Cryptographic Boolean functions and applications.

*(English)*Zbl 1173.94002
Amsterdam: Elsevier/Academic Press (ISBN 978-0-12-374890-4/hbk). xii, 232 p. (2009).

Editorial remark: Accidentally, this book has been issued twice for reviewing. Therefore, we display both reviews, by adding the second one to the originally published review:

First review:

The content of the book can be separated into three parts:

A (Chapters 2–5). An overview of cryptographic properties provided by boolean functions. Detailed analysis are dedicated to (i) Fourier and Walsh (Hadamard in some references) transformations, (2) some correlation immunity between boolean functions, (3) SAC (Strict Avalanche Criterion), PC (Propagation Criterion) and bent (very useful in providing diffusion and confusion characteristics of a cryptosystem) properties.

Moreover, theoretical notions related with Coding Theory and Cryptography (like linear feedback shift registers, or the Berlekamp - Massey algorithm) are shortly presented. The authors take into account the idea that “a Boolean function cannot simultaneously have too many cryptographic desirable properties.” So, some tradeoffs between different properties (immunity, nonlinearity etc) are considered.

B. (Chapters 6-7): Here are the only parts of the book focused on the cryptographic tools: Stream cipher design and standard Block ciphers (DES, AES). Some security requirements of these tools related to boolean functions are detailed. A cryptanalysis like algebraic attack (pp. 147–154) can be very useful in repairing the weaknesses of cryptosystems used in mobile phones area. These two chapters can be completely separated from the book; they are in fact just a study of the case when the boolean functions with cryptographic properties can be applied in information security.

C. (Chapter 8). A special technique – based on Cayley graphs – for dealing with Boolean functions is presented. Although it follows the first part of the book, this chapter can be easily considered as a separate part.

If we add to these components (D) Some historical notes about biographies of two personalities who states the domain: George Boole and Claude Shannon (Chapter 1), and an immpressive bibliography (489 scientific papers, reports, conference proceedings, books, notes), will result a very good monograph having as subject the security properties of Boolean functions.

Readers who do not have sufficient mathematical background will find this book particularly difficult to follow, The book requires enough knowledge of mathematics to be able to understand various logical concepts and their possible applications in security. But, of course, we cannot remain always at the beginning stage. Finally, the topics are covered in an enough meaningful way.

This is a highly recommended book for anyone who wants to develop a real security tools and for anyone who wants to understand the theoretical details that the good security tool writers know.

The content of the book can be separated into three parts:

A (Chapters 2–5). An overview of cryptographic properties provided by boolean functions. Detailed analysis are dedicated to (i) Fourier and Walsh (Hadamard in some references) transformations, (2) some correlation immunity between boolean functions, (3) SAC (Strict Avalanche Criterion), PC (Propagation Criterion) and bent (very useful in providing diffusion and confusion characteristics of a cryptosystem) properties.

Moreover, theoretical notions related with Coding Theory and Cryptography (like linear feedback shift registers, or the Berlekamp-Massey algorithm) are

Reviewer: Adrian Atanasiu (Bucureşti)

Second review

After an introduction in Chapter 1, Chapter 2 describes the Fourier analysis of Boolean functions. It focuses on the (fast) Walsh transform, linear transformations, the Parseval equation and Hadamard matrices. Chapter 3 considers avalanche and propagation criteria as main tools for building block ciphers nowadays. Various classes of functions satysfying SAC (Strong Avalanche Criterion) and/or PC (Propagation Criterion) are listed and analysed. Chapter 4 deals with correlation immune functions, their construction and applications in LFSRs design. Chapter 5 introduces bent Boolean functions – functions for which the Walsh transform coefficients \(\hat{f}\) are all \(\pm 2^{n/2}\) if \(f\) is a function of \(n\) variables. Various types of their construction are described in detail and some variations of bent functions (partial, semi, symmetric, rotation symmetric) are considered at the end of this chapter. Finally, Chapter 6 is concerned with the stream cipher design. Applications of Boolean functions in pseudorandom bit generators (nonlinear combination, nonlinear filter, multiplexer, irregularly clocked) are described. Chapter 7 is on block ciphers, their design approaches and DES/AES ciphers are defined in detail. The rest of that chapter is devoted to algebraic representation of AES. The last chapter concludes with a discussion about Cayley graphs, their coloring, avalanche features and affine transformations.

Beside being extremely useful to those who are interested in new block cipher designs based on Boolean functions, the book is also of great interest to anyone interested in Boolean functions theory. It is written in highly scientific language so the reader should be familiar with Boolean algebra and related notation.

