Symmetric Boolean Functions Research Articles

Equivalence classes for cubic rotation symmetric functions

Rotation symmetric Boolean functions, that is, Boolean functions which are invariant under any cyclic permutation of the variables, have been extensively studied in the last dozen years or so due to their importance in cryptography and coding theory. Little was known about the basic question of when two such functions are affine equivalent until very recently. The case of affine equivalence of quadratic rotation symmetric functions was solved in 2009 by Kim, Park and Hahn. In 2010, Cusick introduced the concept of patterns to study the more complicated case of cubic functions under permutations. This method made possible a detailed analysis of some special classes of cubic rotation symmetric functions in n variables where n has a special form, for example, n equal to a prime power. Here the affine equivalence classes under permutations in n variables for any n are examined. In particular, all possible sizes of these classes are identified and an exact count of the functions in the largest classes is given. It is conjectured that the classes of the largest size contain a positive proportion of all of the distinct functions in n variables.

Finding Hamming weights without looking at truth tables

This paper studies degree 3 Boolean functions in n variables x1, ..., xn which are rotation symmetric, that is, invariant under any cyclic shift of the indices of the variables. These rotation symmetric functions have been extensively studied in the last dozen years or so because of their importance in cryptography. We start from the 2012 paper of Bileschi, Cusick and Padgett, which gave an algorithm for finding a recursion for the truth table of any n-variable cubic rotation symmetric Boolean function generated by a monomial, as well as a homogeneous recursion for its (Hamming) weight as n increases. This greatly reduced the computational complexity of computing the weights of such functions for large n, but it was still necessary to calculate the truth tables of the functions for the values of n needed to give the initial conditions for the recursion. This computation could be infeasible if the recursion order is large, since the truth tables have 2n entries. The present paper shows how to use the roots of the characteristic polynomial of the recursion to find the initial conditions without looking at any truth tables, given the mild and plausible assumption that these roots are distinct. This results in a huge decrease in the computational complexity (including the time needed to find the roots) to something linear in n, apart from logarithmic factors.

On $2k$-Variable Symmetric Boolean Functions With Maximum Algebraic Immunity $k$

Given a positive even integer n, it is found that the weight distribution of any n-variable symmetric Boolean function with maximum algebraic immunity (AI) n/2 is determined by the binary expansion of n . Based on the foregoing, all n-variable symmetric Boolean functions with maximum AI are constructed. The amount is (2 wt(n)+1)2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[log</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n]</sup> .

Construction of rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity

Recent research shows that the class of rotation symmetric Boolean functions is potentially rich in functions of cryptographic significance. In this paper, based on the knowledge of compositions of an integer, we present two new kinds of construction of rotation symmetric Boolean functions having optimal algebraic immunity on either odd variables or even variables. Our new functions are of much better nonlinearity than all the existing theoretical constructions of rotation symmetric Boolean functions with optimal algebraic immunity. Further, the algebraic degree of our rotation symmetric Boolean functions are also high enough.

Optimal Function Computation in Directed and Undirected Graphs

We consider the problem of information aggregation in sensor networks, where one is interested in computing a function of the sensor measurements. We allow for block processing and study in-network function computation in directed graphs and undirected graphs. We study how the structure of the function affects the encoding strategies and the effect of interactive information exchange. Depending on the application, there could be a designated collector node, or every node might want to compute the function. We begin by considering a directed graph C = (γ. ε) on the sensor nodes, where the goal is to determine the optimal encoders on each edge which achieve function computation at the collector node. Our goal is to characterize the rate region in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|ε|</sup> , i.e., the set of points for which there exist feasible encoders with given rates which achieve zero-error computation for asymptotically large block length. We determine the solution for directed trees, specifying the optimal encoder and decoder for each edge. For general directed acyclic graphs, we provide an outer bound on the rate region by finding the disambiguation requirements for each cut, and describe examples where this outer bound is tight. Next, we address the scenario where nodes are connected in an undirected tree network, and every node wishes to compute a given symmetric Boolean function of the sensor data. Undirected edges permit interactive computation, and we therefore study the effect of interaction on the aggregation and communication strategies. We focus on sum-threshold functions and determine the minimum worst case total number of bits to be exchanged on each edge. The optimal strategy involves recursive in-network aggregation which is reminiscent of message passing. In the case of general graphs, we present a cut-set lower bound and an achievable scheme based on aggregation along trees. For complete graphs, we prove that the complexity of this scheme is no more than twice that of the optimal scheme.

