Rotation Symmetric Boolean Functions Research Articles

Constructions of 2-resilient rotation symmetric Boolean functions with odd number of variables

In this paper, a new method for constructing 2-resilient rotation symmetric Boolean functions with odd number of variables is presented. Based on an equivalent characterization of this class of functions and the relation between the orbit matrices and their complements, a system of equations about 2-tuples distribution matrix is established. Then the constructions of some 2-resilient rotation symmetric Boolean functions with odd number of variables can be converted into the solutions of the system of equations. We also give a sufficient condition for the constructed functions to have the maximum algebraic degree. Moreover, we provide a lower bound on the number of nonlinear 2-resilient rotation symmetric Boolean functions with odd number of variables. Particularly, the lower bound is 152 for seven number of variables.

Constructions of 2-correlation immune Rotation Symmetric Boolean functions

Constructions of 2-correlation immune Rotation Symmetric Boolean functions

Short [formula omitted]-rotation symmetric Boolean functions

Rotation symmetric Boolean functions were introduced by Pieprzyk and Qu in the late 1990’s. They showed that these functions are useful, among other things, in the design of fast hashing algorithms with strong cryptographic properties. The concept of rotation symmetric Boolean functions has been generalized to a class of functions known as k-rotation Boolean functions, where k divides n and n is the number of variables of the Boolean function. Analogous to the case of regular rotation symmetric Boolean functions, a monomial k-rotation Boolean function is called long k-cycle if the number of terms coincides with n/k and short k-cycle if the number of terms is less than n/k. In this work we characterize short k-cycles by providing specific generators for them. We also provide a count for the number of short k-cycles in n variables for a fixed degree and for a specified length.

A New Characterization of 2-Resilient Rotation Symmetric Boolean Functions

A New Characterization of 2-Resilient Rotation Symmetric Boolean Functions

Balanced Even-Variable Rotation Symmetric Boolean Functions with Optimal Algebraic Immunity, Maximum Algebraic Degree and Higher Nonlinearity

Rotation symmetric Boolean functions are good candidates for stream ciphers because they have such advantages as simple structure, high operational speed and low implement cost. Recently, Mesnager et al. proposed for the first time an efficient method to construct balanced rotation symmetric Boolean functions with optimal algebraic immunity and good nonlinearity for an arbitrary even number of variables. However, the algebraic degree of their constructed [Formula: see text]-variable ([Formula: see text]) function is always less than the maximum value [Formula: see text]. In this paper, by modifying the support of Boolean functions from Mesnager et al.’s construction, we present two new constructions of balanced even-variable rotation symmetric Boolean functions with optimal algebraic immunity as well as higher algebraic degree and nonlinearity. The algebraic degree of Boolean functions in the first construction reaches the maximum value [Formula: see text] if [Formula: see text] is odd and [Formula: see text] or [Formula: see text] for integer [Formula: see text], while that of the second construction reaches the maximum value for all [Formula: see text]. Moreover, the nonlinearities of Boolean functions in both two constructions are higher than that of Mesnager et al.’s construction.

Balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and higher nonlinearity

By modifying the support of the majority function, we propose a construction of balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity. For n ≥ 17 , the nonlinearity of these constructed functions is the highest among all known rotation symmetric Boolean functions with optimal algebraic immunity. The algebraic degree can reach the maximum for some special numbers of variables. Moreover, simulations show that such functions have good fast algebraic immunity for some small numbers of variables.

Recursions for modified Walsh transforms of some families of Boolean functions

We show that, under certain conditions, restricted and biased exponential sums and Walsh transforms of symmetric and rotation symmetric Boolean functions are, as in the case of nonbiased domain, C-finite sequences. We also prove that under other conditions, these sequences are P-finite, which is a somewhat different behavior than their nonbiased counterparts. We further show that exponential sums and Walsh transforms of a family of rotation symmetric monomials over the restricted domain En,j={x∈𝔽2n:wt (x)=j} (wt (x) is the weight of the vector x) are given by polynomials of degree at most j, and so, they are also C-finite sequences. Finally, we also present a study of the behavior of symmetric Boolean functions under these biased transforms.

