Mathematical methods in solutions of the problems from the Third International Students' Olympiad in Cryptography

The mathematical problems and their solutions of the Third International Students' Olympiad in Cryptography NSUCRYPTO'2016 are presented. We consider mathematical problems related to the construction of algebraic immune vectorial Boolean functions and big Fermat numbers, problems about secrete sharing schemes and pseudorandom binary sequences, biometric cryptosystems and the blockchain technology, etc. Two open problems in mathematical cryptography are also discussed and a solution for one of them proposed by a participant during the Olympiad is described. It was the first time in the Olympiad history

A Novel Application of Boolean Functions with High Algebraic Immunity in Minimal Codes

Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing binary minimal codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension $m$ and length less than or equal to $m(m+1)/2$ are obtained. In addition, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields such as symmetric cryptography, coding theory and secret sharing schemes

On Equivalence of Known Families of APN Functions in Small Dimensions

In this extended abstract, we computationally check and list the CCZ-inequivalent APN functions from infinite families on $\mathbb{F}_2^n$ for n from 6 to 11. These functions are selected with simplest coefficients from CCZ-inequivalent classes. This work can simplify checking CCZ-equivalence between any APN function and infinite APN families.Comment: This paper is already in "PROCEEDING OF THE 20TH CONFERENCE OF FRUCT ASSOCIATION

Algorithms for Computing the Linearity and Degree of Vectorial Boolean Functions

In this article, we study two representations of a Boolean function which are very important in the context of cryptography. We describe Möbius and Walsh Transforms for Boolean functions in details and present effective algorithms for their implementation. We combine these algorithms with the Gray code to compute the linearity, nonlinearity and algebraic degree of a vectorial Boolean function. Such a detailed consideration will be very helpful for students studying the design of block ciphers, including PhD students in the beginning of their research. ACM Computing Classification System (1998): F.2.1, F.2.2