Cryptanalysis of Number Theoretic Ciphers

Author: Samuel S. Wagstaff, Jr.

Publisher: CRC Press

ISBN: 9781584881537

Category: Mathematics

Page: 336

View: 9368

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At the heart of modern cryptographic algorithms lies computational number theory. Whether you're encrypting or decrypting ciphers, a solid background in number theory is essential for success. Written by a number theorist and practicing cryptographer, Cryptanalysis of Number Theoretic Ciphers takes you from basic number theory to the inner workings of ciphers and protocols. First, the book provides the mathematical background needed in cryptography as well as definitions and simple examples from cryptography. It includes summaries of elementary number theory and group theory, as well as common methods of finding or constructing large random primes, factoring large integers, and computing discrete logarithms. Next, it describes a selection of cryptographic algorithms, most of which use number theory. Finally, the book presents methods of attack on the cryptographic algorithms and assesses their effectiveness. For each attack method the author lists the systems it applies to and tells how they may be broken with it. Computational number theorists are some of the most successful cryptanalysts against public key systems. Cryptanalysis of Number Theoretic Ciphers builds a solid foundation in number theory and shows you how to apply it not only when breaking ciphers, but also when designing ones that are difficult to break.

Mathematical Foundations of Public Key Cryptography

Author: Xiaoyun Wang,Guangwu Xu,Mingqiang Wang,Xianmeng Meng

Publisher: CRC Press

ISBN: 1498702244

Category: Computers

Page: 220

View: 5521

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In Mathematical Foundations of Public Key Cryptography, the authors integrate the results of more than 20 years of research and teaching experience to help students bridge the gap between math theory and crypto practice. The book provides a theoretical structure of fundamental number theory and algebra knowledge supporting public-key cryptography. Rather than simply combining number theory and modern algebra, this textbook features the interdisciplinary characteristics of cryptography—revealing the integrations of mathematical theories and public-key cryptographic applications. Incorporating the complexity theory of algorithms throughout, it introduces the basic number theoretic and algebraic algorithms and their complexities to provide a preliminary understanding of the applications of mathematical theories in cryptographic algorithms. Supplying a seamless integration of cryptography and mathematics, the book includes coverage of elementary number theory; algebraic structure and attributes of group, ring, and field; cryptography-related computing complexity and basic algorithms, as well as lattice and fundamental methods of lattice cryptanalysis. The text consists of 11 chapters. Basic theory and tools of elementary number theory, such as congruences, primitive roots, residue classes, and continued fractions, are covered in Chapters 1-6. The basic concepts of abstract algebra are introduced in Chapters 7-9, where three basic algebraic structures of groups, rings, and fields and their properties are explained. Chapter 10 is about computational complexities of several related mathematical algorithms, and hard problems such as integer factorization and discrete logarithm. Chapter 11 presents the basics of lattice theory and the lattice basis reduction algorithm—the LLL algorithm and its application in the cryptanalysis of the RSA algorithm. Containing a number of exercises on key algorithms, the book is suitable for use as a textbook for undergraduate students and first-year graduate students in information security programs. It is also an ideal reference book for cryptography professionals looking to master public-key cryptography.

A Course in Number Theory and Cryptography

Author: Neal Koblitz

Publisher: Springer Science & Business Media

ISBN: 1441985921

Category: Mathematics

Page: 235

View: 2669

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This is a substantially revised and updated introduction to arithmetic topics, both ancient and modern, that have been at the centre of interest in applications of number theory, particularly in cryptography. As such, no background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasising estimates of the efficiency of the techniques that arise from the theory, and one special feature is the inclusion of recent applications of the theory of elliptic curves. Extensive exercises and careful answers are an integral part all of the chapters.

Computational Number Theory

Author: Abhijit Das

Publisher: CRC Press

ISBN: 1482205823

Category: Computers

Page: 614

View: 2656

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Developed from the author’s popular graduate-level course, Computational Number Theory presents a complete treatment of number-theoretic algorithms. Avoiding advanced algebra, this self-contained text is designed for advanced undergraduate and beginning graduate students in engineering. It is also suitable for researchers new to the field and practitioners of cryptography in industry. Requiring no prior experience with number theory or sophisticated algebraic tools, the book covers many computational aspects of number theory and highlights important and interesting engineering applications. It first builds the foundation of computational number theory by covering the arithmetic of integers and polynomials at a very basic level. It then discusses elliptic curves, primality testing, algorithms for integer factorization, computing discrete logarithms, and methods for sparse linear systems. The text also shows how number-theoretic tools are used in cryptography and cryptanalysis. A dedicated chapter on the application of number theory in public-key cryptography incorporates recent developments in pairing-based cryptography. With an emphasis on implementation issues, the book uses the freely available number-theory calculator GP/PARI to demonstrate complex arithmetic computations. The text includes numerous examples and exercises throughout and omits lengthy proofs, making the material accessible to students and practitioners.