Weight and nonlinearity of Boolean functions

In this paper we analyze the weight and the nonlinearity of various types of Boolean functions. We give some general results related to rotation symmetric Boolean functions, and in particular, we prove partially a conjecture stated by Cusick and Stanica in [3].

Constructing Rotation Symmetric Boolean Functions with Maximum Algebraic Immunity on an Odd Number of Variables

Constructing Rotation Symmetric Boolean Functions with Maximum Algebraic Immunity on an Odd Number of Variables

Weights of Boolean cubic monomial rotation symmetric functions

This paper studies degree 3 Boolean functions in n variables x 1, ..., x n which are rotation symmetric, that is, invariant under any cyclic shift of the indices of the variables. These rotation symmetric functions have been extensively studied in the last dozen years or so because of their importance in cryptography. Some of the cryptographic applications are described in a 2002 paper of Cusick and St?nic?, which gave a recursion for the truth table and a nonhomogeneous recursion for the (Hamming) weight of the homogeneous cubic rotation symmetric function generated by the monomial x 1 x 2 x 3. Until now, this was the only investigation of the recursive structure of such functions. Here we provide an algorithm for finding a recursion for the truth table of any cubic rotation symmetric Boolean function generated by a monomial, as well as a homogeneous recursion for its weight as n increases; in doing so we greatly reduce the computational complexity of a problem that appeared to be exponential in the number of variables, as well as provide a new way of studying the structure of the functions. The method makes some computations practically accessible that were previously entirely unfeasible. Once the weights have been computed for the initial small values of n, the further weights can be computed from the recursion, without looking at the truth table at all.

Affine equivalence for rotation symmetric Boolean functions with [formula omitted] variables

Rotation symmetric Boolean functions have been extensively studied in the last dozen years or so because of their importance in cryptography and coding theory. Until recently, very little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in a 2009 paper of Kim, Park and Hahn. The much more complicated analogous problem for cubic functions was solved for permutations using a new concept of patterns in a 2010 paper of Cusick, and it is conjectured that, as in the quadratic case, this solution actually applies for all affine transformations. The patterns method enables a detailed analysis of the affine equivalence classes for various special classes of cubic rotation symmetric functions in n variables. Here the case of functions generated by a single monomial and having pk variables, where p>3 is prime, is examined in detail, and in particular, a formula for the number of classes is proved.

Construction of even-variable rotation symmetric Boolean functions with maximum algebraic immunity

Rotation symmetric Boolean functions (RSBFs) have been used as components of different cryptosystems. In this paper, we investigate n-variable (n even and n ⩾ 12) RSBFs to achieve maximum algebraic immunity (AI), and provide a construction of RSBFs with maximum AI and nonlinearity. These functions have higher nonlinearity than the previously known nonlinearity of RSBFs with maximum AI. We also prove that our construction provides high algebraic degree in some case.

A recursive formula for weights of Boolean rotation symmetric functions

For the last dozen years or so, there has been much research on the applications of rotation symmetric Boolean functions with n variables in cryptography. In particular, the Hamming weights of these functions have been studied, because knowledge of these weights is important if the functions are to be useful in cryptography. Only in 2009, in a paper by Kim et al., there was a closed formula for the weights as a function of n obtained for some of these functions in the simplest case of quadratic functions. In this paper, we present a method for recursively computing the weights of certain kinds of rotation symmetric Boolean functions with arbitrary degree. Using some recent work of Cusick on the affine equivalence classes of certain cubic rotation symmetric functions, we obtain some detailed information on relationships between the weights of some of these cubic functions as n increases. This leads to some very specific information about previously unsuspected connections between the truth tables of various cubic rotation symmetric functions.

On Symmetric Boolean Functions With High Algebraic Immunity on Even Number of Variables

In this paper, we put forward an efficient method to study the symmetric Boolean functions with high algebraic immunity on even number of variables. We obtain some powerful necessary conditions for symmetric Boolean functions to achieve high algebraic immunity by studying the weight support of some specific types of Boolean functions of low degrees. With these results, we prove that the algebraic immunity of a large class of symmetric correlation immune Boolean functions, namely the symmetric palindromic functions, is not high. Besides, we construct all symmetric Boolean functions with maximum algebraic immunity and give a description for those with submaximum algebraic immunity. We also determine the Hamming weight, degrees and nonlinearity of the symmetric Boolean functions with maximum algebraic immunity.

