Construction and Analysis of Cryptographic Functions

Lilya Budaghyan
 This book covers novel research on construction and analysis of optimal cryptographic functions such as almost perfect nonlinear (APN), almost bent (AB), planar and bent functions. These functions have optimal resistance to linear and/or differential attacks, which are the two most powerful attacks on symmetric cryptosystems. Besides cryptographic applications, these functions are significant in many branches of mathematics and information theory including coding theory, combinatorics, commutative algebra, finite geometry, sequence design and quantum information theory. The author analyzes equivalence relations for these functions and develops several new methods for construction of their infinite families. In addition, the book offers solutions to two longstanding open problems, including the problem on characterization of APN and AB functions via Boolean, and the problem on the relation between two classes of bent functions. 

Boolean Models and Methods in Mathematics, Computer Science, and Engineering

Yves Crama and Peter L. Hammer editors
 Contains Claude Carlet’s two books among other things. 

Boolean Functions for Cryptography and Error Correcting Codes

Claude Carlet (Chapter in “Boolean Models and Methods in Mathematics, Computer Science, and Engineering”)
 Amazing book! 

Vectorial Boolean Functions for Cryptography

Claude Carlet (Chapter in “Boolean Models and Methods in Mathematics, Computer Science, and Engineering”)
 Also amazing! 

The Equivalence of AB and APN Functions and Their Generalizations

Lilya Budaghyan
 Vectorial Boolean functions are used in cryptography, in particular in block ciphers. An important condition on these functions is a high resistance to differential and linear cryptanalysis, which are the main attacks on block ciphers. The functions which possess the best resistance to the differential attack are called almost perfect nonlinear (APN). Almost bent (AB) functions are those mappings which oppose an optimum resistance to both linear and differential attacks. Before this work, only a few classes of APN and AB functions had been known and all these classes happened to be extended affine equivalent (EA-equivalent) to power functions. In this work we construct the first classes of APN and AB polynomials EA-inequivalent to power mappings by using the equivalence relation of functions (which we call CCZ-equivalence) introduced by Carlet, Charpin and Zinoviev (1998). Then the constructed APN and AB functions are used to solve other related problems. 

Bent Functions: Fundamentals and Results

Sihem Mesnager
 This book gives a detailed survey of the main results on bent functions over finite fields, presents a systematic overview of their generalizations, variations and applications, considers open problems in classification and systematization of bent functions, and discusses proofs of several results. This book uniquely provides a necessary comprehensive coverage of bent functions.It serves as a useful reference for researchers in discrete mathematics, coding and cryptography. Students and professors in mathematics and computer science will also find the content valuable, especially those interested in mathematical foundations of cryptography. It can be used as a supplementary text for university courses on discrete mathematics, Boolean functions, or cryptography, and is appropriate for both basic classes for under-graduate students and advanced courses for specialists in cryptography and mathematics. 

Introduction to Finite Fields and their Applications

Rudolf Lidl and Harald Niederreiter
 A standard textbook on the theory of finite fields and other elementary topics. 

Handbook of Finite Fields

Gary L. Mullen and Daniel Panario
 Another amazing reference!