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.

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.

This book constitutes the thoroughly refereed post-workshop proceedings of the 7th International Workshop on the Arithmetic of Finite Field, WAIFI 2018, held in Bergen, Norway, in June 2018. The 14 revised full papers and six invited talks presented were carefully reviewed and selected from 26 submissions. The papers are organized in topical sections on invited talks; elliptic curves; hardware implementations; arithmetic and applications of finite fields and cryptography.

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.