New constructions of even-variable rotation symmetric Boolean functions with maximum algebraic immunity

Rotation symmetric Boolean functions (RSBFs), which are invariant under circular translations, have been used as components of some different cryptosystems. In this paper, we study the constructions of even-variable RSBFs with maximum algebraic immunity (AI). First, we present a new construction class of even-variable RSBFs with maximum AI, which has totally n / 2 − 1 different constructions. And then in order to get even-variable RSBFs which achieve both maximum AI and high nonlinearity, an improved construction is proposed.

Affine equivalence for rotation symmetric Boolean functions with 2 k variables

Rotation symmetric Boolean functions have been extensively studied in the last 10 years or so because of their importance in cryptography and coding theory. Until recently, very little was known about the basic question of when two such functions are affine equivalent. Even the case of quadratic functions is nontrivial, and this was only completely settled in a 2009 paper of Kim, Park and Hahn. The much more complicated case of cubic functions was solved for permutations using a new concept of patterns in a 2010 paper of Cusick, and it is conjectured that, as in the quadratic case, this solution actually applies for all affine transformations. The patterns method enables a detailed analysis of the affine equivalence classes for various special classes of cubic rotation symmetric functions in n variables. Here the case of functions with 2 k variables (this number is especially relevant in computer applications) and generated by a single monomial is examined in detail, and in particular a formula for the number of classes is proved.

Affine equivalence of cubic homogeneous rotation symmetric functions

Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009. This paper studies the much more complicated cubic case for such functions. A new concept of patterns is introduced, by means of which the structure of the smallest group G n , whose action on the set of all such cubic functions in n variables gives the affine equivalence classes for these functions under permutation of the variables, is determined. We conjecture that the equivalence classes are the same if all nonsingular affine transformations, not just permutations, are allowed. Our method gives much more information about the equivalence classes; for example, in this paper we give a complete description of the equivalence classes when n is a prime or a power of 3.

The Degree of Balanced Elementary Symmetric Boolean Functions of &lt;formula formulatype="inline"&gt; &lt;tex Notation="TeX"&gt;${{\bf 4k}+{\bf 3}}$&lt;/tex&gt; &lt;/formula&gt; Variables

In this paper, we consider the conjecture that σ2t+1l-1,2t are the only nonlinear balanced elementary symmetric Boolean functions where t and l are positive integers. We prove if n=2t+1l-1, l odd and 2t+1\nmid d, σn,d is balanced if and only if d=2k, 1 ≤ k ≤ t. Our results verify most cases of the conjecture for n ≡ 3 (mod 4) .

Balanced [formula omitted]-variable rotation symmetric Boolean functions with maximum algebraic immunity

In this paper, we study the construction of Rotation Symmetric Boolean Functions (RSBFs) which achieve a maximum algebraic immunity (AI). For the first time, a construction of balanced 2 p -variable ( p is an odd prime) RSBFs with maximum AI was provided, and the nonlinearity of the constructed RSBFs is not less than 2 2 p − 1 − ( 2 p − 1 p ) + ( p − 2 ) ( p − 3 ) + 2 ; this nonlinearity result is significantly higher than the previously best known nonlinearity of RSBFs with maximum AI.

Cryptographic Properties of Several Classes of Rotation Symmetric Boolean Functions

Sumanta Sarkar,et al.give a class of rotation symmetric Boolean functions with maximum algebraic immunity,but only consider the nonlinearity of the functions and did not study other cryptographic properties.In this paper,other cryptographic properties of the class of Boolean functions are studied,such as,algebraic degree,linear structure,propagation,correlation immunity etc.The results,unfortunately,show that their other cryptographic properties are not good even though their algebraic immunity is optimum.Hence,the class of Boolean functions cannot be applied in cryptography.

Linear Recurrences and Asymptotic Behavior of Exponential Sums of Symmetric Boolean Functions

In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In particular, when the degree of the symmetric function is a power of two, then the exponential sum is much smaller than $2^n$.

Some results on the algebraic immunity of Boolean functions

From the motivation of algebraic attacks on stream and block ciphers, the concept of algebraic immunity (AI) of a Boolean function was introduced and studied extensively. High algebraic immunity is a necessary condition for resisting algebraic attacks. In this paper, we give some lower bounds on the algebraic immunity of Boolean functions. The results are applied to give lower bounds on the AI of symmetric Boolean functions and rotation symmetric Boolean functions. Some balanced rotation symmetric Boolean functions with their AI near the maximum possible value ( n/2 (are constructed.