Cryptography ,Cryptology and Coding

Symbolic dynamics is a mature yet rapidly developing area of dynamical systems. It has established strong connections with many areas, including linear algebra, graph theory, probability, group theory, and the theory of computation, as well as data storage, statistical mechanics, and $C^*$-algebras. This Second Edition maintains the introductory character of the original 1995 edition as a general textbook on symbolic dynamics and its applications to coding. It is written at an elementary level and aimed at students, well-established researchers, and experts in mathematics, electrical engineering, and computer science. Topics are carefully developed and motivated with many illustrative examples. There are more than 500 exercises to test the reader's understanding. In addition to a chapter in the First Edition on advanced topics and a comprehensive bibliography, the Second Edition includes a detailed Addendum, with companion bibliography, describing major developments and new research directions since publication of the First Edition.

Boolean functions are essential to systems for secure and reliable communication. This comprehensive survey of Boolean functions for cryptography and coding covers the whole domain and all important results, building on the author's influential articles with additional topics and recent results. A useful resource for researchers and graduate students, the book balances detailed discussions of properties and parameters with examples of various types of cryptographic attacks that motivate the consideration of these parameters. It provides all the necessary background on mathematics, cryptography, and coding, and an overview on recent applications, such as side channel attacks on smart cards, cloud computing through fully homomorphic encryption, and local pseudo-random generators. The result is a complete and accessible text on the state of the art in single and multiple output Boolean functions that illustrates the interaction between mathematics, computer science, and telecommunications.

Summary

An early version of this book was a handwritten manuscript from around 1990. In June 2001 a synopsis was written by invitation of Cambridge University Press with the working title “The construction and decoding of algebraic geometry codes,” or “Algebraic geometry and its applications (in error-correcting codes and cryptography).” That proposal was accepted, but with no indication of a deadline. So originally its aim was a book on algebraic geometry codes. As time passed more and more co-authors joined the team: Xin-Wen Wu in 2004, Stanislav Bulygin in 2007 and finally Relinde Jurrius in 2012.

Early versions of chapters were written on algebraic geometry codes, elementary coding theory, list decoding Reed–Muller codes, decoding algorithms and Gröbner bases, cryptography, weight enumerators and its generalizations and relations with matroid theory that appeared in books and journals [68, 69, 70, 71, 177, 184, 186, 266].

The prerequisites of this book are: elementary logical reasoning and naive set theory, some combinatorics and probability theory. Furthermore: linear algebra, the beginnings of group theory, the algebra of rings and fields. We will go into the details of polynomial rings and finite fields in several chapters.

The first six chapters of the book on the construction, properties and decoding of error-correcting codes are self-contained. This can be used as a course in Coding Theory of four hours a week during a semester in the first year of the Masters. It is advised to use Chapter 12 from the start to practise the theory with computer algebra systems.

The second half of the book on complexity theory, cryptology, Gröbner bases applied to codes and cryptosystems and algebraic geometry codes is more advanced. This can be used for a course in the second year of a Masters degree or can be read individually as a Capita Selecta. It is also a good starting point for a project or assignment. In the Notes at the end of every chapter we give ample references to further reading.

We thank the editors of Cambridge University Press: David Tranah, Jonathan Walthoe, Roger Astley, Clare Dennison and Abigail Walkington for their advice and patience.

Summary

In this chapter we give a brief overview of four computer algebra systems: Singular, Magma, GAP and Sage. We show how these computer algebra systems are able to deal with the objects considered in this book: error-correcting codes and their parameters, cryptosystems, as well as Gröbner bases and their applications. We work out specific examples that help to understand specifics that sometimes go beyond basic constructions with small parameters. This chapter also aims at invoking interest with those readers who have not worked with computer algebra systems before, but would like to do so having specific interesting examples to enter the field.

We start with a short overview for each of the four systems and then dive into hands-on examples.

SINGULAR

As is stated at www.singular.uni–kl.de: “Singular is a computer algebra system for polynomial computations, with special emphasis on commutative and non-commutative algebra, algebraic geometry, and singularity theory.” The system started as a tool for computing Gröbner bases with a focus on applications in singularity theory, but it grew into a tool for applications in many other fields. Relevant for us will be the method of Gröbner bases for decoding error-correcting codes, see Section 12.7. Singular also provides some basic functionality for cryptology, mainly for teaching purposes. We use a powerful kernel functionality of Singular for relevant Gröbner bases computations.

