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 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

On the algebraic immunity of weightwise perfectly balanced functions

In this article we study the Algebraic Immunity (AI) of Weightwise Perfectly Balanced (WPB) functions. After showing a lower bound on the AI of two classes of WPB functions from the previous literature, we prove that the minimal AI of a WPB $n$-variables function is constant, equal to $2$ for $n\ge 4$ . Then, we compute the distribution of the AI of WPB function in $4$ variables, and estimate the one in $8$ and $16$ variables. For these values of $n$ we observe that a large majority of WPB functions have optimal AI, and that we could not obtain an AI-$2$ WPB function by sampling at random. Finally, we address the problem of constructing WPB functions with bounded algebraic immunity, exploiting a construction from 2022 by Gini and Méaux. In particular, we present a method to generate multiple WPB functions with minimal AI, and we prove that the WPB functions with high nonlinearity exhibited by Gini and Méaux also have minimal AI. We conclude with a construction giving WPB functions with lower bounded AI, and give as example a family with all elements with AI at least $n/2-\log(n)+1$

Affine-Power S-Boxes over Galois Fields with Area-Optimized Logic Implementations

Cryptographic S-boxes are fundamental in key-iterated sub- stitution permutation network (SPN) designs for block ciphers. As a natural way for realizing Shannon’s confusion and diffusion properties in cryptographic primitives through nonlinear and linear behavior, re- spectively, SPN designs served as the basis for the Advanced Encryption Standard and a variety of other block ciphers. In this work we present a methodology for minimizing the logic resources for n-bit affine-power S- boxes over Galois fields based on measurable security properties and find- ing corresponding area-efficient combinational implementations in hard- ware. Motivated by the potential need for new and larger S-boxes, we use our methodology to find area-optimized circuits for 8- and 16-bit S-boxes. Our methodology is capable of finding good upper bounds on the number of XOR and AND gate equivalents needed for these circuits, which can be further optimized using modern CAD tools

A Conjecture on Binary String and Its Applications on Constructing Boolean Functions of Optimal Algebraic Immunity

In this paper, we propose a combinatoric conjecture on binary string, on the premise that our conjecture is correct we mainly obtain two classes of functions which are both algebraic immunity optimal: the first class of functions are also bent, moreover, from this fact we conclude that the algebraic immunity of bent functions can take all possible values except one. The second class are balanced functions, which have optimal algebraic degree and the best nonlinearity up to now