Computational Number Theory and Modern Cryptography

Author: Song Y. Yan

Publisher: John Wiley & Sons

ISBN: 1118188616

Category: Computers

Page: 432

View: 4962

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The only book to provide a unified view of the interplay between computational number theory and cryptography Computational number theory and modern cryptography are two of the most important and fundamental research fields in information security. In this book, Song Y. Yang combines knowledge of these two critical fields, providing a unified view of the relationships between computational number theory and cryptography. The author takes an innovative approach, presenting mathematical ideas first, thereupon treating cryptography as an immediate application of the mathematical concepts. The book also presents topics from number theory, which are relevant for applications in public-key cryptography, as well as modern topics, such as coding and lattice based cryptography for post-quantum cryptography. The author further covers the current research and applications for common cryptographic algorithms, describing the mathematical problems behind these applications in a manner accessible to computer scientists and engineers. Makes mathematical problems accessible to computer scientists and engineers by showing their immediate application Presents topics from number theory relevant for public-key cryptography applications Covers modern topics such as coding and lattice based cryptography for post-quantum cryptography Starts with the basics, then goes into applications and areas of active research Geared at a global audience; classroom tested in North America, Europe, and Asia Incudes exercises in every chapter Instructor resources available on the book’s Companion Website Computational Number Theory and Modern Cryptography is ideal for graduate and advanced undergraduate students in computer science, communications engineering, cryptography and mathematics. Computer scientists, practicing cryptographers, and other professionals involved in various security schemes will also find this book to be a helpful reference.

Elliptic Curves

Number Theory and Cryptography, Second Edition

Author: Lawrence C. Washington

Publisher: CRC Press

ISBN: 9781420071474

Category: Mathematics

Page: 536

View: 3856

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Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and applications of elliptic curves. New to the Second Edition Chapters on isogenies and hyperelliptic curves A discussion of alternative coordinate systems, such as projective, Jacobian, and Edwards coordinates, along with related computational issues A more complete treatment of the Weil and Tate–Lichtenbaum pairings Doud’s analytic method for computing torsion on elliptic curves over Q An explanation of how to perform calculations with elliptic curves in several popular computer algebra systems Taking a basic approach to elliptic curves, this accessible book prepares readers to tackle more advanced problems in the field. It introduces elliptic curves over finite fields early in the text, before moving on to interesting applications, such as cryptography, factoring, and primality testing. The book also discusses the use of elliptic curves in Fermat’s Last Theorem. Relevant abstract algebra material on group theory and fields can be found in the appendices.

An Introduction to Mathematical Cryptography

Author: Jeffrey Hoffstein,Jill Pipher,Joseph H. Silverman

Publisher: Springer

ISBN: 1493917110

Category: Mathematics

Page: 538

View: 3717

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This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. This text provides an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online. The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include: classical cryptographic constructions, such as Diffie–Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures; fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms; an in-depth treatment of important cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem. The second edition of An Introduction to Mathematical Cryptography includes a significant revision of the material on digital signatures, including an earlier introduction to RSA, Elgamal, and DSA signatures, and new material on lattice-based signatures and rejection sampling. Many sections have been rewritten or expanded for clarity, especially in the chapters on information theory, elliptic curves, and lattices, and the chapter of additional topics has been expanded to include sections on digital cash and homomorphic encryption. Numerous new exercises have been included.

