Fast multiplication and its applications , 2008 .

[6]  Charles H. Bennett.  Time/Space Trade-Offs for Reversible Computation , 1989, SIAM J. Comput..

[7]  Giovanni Cesari,et al.  Performance Analysis of the Parallel Karatsuba Multiplication Algorithm for Distributed Memory Architectures , 1996, J. Symb. Comput..

[8]  Ming Li,et al.  Reversible simulation of irreversible computation , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[9]  Preskill,et al.  Efficient networks for quantum factoring. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[10]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[11]  T. Toffoli,et al.  Conservative logic , 2002, Collision-Based Computing.

[12]  Jongin Lim,et al.  A Non-redundant and Efficient Architecture for Karatsuba-Ofman Algorithm , 2005, ISC.

[13]  Christof Paar,et al.  Generalizations of the Karatsuba Algorithm for Efficient Implementations , 2006, IACR Cryptol. ePrint Arch..

[14]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[15]  Tudor Jebelean,et al.  Using the Parallel Karatsuba Algorithm for Long Integer Multiplication and Division , 1997, Euro-Par.

[16]  Dan Zuras.  More On Squaring and Multiplying Large Integers , 1994, IEEE Trans. Computers.

[17]  R. Landauer,et al.  Irreversibility and heat generation in the computing process , 1961, IBM J. Res. Dev..

[18]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[19]  Anatolij A. Karatsuba,et al.  Multiplication of Multidigit Numbers on Automata , 1963 .

[20]  Barenco,et al.  Quantum networks for elementary arithmetic operations. , 1995, Physical review. A, Atomic, molecular, and optical physics.