The linear structures and fast points of rotation symmetric Boolean functions

The linear structures and fast points of rotation symmetric Boolean functions

Symbolic dynamics and rotation symmetric Boolean functions

We identify the weights wt(fn) of a family {fn} of rotation symmetric Boolean functions with the cardinalities of the sets of n-periodic points of a finite-type shift, recovering the second author’s result that said weights satisfy a linear recurrence. Similarly, the weights of idempotent functions fn defined on finite fields can be recovered as the cardinalities of curves over those fields and hence satisfy a linear recurrence as a consequence of the rationality of curves’ zeta functions. Weil’s Riemann hypothesis for curves then provides additional information about wt(fn). We apply our results to the case of quadratic functions and considerably extend the results in an earlier paper of ours.

Constructions of 2-resilient rotation symmetric Boolean functions through symbol transformations of cyclic Hadamard matrix

In this paper, the properties of symbol transformations of cyclic Hadamard matrices are studied. An infinite class of (n−1)-variable 2-resilient rotation symmetric Boolean functions are constructed, and the nonlinearity of the constructed functions is 2n(n−1). The crucial technique of this method is to determine a subset T⊆F2n−1 satisfying a correspondent condition. This is a new construction of 2-resilient rotation symmetric Boolean functions via switching the supports of (n−1)-variable rotation symmetric Boolean functions of degree one, i.e., f0n−1(x1,x2,⋯,xn−1)=⊕i=1n−1xi, where n=4t.

Searching for Cryptographically Significant Rotation Symmetric Boolean Functions by Designing Heuristic Algorithms

It has been proved that the set of rotation symmetric Boolean functions (RSBFs) is abundant in cryptographically strong functions with multiple criteria. In this study, we design two genetic algorithms and apply them to search for balanced RSBFs with high nonlinearity. The experimental results show that our methods can generate cryptographically strong Boolean functions with high nonlinearity, 1-resilient functions, and optimal algebraic immunity. It shows that these functions have superiority from the view point of practical application in cryptosystems compared with known ones which are obtained by other heuristics.

A Novel Construction of 2-Resilient Rotation Symmetric Boolean Functions

A Novel Construction of 2-Resilient Rotation Symmetric Boolean Functions

Several classes of even-variable 1-resilient rotation symmetric Boolean functions with high algebraic degree and nonlinearity

Recent research shows that the class of rotation symmetric Boolean functions is potentially rich in functions of cryptographic significance. In this paper, some classes of 2m-variable (m is an odd integer) 1-resilient rotation symmetric Boolean functions are got, whose nonlinearity and algebraic degree are studied. For the first time, we obtain 2m-variable 1-resilient rotation symmetric Boolean functions having high nonlinearity and optimal algebraic degree. In addition, we obtain a class of non-linear rotation symmetric 1-resilient function for every n≥5, and a class of quadratic rotation symmetric (k−1)-resilient function of n=3k variables, where k is an integer.

A new construction of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and higher nonlinearity

Rotation symmetric Boolean functions are potentially rich in functions of cryptographic significance. In this paper, a new construction of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity is presented. By a direct calculation, the nonlinearity of the newly constructed functions is higher than the nonlinearities of all the known odd-variable rotation symmetric Boolean functions with optimal algebraic immunity. The algebraic degree and the fast algebraic immunity of our functions are also considered.

Balanced Rotation Symmetric Boolean Functions With Good Autocorrelation Properties

Rotation symmetric Boolean functions (RSBFs) are used widely in symmetric cryptography. In this paper, a systematic construction of balanced odd-variable RSBFs satisfying strict avalanche criterion is proposed. Hence, a class of $(6k+3)$ -variable resilient RSBFs satisfying strict avalanche criterion is also presented for any $k\geq 2$ . Some of the obtained RSBFs have many other good cryptographic properties at the same time, that is, optimal algebraic degree, good global avalanche characteristics, high nonlinearity and nonexistence of nonzero linear structures. Moreover, we obtain some count results of RSBFs satisfying strict avalanche criterion. This is the first time that the autocorrelation properties of RSBFs are investigated systemically.