Introduction to Cryptography with Maple

Author: José Luis Gómez Pardo

Publisher: Springer Science & Business Media

ISBN: 3642321666

Category: Computers

Page: 706

View: 8558

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This introduction to cryptography employs a programming-oriented approach to study the most important cryptographic schemes in current use and the main cryptanalytic attacks against them. Discussion of the theoretical aspects, emphasizing precise security definitions based on methodological tools such as complexity and randomness, and of the mathematical aspects, with emphasis on number-theoretic algorithms and their applications to cryptography and cryptanalysis, is integrated with the programming approach, thus providing implementations of the algorithms and schemes as well as examples of realistic size. A distinctive feature of the author's approach is the use of Maple as a programming environment in which not just the cryptographic primitives but also the most important cryptographic schemes are implemented following the recommendations of standards bodies such as NIST, with many of the known cryptanalytic attacks implemented as well. The purpose of the Maple implementations is to let the reader experiment and learn, and for this reason the author includes numerous examples. The book discusses important recent subjects such as homomorphic encryption, identity-based cryptography and elliptic curve cryptography. The algorithms and schemes which are treated in detail and implemented in Maple include AES and modes of operation, CMAC, GCM/GMAC, SHA-256, HMAC, RSA, Rabin, Elgamal, Paillier, Cocks IBE, DSA and ECDSA. In addition, some recently introduced schemes enjoying strong security properties, such as RSA-OAEP, Rabin-SAEP, Cramer--Shoup, and PSS, are also discussed and implemented. On the cryptanalysis side, Maple implementations and examples are used to discuss many important algorithms, including birthday and man-in-the-middle attacks, integer factorization algorithms such as Pollard's rho and the quadratic sieve, and discrete log algorithms such as baby-step giant-step, Pollard's rho, Pohlig--Hellman and the index calculus method. This textbook is suitable for advanced undergraduate and graduate students of computer science, engineering and mathematics, satisfying the requirements of various types of courses: a basic introductory course; a theoretically oriented course whose focus is on the precise definition of security concepts and on cryptographic schemes with reductionist security proofs; a practice-oriented course requiring little mathematical background and with an emphasis on applications; or a mathematically advanced course addressed to students with a stronger mathematical background. The main prerequisite is a basic knowledge of linear algebra and elementary calculus, and while some knowledge of probability and abstract algebra would be helpful, it is not essential because the book includes the necessary background from these subjects and, furthermore, explores the number-theoretic material in detail. The book is also a comprehensive reference and is suitable for self-study by practitioners and programmers.

Algebraic Cryptanalysis

Author: Gregory Bard

Publisher: Springer Science & Business Media

ISBN: 0387887571

Category: Computers

Page: 356

View: 7596

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Algebraic Cryptanalysis bridges the gap between a course in cryptography, and being able to read the cryptanalytic literature. This book is divided into three parts: Part One covers the process of turning a cipher into a system of equations; Part Two covers finite field linear algebra; Part Three covers the solution of Polynomial Systems of Equations, with a survey of the methods used in practice, including SAT-solvers and the methods of Nicolas Courtois. Topics include: Analytic Combinatorics, and its application to cryptanalysis The equicomplexity of linear algebra operations Graph coloring Factoring integers via the quadratic sieve, with its applications to the cryptanalysis of RSA Algebraic Cryptanalysis is designed for advanced-level students in computer science and mathematics as a secondary text or reference book for self-guided study. This book is suitable for researchers in Applied Abstract Algebra or Algebraic Geometry who wish to find more applied topics or practitioners working for security and communications companies.

Cryptological Mathematics

Author: Robert Lewand

Publisher: MAA

ISBN: 9780883857199

Category: Mathematics

Page: 199

View: 3908

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Introduction to the mathematics of cryptology suitable for beginning undergraduates.

Stream Ciphers and Number Theory

Author: Thomas W. Cusick,Cunsheng Ding,Ari Renvall

Publisher: Elsevier

ISBN: 0444828737

Category: Mathematics

Page: 431

View: 9126

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Hardbound. This book is almost entirely concerned with stream ciphers, concentrating on a particular mathematical model for such ciphers which are called additive natural stream ciphers. These ciphers use a natural sequence generator to produce a periodic keystream. Full definitions of these concepts are given in Chapter 2.This book focuses on keystream sequences which can be analysed using number theory. It turns out that a great deal of information can be deducted about the cryptographic properties of many classes of sequences by applying the terminology and theorems of number theory. These connections can be explicitly made by describing three kinds of bridges between stream ciphering problems and number theory problems. A detailed summary of these ideas is given in the introductory Chapter 1.Many results in the book are new, and over seventy percent of these results described in this book are based on recent research

Algorithmic Cryptanalysis

Author: Antoine Joux

Publisher: CRC Press

ISBN: 9781420070033

Category: Computers

Page: 520

View: 6380

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Illustrating the power of algorithms, Algorithmic Cryptanalysis describes algorithmic methods with cryptographically relevant examples. Focusing on both private- and public-key cryptographic algorithms, it presents each algorithm either as a textual description, in pseudo-code, or in a C code program. Divided into three parts, the book begins with a short introduction to cryptography and a background chapter on elementary number theory and algebra. It then moves on to algorithms, with each chapter in this section dedicated to a single topic and often illustrated with simple cryptographic applications. The final part addresses more sophisticated cryptographic applications, including LFSR-based stream ciphers and index calculus methods. Accounting for the impact of current computer architectures, this book explores the algorithmic and implementation aspects of cryptanalysis methods. It can serve as a handbook of algorithmic methods for cryptographers as well as a textbook for undergraduate and graduate courses on cryptanalysis and cryptography.