SINGULAR can be downloaded free of charge under the GNU public license from www.singular.uni–kl.de/index.php/singular-download.html for different platforms (Linux/Unix, Windows, OS X). The web site provides an online manual at www.singular.uni–kl.de/Manual/latest/index.htm. Also an online version of the system for trying out without a download is available at https://www.singular.uni–kl.de:8003/

In the context of this book, we use functionality provided for decoding linear codes via polynomial system solving (decodegb.lib). Interesting also, and therefore mentioned here, are the libraries:

Summary

This chapter provides a brief overview of selected topics in the exciting field of cryptology. In particular, we cover symmetric and asymmetric encryption cryptosystems. When talking about symmetric cryptography, we deal with the notion of a block cipher as one of the main cryptographic primitives, i.e. building blocks, for modern practical applications. Asymmetric cryptography is illustrated by the RSA and El Gamal encryption cryptosystems. Later, we present code-based cryptosystems due to McEliece and Niederreiter. In this chapter we will use the terms “cryptosystem” and “encryption schemes” as synonyms for simplicity. These schemes are good examples for a connection between the theory of error-correcting codes and cryptography. Whereas in the theory of error-correcting codes one deals with correcting errors in noisy channels, in the cryptographic applications a crypto designer intentionally adds noise to confuse an attacker and only a legitimate counterpart can remove the noise from a transmitted message.

We also take a look at other interesting topics such as authentication codes, secret sharing, and linear feedback shift registers. Here, we especially discuss connections to codes and related structures. Therewith we show how error-correcting codes can be used in other parts of cryptography as well.

Symmetric Encryption Schemes and Block Ciphers

Symmetric Encryption Schemes

This section is devoted to symmetric cryptosystems, symmetric encryption schemes to be more precise. The idea behind these is quite simple and thus has been known for ages. The problem to be addressed is to convey a secret message between two parties, called traditionally Alice and Bob, so that figuring out the secret message is not possible without knowledge of some additional information. This additional information is called a secret key and is supposed to be known only to the two communicating parties. The secrecy of the transmitted message rests entirely upon the knowledge of this secret key, and thus if an adversary or an eavesdropper, traditionally called Eve, is able to find out the key, then the whole secret communication is corrupted. Now let us take a look at the formal definition.

Summary

The idea of redundant information is a well-known phenomenon in reading a newspaper. Misspellings usually go unnoticed for a casual reader, while the meaning is still grasped. In Semitic languages such as Hebrew, and even older in the hieroglyphics in the tombs of the pharaohs of Egypt, only the consonants are written while the vowels are left out so that we do not know for sure how to pronounce these words nowadays. The letter “e” is the most frequent occurring symbol in the English language, and leaving out all these letters would still give in almost all cases an understandable text to the expense of greater attention of the reader.

The science of deleting redundant information in a clever way such that it can be stored in less memory or space and still can be expanded to the original message is called data compression or source coding. It is not the topic of this book. So we can compress data, but an error made in a compressed text would give a different message that is most of the time utterly meaningless.

The idea in error-correcting codes is the converse. One adds redundant information in such a way that it is possible to detect or even correct errors after transmission. In radio contacts between pilots and radar controls the letters in the alphabet are spoken phonetically as “Alpha, Bravo, Charlie, …” but “Adams, Boston, Chicago, …” is more commonly used for spelling in a telephone conversation. The addition of a parity check symbol enables one to detect an error, such as on the former punch cards that were fed into a computer, in the ISBN code for books, the European Article Numbering (EAN) and the Universal Product Code (UPC) for articles. Error-correcting codes are common in numerous modern applications where data are required to be stored, processed and transmitted in a reliable manner, such as in audiovisual media, quick response (QR) codes, fault-tolerant computer systems and deep space telecommunication, to name but a few.

In this chapter, we present an introduction to error-correcting codes focusing on the most commonly used codes, namely, block codes and linear codes, including fundamental concepts and procedures for code construction, encoding and decoding.

Cryptography ,Cryptology and Coding

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8 - Functions satisfying SAC, PC, and EPC, or having good GAC

Index

An Introduction to Symbolic Dynamics and Coding

Boolean Functions for Cryptography and Coding Theory

3 - Weight Enumeration

Preface

Summary

2 - Code Constructions and Bounds on Codes

12 - Coding and Cryptology with Computer Algebra

Summary

7 - Complexity and Decoding

Frontmatter

Index

Contents

5 - Polynomial Codes

Dedication

11 - Codes on Curves

10 - Gröbner Bases for Coding and Cryptology

6 - Algebraic Decoding

9 - Cryptology

Summary

References

1 - Error-correcting Codes

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An Introduction to Symbolic Dynamics and Coding

Boolean Functions for Cryptography and Coding Theory

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