Fast Algebraic Attacks and Decomposition of Symmetric Boolean Functions

Algebraic and fast algebraic attacks are power tools to analyze stream ciphers. A class of symmetric Boolean functions with maximum algebraic immunity were found vulnerable to fast algebraic attacks at EUROCRYPT'06. Recently, the notion of AAR (algebraic attack resistant) functions was introduced as a unified measure of protection against both classical algebraic and fast algebraic attacks. In this correspondence, we first give a decomposition of symmetric Boolean functions, then we show that almost all symmetric Boolean functions, including these functions with good algebraic immunity, behave badly against fast algebraic attacks, and we also prove that no symmetric Boolean functions are AAR functions. Besides, we improve the relations between algebraic degree and algebraic immunity of symmetric Boolean functions.Comment: 13 pages, submitted to IEEE Transactions on Information Theor

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

Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any $m$, one can construct infinitely many $n$-variable ($n$ even), $m$-resilient functions with nonlinearity $>2^{n-1}-2^{n/2}$. A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.Comment: 14 pages, 2 table