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Journal Article

Multiplication of Multidigit Numbers on Automata

About: This article is published in Soviet physics. Doklady.The article was published on 1963-01-01 and is currently open access. It has received 849 citations till now. The article focuses on the topics: Multiplication.
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Book
01 Jan 1996
TL;DR: A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols.
Abstract: From the Publisher: A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.

13,597 citations


Cites background or methods from "Multiplication of Multidigit Number..."

  • ...18 was known to Karatsuba and Ofman [661]....

    [...]

  • ...A recursive algorithm due to Karatsuba and Ofman [661] reduces the complexity of multiplying two n-bit integers to O(n)....

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Book
01 Jan 1974
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Abstract: From the Publisher: With this text, you gain an understanding of the fundamental concepts of algorithms, the very heart of computer science. It introduces the basic data structures and programming techniques often used in efficient algorithms. Covers use of lists, push-down stacks, queues, trees, and graphs. Later chapters go into sorting, searching and graphing algorithms, the string-matching algorithms, and the Schonhage-Strassen integer-multiplication algorithm. Provides numerous graded exercises at the end of each chapter. 0201000296B04062001

9,262 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered factoring integers and finding discrete logarithms on a quantum computer and gave an efficient randomized algorithm for these two problems, which takes a number of steps polynomial in the input size of the integer to be factored.
Abstract: A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

7,427 citations

Book
01 Jan 2004
TL;DR: This guide explains the basic mathematics, describes state-of-the-art implementation methods, and presents standardized protocols for public-key encryption, digital signatures, and key establishment, as well as side-channel attacks and countermeasures.
Abstract: After two decades of research and development, elliptic curve cryptography now has widespread exposure and acceptance. Industry, banking, and government standards are in place to facilitate extensive deployment of this efficient public-key mechanism. Anchored by a comprehensive treatment of the practical aspects of elliptic curve cryptography (ECC), this guide explains the basic mathematics, describes state-of-the-art implementation methods, and presents standardized protocols for public-key encryption, digital signatures, and key establishment. In addition, the book addresses some issues that arise in software and hardware implementation, as well as side-channel attacks and countermeasures. Readers receive the theoretical fundamentals as an underpinning for a wealth of practical and accessible knowledge about efficient application. Features & Benefits: * Breadth of coverage and unified, integrated approach to elliptic curve cryptosystems * Describes important industry and government protocols, such as the FIPS 186-2 standard from the U.S. National Institute for Standards and Technology * Provides full exposition on techniques for efficiently implementing finite-field and elliptic curve arithmetic* Distills complex mathematics and algorithms for easy understanding* Includes useful literature references, a list of algorithms, and appendices on sample parameters, ECC standards, and software toolsThis comprehensive, highly focused reference is a useful and indispensable resource for practitioners, professionals, or researchers in computer science, computer engineering, network design, and network data security.

2,893 citations


Cites methods from "Multiplication of Multidigit Number..."

  • ...268 B.2 URLs for standards bodies and working groups. . . . . . . . . . . . . . . 268 List of Figures 1.1 Basic communications model . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Symmetric-key versus public-key cryptography . . . . . . . . . . . . . . 4 2.1 Representing a prime-field element as an array of words . . . . . . . . . . 29 2.2 Depth-2 splits for 224-bit integers (Karatsuba-Ofman multiplication) . . . 33 2.3 Depth-2 splits for 192-bit integers (Karatsuba-Ofman multiplication) . . . 34 2.4 Representing a binary-field element as an array of words . . . . . . . . . 47 2.5 Right-to-left comb method for polynomial multiplication . . . . . . . . . 49 2.6 Left-to-right comb method for polynomial multiplication . . . . . . . . . 49 2.7 Left-to-right comb method with windows of width w . . . . . . . . . . ....

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  • ...The multiplication method of Karatsuba-Ofman is due to Karatsuba and Ofman [239]....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the authors considered factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems.
Abstract: A digital computer is generally believed to be an efficient universal computing device; that is, it is believed to be able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, for example, the number of digits of the integer to be factored.

2,856 citations