Reversible Karatsuba's Algorithm

  title={Reversible Karatsuba's Algorithm},
  author={Luis Antonio Brasil Kowada and Renato Portugal and Celina M. H. de Figueiredo},
  journal={J. Univers. Comput. Sci.},
Karatsuba discovered the first algorithm that accomplishes multiprecision integer multiplication with complexity below that of the grade-school method. [...] Key Method 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.Expand
Improved reversible and quantum circuits for Karatsuba-based integer multiplication
A reversible circuit for integer multiplication that is inspired by Karatsuba's recursive method is presented, with the main improvement over circuits that have been previously reported in the literature is an asymptotic reduction of the amount of space required. Expand
Efficient hardware architecture of recursive Karatsuba-Ofman multiplier
Two Sequential/Parallel architectures of Recursive Karatsuba-Ofman multiplier are presented, developed and implemented on the Spartan 3 FPGA platform and Mathematical Performances models for large number (n) are elaborated for the proposed architectures. Expand
Garbage-Free Reversible Integer Multiplication with Constants of the Form 2 k ±2 l ±1
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 beExpand
The Compilation of Reversible Circuits and a New Optimization Game
The usefulness of the pebble games for circuit analysis is demonstrated by finding a new space bound for the Karatsuba algorithm and more generally for any similar algorithm based on recurrence relations. Expand
Self-Inverse Functions and Palindromic Circuits
We investigate the subclass of reversible functions that are self-inverse and relate them to reversible circuits that are equal to their reverse circuit, which are called palindromic circuits. WeExpand
Quantum Circuit Design of Toom 3-Way Multiplication
In classical computation, Toom–Cook is one of the multiplication methods for large numbers which offers faster execution time compared to other algorithms such as schoolbook and KaratsubaExpand
High Performance Quantum Modular Multipliers
We present a novel set of reversible modular multipliers applicable to quantum computing, derived from three classical techniques: 1) traditional integer division, 2) Montgomery residue arithmetic,Expand
Efficient Large Numbers Karatsuba-Ofman Multiplier Designs for Embedded Systems
A mixture of sequential and combinational system design techniques involving pipelining is applied to the proposed designs of implementing Karatsuba-Ofman multiplier and these designs are compared to other existing techniques. Expand
Synthesizing multiplier in reversible logic
Three methods are presented that address the drawbacks of previous approaches to synthesis of multiplier circuits in reversible logic, including the large number of circuit lines in the resulting realizations as well as the poor scalability. Expand
A reversible circuit synthesis algorithm with progressive increase of controls in generalized Toffoli gates
We present a new algorithm for synthesis of reversible circuits for arbitrary n-bit bijective functions. This algorithm uses generalized Toffoli gates, which include positive and negative controls.Expand


Generalizations of the Karatsuba Algorithm for Efficient Implementations
In this work we generalize the classical Karatsuba Algorithm (KA) for polynomial multiplication to (i) polynomials of arbitrary degree and (ii) recursive use. We determine exact complexityExpand
Using the Parallel Karatsuba Algorithm for Long Integer Multiplication and Division
The Karatsuba algorithm is used within long integer division, by a recent divide-and-conquer technique, and exhibits modest speed-ups, however the combined speed-up over the classical sequential method is more than 10 at 500 words. Expand
A Non-redundant and Efficient Architecture for Karatsuba-Ofman Algorithm
A Non-Redundant Karatsuba-Ofman algorithm (NRKOA) with removing redundancy operations is proposed, and a parallel hardware architecture based on the proposed algorithm reduces the area complexity. Expand
Performance Analysis of the Parallel Karatsuba Multiplication Algorithm for Distributed Memory Architectures
It is shown that the theoretically best speed-up and efficiency can be obtained with two of the algorithms for sufficient problem size, and the experimental results confirm the analysis. Expand
Algorithms for quantum computation: discrete logarithms and factoring
  • P. Shor
  • Mathematics, Computer Science
  • Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given. Expand
Reversible simulation of irreversible computation
  • Ming Li, P. Vitányi
  • Computer Science
  • Proceedings of Computational Complexity (Formerly Structure in Complexity Theory)
  • 1996
It is shown that among all simulations which can be modelled by the pebble game, Bennett's simulation is optimal in that it uses the least auxiliary space for the greatest number of simulated steps. Expand
Schnelle Multiplikation großer Zahlen
ZusammenfassungEs wird ein Algorithmus zur Berechnung des Produktes von zweiN-stelligen Dualzahlen angegeben. Zwei Arten der Realisierung werden betrachtet: Turingmaschinen mit mehreren Bändern undExpand
Efficient networks for quantum factoring.
The number of memory quantum bits (qubits) and the number of operations required to perform factorization, using the algorithm suggested by Shor are estimated. Expand
Logical reversibility of computation
This result makes plausible the existence of thermodynamically reversible computers which could perform useful computations at useful speed while dissipating considerably less than kT of energy per logical step. Expand
Fast versions of Shor's quantum factoring algorithm
Fast and highly parallelized versions of Shor's algorithm are presented, which with a sizable quantum computer it would then be possible to factor numbers with millions of digits. Expand