Modern Cryptanalysis

Techniques for Advanced Code Breaking

Author: Christopher Swenson

Publisher: John Wiley & Sons

ISBN: 1118428625

Category: Computers

Page: 264

View: 2867

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As an instructor at the University of Tulsa, Christopher Swenson could find no relevant text for teaching modern cryptanalysis?so he wrote his own. This is the first book that brings the study of cryptanalysis into the 21st century. Swenson provides a foundation in traditional cryptanalysis, examines ciphers based on number theory, explores block ciphers, and teaches the basis of all modern cryptanalysis: linear and differential cryptanalysis. This time-honored weapon of warfare has become a key piece of artillery in the battle for information security.

Making, Breaking Codes

An Introduction to Cryptography

Author: Paul B. Garrett

Publisher: Pearson

ISBN: 9780130303691

Category: Business & Economics

Page: 524

View: 695

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This unique book explains the basic issues of classical and modern cryptography, and provides a self contained essential mathematical background in number theory, abstract algebra, and probability—with surveys of relevant parts of complexity theory and other things. A user-friendly, down-to-earth tone presents concretely motivated introductions to these topics.More detailed chapter topics include simple ciphers; applying ideas from probability; substitutions, transpositions, permutations; modern symmetric ciphers; the integers; prime numbers; powers and roots modulo primes; powers and roots for composite moduli; weakly multiplicative functions; quadratic symbols, quadratic reciprocity; pseudoprimes; groups; sketches of protocols; rings, fields, polynomials; cyclotomic polynomials, primitive roots; pseudo-random number generators; proofs concerning pseudoprimality; factorization attacks finite fields; and elliptic curves. For personnel in computer security, system administration, and information systems.

Group-based Cryptography

Author: Alexei Myasnikov,Vladimir Shpilrain,Alexander Ushakov

Publisher: Springer Science & Business Media

ISBN: 3764388277

Category: Mathematics

Page: 183

View: 920

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Covering relations between three different areas of mathematics and theoretical computer science, this book explores how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public key cryptography.

RC4 Stream Cipher and Its Variants

Author: Goutam Paul,Subhamoy Maitra

Publisher: CRC Press

ISBN: 1439831378

Category: Computers

Page: 311

View: 7885

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RC4 Stream Cipher and Its Variants is the first book to fully cover the popular software stream cipher RC4. With extensive expertise in stream cipher cryptanalysis and RC4 research, the authors focus on the analysis and design issues of RC4. They also explore variants of RC4 and the eSTREAM finalist HC-128. After an introduction to the vast field of cryptology, the book reviews hardware and software stream ciphers and describes RC4. It presents a theoretical analysis of RC4 KSA, discussing biases of the permutation bytes toward secret key bytes and absolute values. The text explains how to reconstruct the secret key from known state information and analyzes the RC4 PRGA in detail, including a sketch of state recovery attacks. The book then describes three popular attacks on RC4: distinguishing attacks, Wired Equivalent Privacy (WEP) protocol attacks, and fault attacks. The authors also compare the advantages and disadvantages of several variants of RC4 and examine stream cipher HC-128, which is the next level of evolution after RC4 in the software stream cipher paradigm. The final chapter emphasizes the safe use of RC4. With open research problems in each chapter, this book offers a complete account of the most current research on RC4.

Cryptography, a primer

Author: Alan G. Konheim

Publisher: John Wiley & Sons Inc

ISBN: N.A

Category: Computers

Page: 432

View: 2606

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Foundations of cryptography. Secrety systems. Monalphabetic sasubstitution. Polyalphabetic systems. Rotor systems. Block ciphers and the data encryption standard. Key management. Public key systems. Digital signatures and authentications. File security. References. Appendixes: Probability theory. The variance ...

Public-key Cryptography

Theory and Practice

Author: Abhijit Das,C. E. Veni Madhavan

Publisher: Pearson Education India

ISBN: 9788131708323

Category: Computers

Page: 562

View: 2415

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Public-key Cryptography provides a comprehensive coverage of the mathematical tools required for understanding the techniques of public-key cryptography and cryptanalysis. Key topics covered in the book include common cryptographic primitives and symmetric techniques, quantum cryptography, complexity theory, and practical cryptanalytic techniques such as side-channel attacks and backdoor attacks. Organized into eight chapters and supplemented with four appendices, this book is designed to be a self-sufficient resource for all students, teachers and researchers interested in the field of cryptography.