Reviewer: Vladimír Lacko (Košice)

First review:

The content of the book can be separated into three parts:

A (Chapters 2–5). An overview of cryptographic properties provided by boolean functions. Detailed analysis are dedicated to (i) Fourier and Walsh (Hadamard in some references) transformations, (2) some correlation immunity between boolean functions, (3) SAC (Strict Avalanche Criterion), PC (Propagation Criterion) and bent (very useful in providing diffusion and confusion characteristics of a cryptosystem) properties.

Moreover, theoretical notions related with Coding Theory and Cryptography (like linear feedback shift registers, or the Berlekamp - Massey algorithm) are shortly presented. The authors take into account the idea that “a Boolean function cannot simultaneously have too many cryptographic desirable properties.” So, some tradeoffs between different properties (immunity, nonlinearity etc) are considered.

B. (Chapters 6-7): Here are the only parts of the book focused on the cryptographic tools: Stream cipher design and standard Block ciphers (DES, AES). Some security requirements of these tools related to boolean functions are detailed. A cryptanalysis like algebraic attack (pp. 147–154) can be very useful in repairing the weaknesses of cryptosystems used in mobile phones area. These two chapters can be completely separated from the book; they are in fact just a study of the case when the boolean functions with cryptographic properties can be applied in information security.

C. (Chapter 8). A special technique – based on Cayley graphs – for dealing with Boolean functions is presented. Although it follows the first part of the book, this chapter can be easily considered as a separate part.

If we add to these components (D) Some historical notes about biographies of two personalities who states the domain: George Boole and Claude Shannon (Chapter 1), and an immpressive bibliography (489 scientific papers, reports, conference proceedings, books, notes), will result a very good monograph having as subject the security properties of Boolean functions.

Readers who do not have sufficient mathematical background will find this book particularly difficult to follow, The book requires enough knowledge of mathematics to be able to understand various logical concepts and their possible applications in security. But, of course, we cannot remain always at the beginning stage. Finally, the topics are covered in an enough meaningful way.

This is a highly recommended book for anyone who wants to develop a real security tools and for anyone who wants to understand the theoretical details that the good security tool writers know.

The content of the book can be separated into three parts:

A (Chapters 2–5). An overview of cryptographic properties provided by boolean functions. Detailed analysis are dedicated to (i) Fourier and Walsh (Hadamard in some references) transformations, (2) some correlation immunity between boolean functions, (3) SAC (Strict Avalanche Criterion), PC (Propagation Criterion) and bent (very useful in providing diffusion and confusion characteristics of a cryptosystem) properties.

Moreover, theoretical notions related with Coding Theory and Cryptography (like linear feedback shift registers, or the Berlekamp-Massey algorithm) are

Reviewer: Adrian Atanasiu (Bucureşti)

Second review

After an introduction in Chapter 1, Chapter 2 describes the Fourier analysis of Boolean functions. It focuses on the (fast) Walsh transform, linear transformations, the Parseval equation and Hadamard matrices. Chapter 3 considers avalanche and propagation criteria as main tools for building block ciphers nowadays. Various classes of functions satysfying SAC (Strong Avalanche Criterion) and/or PC (Propagation Criterion) are listed and analysed. Chapter 4 deals with correlation immune functions, their construction and applications in LFSRs design. Chapter 5 introduces bent Boolean functions – functions for which the Walsh transform coefficients \(\hat{f}\) are all \(\pm 2^{n/2}\) if \(f\) is a function of \(n\) variables. Various types of their construction are described in detail and some variations of bent functions (partial, semi, symmetric, rotation symmetric) are considered at the end of this chapter. Finally, Chapter 6 is concerned with the stream cipher design. Applications of Boolean functions in pseudorandom bit generators (nonlinear combination, nonlinear filter, multiplexer, irregularly clocked) are described. Chapter 7 is on block ciphers, their design approaches and DES/AES ciphers are defined in detail. The rest of that chapter is devoted to algebraic representation of AES. The last chapter concludes with a discussion about Cayley graphs, their coloring, avalanche features and affine transformations.

Beside being extremely useful to those who are interested in new block cipher designs based on Boolean functions, the book is also of great interest to anyone interested in Boolean functions theory. It is written in highly scientific language so the reader should be familiar with Boolean algebra and related notation.

Reviewer: Vladimír Lacko (Košice)

##### MSC:

94-02 | Research exposition (monographs, survey articles) pertaining to information and communication theory |

06-02 | Research exposition (monographs, survey articles) pertaining to ordered structures |

94A60 | Cryptography |

06E30 | Boolean functions |