A new construction of odd-variable rotation symmetric boolean functions with good cryptographic properties

<p style='text-indent:20px;'>Rotation symmetric Boolean functions constitute a class of cryptographically significant Boolean functions. In this paper, based on the theory of ordered integer partitions, we present a new class of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity by modifying the support of the majority function. Compared with the existing rotation symmetric Boolean functions on odd variables, the newly constructed functions have the highest nonlinearity.</p>

A construction method of balanced rotation symmetric Boolean functions on arbitrary even number of variables with optimal algebraic immunity

Rotation symmetric Boolean functions incorporate a super-class of symmetric functions which represent an attractive corpus for computer investigation. These functions have been investigated from the viewpoints of bentness and correlation immunity and have also played a role in the study of nonlinearity. In the literature, many constructions of balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity have been derived. While it seems that the construction of balanced even-variable rotation symmetric Boolean functions with optimal algebraic immunity is very hard work to breakthrough. In this paper, we present for the first time a construction of balanced rotation symmetric Boolean functions on an arbitrary even number of variables with optimal algebraic immunity by modifying the support of the majority function. The nonlinearity of the newly constructed rotation symmetric Boolean functions is also derived.

Affine equivalence for quadratic rotation symmetric Boolean functions

Let $$f_n(x_0, x_1, \ldots , x_{n-1})$$ denote the algebraic normal form (polynomial form) of a rotation symmetric (RS) Boolean function of degree d in $$n \ge d$$ variables and let $$wt(f_n)$$ denote the Hamming weight of this function. Let $$(0, a_1, \ldots , a_{d-1})_n$$ denote the function $$f_n$$ of degree d in n variables generated by the monomial $$x_0x_{a_1} \ldots x_{a_{d-1}}.$$ Such a function $$f_n$$ is called monomial rotation symmetric (MRS). It was proved in a 2012 paper that for any MRS $$f_n$$ with $$d=3,$$ the sequence of weights $$\{w_k = wt(f_k):~k = 3, 4, \ldots \}$$ satisfies a homogeneous linear recursion with integer coefficients. This result was gradually generalized in the following years, culminating around 2016 with the proof that such recursions exist for any rotation symmetric function $$f_n.$$ Recursions for quadratic RS functions were not explicitly considered, since a 2009 paper had already shown that the quadratic weights themselves could be given by an explicit formula. However, this formula is not easy to compute for a typical quadratic function. This paper shows that the weight recursions for the quadratic RS functions have an interesting special form which can be exploited to solve various problems about these functions, for example, deciding exactly which quadratic RS functions are balanced.

The nonlinearity and Hamming weights of rotation symmetric Boolean functions of small degree

Let $ e $, $ l $ and $ n $ be integers such that $ 1\le e \lt n $ and $ 3\le l\le n $. Let $ \left\langle{i}\right\rangle $ denote the least nonnegative residue of $ i \mod n $. In this paper, we investigate the following Boolean function $ F_{l, e}^n(x^n) = \sum\limits_{i = 0}^{n-1}x_{i} x_{\left\langle{i+e}\right\rangle}x_{\left\langle{i+2e}\right\rangle}...x_{\left\langle{i+(l-1)e}\right\rangle}, $ which plays an important role in cryptography and coding theory. We introduce some new sub-functions and provide some recursive formulas for the Fourier transform. Using these recursive formulas, we show that the nonlinearity of $ F_{l, e}^n(x^n) $ is the same as its weight for $ 5\leq l\leq 7 $. Our result confirms partially a conjecture of Yang, Wu and Hong raised in 2013. It also gives a partial answer to a conjecture of Castro, Medina and Stănică proposed in 2018. Our result extends the result of Zhang, Guo, Feng and Li for the case $ l = 3 $ and that of Yang, Wu and Hong for the case $ l = 4 $.

Constructing odd-variable RSBFs with optimal algebraic immunity, good nonlinearity and good behavior against fast algebraic attacks

Rotation symmetric Boolean functions have raised widespread attention because of the good properties in cryptosystem. This paper presents a new construction of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity. The nonlinearity of the new functions is higher than some existing theoretical constructions of rotation symmetric Boolean functions with optimal algebraic immunity. Further, it is also checked that such functions have almost optimal immunity against fast algebraic attacks for small variables. Besides, the algebraic degree of the constructed functions is also high enough.