Generalized Mersenne Numbers in Pairing-Based Cryptography
01 Jan 2006-
TL;DR: The author’s home country, the United States, and some of the characters from the film adaptation are fictitious.
Abstract: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter
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01 Jun 1986
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Book•
14 Feb 2002
TL;DR: The underlying mathematics and the wide trail strategy as the basic design idea are explained in detail and the basics of differential and linear cryptanalysis are reworked.
Abstract: 1. The Advanced Encryption Standard Process.- 2. Preliminaries.- 3. Specification of Rijndael.- 4. Implementation Aspects.- 5. Design Philosophy.- 6. The Data Encryption Standard.- 7. Correlation Matrices.- 8. Difference Propagation.- 9. The Wide Trail Strategy.- 10. Cryptanalysis.- 11. Related Block Ciphers.- Appendices.- A. Propagation Analysis in Galois Fields.- A.1.1 Difference Propagation.- A.l.2 Correlation.- A. 1.4 Functions that are Linear over GF(2).- A.2.1 Difference Propagation.- A.2.2 Correlation.- A.2.4 Functions that are Linear over GF(2).- A.3.3 Dual Bases.- A.4.2 Relationship Between Trace Patterns and Selection Patterns.- A.4.4 Illustration.- A.5 Rijndael-GF.- B. Trail Clustering.- B.1 Transformations with Maximum Branch Number.- B.2 Bounds for Two Rounds.- B.2.1 Difference Propagation.- B.2.2 Correlation.- B.3 Bounds for Four Rounds.- B.4 Two Case Studies.- B.4.1 Differential Trails.- B.4.2 Linear Trails.- C. Substitution Tables.- C.1 SRD.- C.2 Other Tables.- C.2.1 xtime.- C.2.2 Round Constants.- D. Test Vectors.- D.1 KeyExpansion.- D.2 Rijndael(128,128).- D.3 Other Block Lengths and Key Lengths.- E. Reference Code.
3,444 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
TL;DR: A method for multiplying two integers modulo N while avoiding division by N, a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms.
Abstract: Let N > 1. We present a method for multiplying two integers (called N-residues) modulo N while avoiding division by N. N-residues are represented in a nonstandard way, so this method is useful only if several computations are done modulo one N. The addition and subtraction algorithms are unchanged. 1. Description. Some algorithms (1), (2), (4), (5) require extensive modular arith- metic. We propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms. Other recent algorithms for modular arithmetic appear in (3), (6). Fix N > 1. Define an A'-residue to be a residue class modulo N. Select a radix R coprime to N (possibly the machine word size or a power thereof) such that R > N and such that computations modulo R are inexpensive to process. Let R~l and N' be integers satisfying 0 N then return t - N else return t ■ To validate REDC, observe mN = TN'N = -Tmod R, so t is an integer. Also, tR = Tmod N so t = TR'X mod N. Thirdly, 0 < T + mN < RN + RN, so 0 < t < 2N. If R and N are large, then T + mN may exceed the largest double-precision value. One can circumvent this by adjusting m so -R < m < 0. Given two numbers x and y between 0 and N - 1 inclusive, let z = REDC(xy). Then z = (xy)R~x mod N, so (xR-l)(yR~x) = zRx mod N. Also, 0 < z < N, so z is the product of x and y in this representation. Other algorithms for operating on N-residues in this representation can be derived from the algorithms normally used. The addition algorithm is unchanged, since xR~x + yR~x = zR~x mod N if and only if x + y = z mod N. Also unchanged are
2,647 citations
"Generalized Mersenne Numbers in Pai..." refers methods in this paper
...Montgomery’s REDC algorithm [54] at the word level (REDC1)....
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01 Jan 2002
TL;DR: This volume is the authoritative guide to the Rijndael algorithm and AES and professionals, researchers, and students active or interested in data encryption will find it a valuable source of information and reference.
Abstract: From the Publisher:
In October 2000, the US National Institute of Standards and Technology selected the block cipher Rijndael as the Advanced Encryption Standard (AES). AES is expected to gradually replace the present Data Encryption Standard (DES) as the most widely applied data encryption technology.|This book by the designers of the block cipher presents Rijndael from scratch. The underlying mathematics and the wide trail strategy as the basic design idea are explained in detail and the basics of differential and linear cryptanalysis are reworked. Subsequent chapters review all known attacks against the Rijndael structure and deal with implementation and optimization issues. Finally, other ciphers related to Rijndael are presented.|This volume is THE authoritative guide to the Rijndael algorithm and AES. Professionals, researchers, and students active or interested in data encryption will find it a valuable source of information and reference.
2,140 citations
Book•
01 Jul 1999TL;DR: In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems as mentioned in this paper, and it has become all pervasive.
Abstract: In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems. Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes. As digital signatures become more and more important in the commercial world the use of elliptic curve-based signatures will become all pervasive. This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. Due to the advanced nature of the mathematics there is a high barrier to entry for individuals and companies to this technology. Hence this book will be invaluable not only to mathematicians wanting to see how pure mathematics can be applied but also to engineers and computer scientists wishing (or needing) to actually implement such systems.
1,697 citations