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

Fast algebraic immunity of Boolean functions and LCD codes

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure the resistance of a Boolean function to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. The binary LCD codes presented in this paper have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems

Balanced Boolean Functions with Optimum Algebraic Immunity and High Nonlinearity

In this paper, three constructions of balanced Boolean functions with optimum algebraic immunity are proposed. The cryptographical properties such as algebraic degree and nonlinearity of the constructed functions are also analyzed

A First Order Recursive Construction of Boolean Function with Optimum Algebraic Immunity

This paper proposed a first order recursive construction of Boolean function with optimum algebraic immunity. We also show that the Boolean functions are balanced and have good algebraic degrees

Constructions of Even-variable Boolean Function with Optimum Algebraic Immunity

This paper proposed an improved construction of even-variable Boolean function with optimum algebraic immunity. Compared with those in~\cite{Carl06}, our Boolean functions are more balance. Specially, for $k{=}2t{+}1$ $(t{>}1)$, the $2k$-variables Boolean function is balanced. Furthermore, we generalized it to a class of constructions, meaning there would be much more constructions

A general conjecture similar to T-D conjecture and its applications in constructing Boolean functions with optimal algebraic immunity

In this paper, we propose two classes of 2k-variable Boolean functions, which have optimal algebraic immunity under the assumption that a general combinatorial conjecture is correct. These functions also have high algebraic degree and high nonlinearity. One class contain more bent functions, and the other class are balanced