Highly Nonlinear Boolean Functions with Optimal Algebraic Immunity and Good Behavior Against Fast Algebraic Attacks

In this paper, we present a new combinatorial conjecture about binary strings. Based on the new conjecture, two classes of Boolean functions of $2k$ variables with optimal algebraic immunity are proposed, where $k\ge 2$. The first class contains unbalanced functions having high algebraic degree and nonlinearity. The functions in the second one are balanced and have maximal algebraic degree and high nonlinearity. It is checked that, at least for small numbers of variables, both classes of functions have a good behavior against fast algebraic attacks. Compared with the known Boolean functions resisting algebraic attacks and fast algebraic attacks, the two classes of functions possess the highest lower bounds on nonlinearity. These bounds are however not enough for ensuring a sufficient nonlinearity for allowing resistance to the fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for small numbers of variables. Moreover, these values are very good and much better than for the previously found functions having all the necessary features for being used in the filter model of pseudo-random generators

Algorithm 959: VBF: A Library of C plus plus Classes for Vector Boolean Functions in Cryptography

VBF is a collection of C++ classes designed for analyzing vector Boolean functions (functions that map a Boolean vector to another Boolean vector) from a cryptographic perspective. This implementation uses the NTL library from Victor Shoup, adding new modules that call NTL functions and complement the existing ones, making it better suited to cryptography. The class representing a vector Boolean function can be initialized by several alternative types of data structures such as Truth Table, Trace Representation, and Algebraic Normal Form (ANF), among others. The most relevant cryptographic criteria for both block and stream ciphers as well as for hash functions can be evaluated with VBF: it obtains the nonlinearity, linearity distance, algebraic degree, linear structures, and frequency distribution of the absolute values of the Walsh Spectrum or the Autocorrelation Spectrum, among others. In addition, operations such as equality testing, composition, inversion, sum, direct sum, bricklayering (parallel application of vector Boolean functions as employed in Rijndael cipher), and adding coordinate functions of two vector Boolean functions are presented. Finally, three real applications of the library are described: the first one analyzes the KASUMI block cipher, the second one analyzes the Mini-AES cipher, and the third one finds Boolean functions with very high nonlinearity, a key property for robustness against linear attacks

1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity

Several factors (e.g., balancedness, good correlation immunity) are considered as important properties of Boolean functions for using in cryptographic primitives. A Boolean function is perfect algebraic immune if it is with perfect immunity against algebraic and fast algebraic attacks. There is an increasing interest in construction of Boolean function that is perfect algebraic immune combined with other characteristics, like resiliency. A resilient function is a balanced correlation-immune function. This paper uses bivariate representation of Boolean function and theory of finite field to construct a generalized and new class of Boolean functions on even variables by extending the Carlet-Feng functions. We show that the functions generated by this construction support cryptographic properties of 1-resiliency and (sub)optimal algebraic immunity and further propose the sufficient condition of achieving optimal algebraic immunity. Compared experimentally with Carlet-Feng functions and the functions constructed by the method of first-order concatenation existing in the literature on even (from 6 to 16) variables, these functions have better immunity against fast algebraic attacks. Implementation results also show that they are almost perfect algebraic immune functions

Fast Algebraic Immunity of 2m+22^m+2 & 2m+32^m+3 variables Majority Function

Boolean functions used in some cryptosystems of stream ciphers should satisfy various criteria simultaneously to resist some known attacks. The fast algebraic attack (FAA) is feasible if one can find a nonzero function $g$ of low algebraic degree and a function $h$ of algebraic degree significantly lower than $n$ such that $f\cdot g=h$. Then one new cryptographic property fast algebraic immunity was proposed, which measures the ability of Boolean functions to resist FAAs. It is a great challenge to determine the exact values of the fast algebraic immunity of an infinite class of Boolean functions with optimal algebraic immunity. In this letter, we explore the exact fast algebraic immunity of two subclasses of the majority function