Optimal Algebraic Degree Research Articles

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.

Constructing differentially 4-uniform involutions over [formula omitted] by using Carlitz form

Differentially 4-uniform involutions on F22k play important roles in the design of substitution boxes (S-boxes). Despite the active researches on differentially 4-uniform permutation, there is not so much research on differentially 4-uniform involutions, especially over the field F2n with 4|n. In this paper, we introduce a new approach to construct differentially 4-uniform involutions by using Carlitz form. With this approach, we explicitly construct two new classes of differentially 4-uniform involutions over F2n with 4|n. We also show that our constructions have high nonlinearity and optimal algebraic degree. With the help of computer, we show that our constructions are CCZ-inequivalent to the known differentially 4-uniform involutions over F28.

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.

New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field, used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and is 1. The constructed balanced variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of for odd , for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.

A construction of bent functions with optimal algebraic degree and large symmetric group

As maximal, nonlinear Boolean functions, bent functions have many theoretical and practical applications in combinatorics, coding theory, and cryptography. In this paper, we present a construction of bent function \begin{document}$ f_{a,S} $\end{document} with \begin{document}$ n = 2m $\end{document} variables for any nonzero vector \begin{document}$ a\in \mathbb{F}_{2}^{m} $\end{document} and subset \begin{document}$ S $\end{document} of \begin{document}$ \mathbb{F}_{2}^{m} $\end{document} satisfying \begin{document}$ a+S = S $\end{document} . We give a simple expression of the dual bent function of \begin{document}$ f_{a,S} $\end{document} and prove that \begin{document}$ f_{a,S} $\end{document} has optimal algebraic degree \begin{document}$ m $\end{document} if and only if \begin{document}$ |S|\equiv 2 (\bmod 4) $\end{document} . This construction provides a series of bent functions with optimal algebraic degree and large symmetric group if \begin{document}$ a $\end{document} and \begin{document}$ S $\end{document} are chosen properly. We also give some examples of those bent functions \begin{document}$ f_{a,S} $\end{document} and their dual bent functions.

New Constructions of balanced Boolean functions with maximum algebraic immunity, high nonlinearity and optimal algebraic degree

New Constructions of balanced Boolean functions with maximum algebraic immunity, high nonlinearity and optimal algebraic degree

Constructing Two Classes of Boolean Functions With Good Cryptographic Properties

Wu et al. proposed a generalized Tu-Deng conjecture over $\mathbb {F}_{2^{rm}}\times {\mathbb {F}_{2^{m}}}$ , and constructed Boolean functions with good properties. However the proof of the generalized conjecture is still open. Based on Wu’s work and assuming that the conjecture is true, we come up with a new class of balanced Boolean functions which has optimal algebraic degree, high nonlinearity and optimal algebraic immunity. The Boolean function also behaves well against fast algebraic attacks. Meanwhile we construct another class of Boolean functions by concatenation, which is 1-resilient and also has other good cryptographic properties.

Constructions of balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity

Rotation symmetric Boolean functions have been used as components of different cryptosystems. In this paper, two classes of balanced rotation symmetric Boolean functions having optimal algebraic immunity on odd number of variables are constructed. We give a lower bound on the algebraic degree of the first class of functions, and prove that the n-variable functions in the second class has optimal algebraic degree if n≠2m+1 for m>2. Moreover, it is shown that both classes of functions have much better nonlinearity than all the previously obtained rotation symmetric Boolean functions with optimal algebraic immunity, and have good behavior against fast algebraic attacks at least for small numbers of input variables.

Constructions of complete permutation polynomials

Based on the Feistel and MISTY structures, this paper presents several new constructions of complete permutation polynomials (CPPs) of the finite field $${\mathbb {F}}_{2^{n}}^2$$ for a positive integer n and three constructions of CPPs over $${\mathbb {F}}_{p^{n}}^m$$ for any prime p and positive integer $$m\ge 2$$ . In addition, we investigate the upper bound on the algebraic degree of these CPPs and show that some of them can have nearly optimal algebraic degree.

A Construction of 1-Resilient Boolean Functions with Good Cryptographic Properties

This paper proposes a general method to construct 1-resilient Boolean functions by modifying the Tu-Deng and Tang-Carlet-Tang functions. Cryptographic properties such as algebraic degree, nonlinearity and algebraic immunity are also considered. A sufficient condition of the modified functions with optimal algebraic degree in terms of the Siegenthaler bound is proposed. The authors obtain a lower bound on the nonlinearity of the Tang-Carlet-Tang functions, which is slightly better than the known result. If the authors do not break the “continuity” of the support and zero sets, the functions constructed in this paper have suboptimal algebraic immunity. Finally, four specific classes of 1-resilient Boolean functions constructed from this construction and with the mentioned good cryptographic properties are proposed. Experimental results show that there are many 1-resilient Boolean functions have higher nonlinearities than known 1-resilient functions modified by Tu-Deng and Tang-Carlet-Tang functions.

Results on highly nonlinear Boolean functions with provably good immunity to fast algebraic attacks

In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSR-based stream ciphers. Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been researched in a large number of papers. However, there are few results with respect to Boolean functions with provably good immunity against fast algebraic attacks. In previous literatures, only Carlet–Feng function was proven to have good immunity to fast algebraic attacks.In this paper, we first study a large family of highly nonlinear Boolean functions in terms of the immunity to fast algebraic attacks, which includes the functions of Tu–Deng, the functions of Tang et al. and the functions of Jin et al. Based on a sufficient and necessary condition for measuring the immunity of Boolean functions against fast algebraic attacks using bivariate polynomial representation, we propose an efficient method for estimating the immunity of the functions of such family. Then we prove that a family of 2k-variable Boolean functions, including the function recently constructed by Tang et al., are almost perfect algebraic immune for any integer k ≥ 3. More exactly, they achieve optimal algebraic immunity and almost perfect immunity to fast algebraic attacks. The functions of such family are balanced and have optimal algebraic degree. Besides, we prove a lower bound on their nonlinearity based on the work of Tang et al. which is better than that of Carlet–Feng function. It is also checked for 3 ≤ k ≤ 9 that the exact nonlinearity of such functions is very good, which is slightly smaller than that of Carlet–Feng function, and some functions of this family even have a slightly larger nonlinearity than Tang’s et al. function. To sum up, among the known functions with provably good immunity against fast algebraic attacks, the functions of this family make a trade-off between the exact value and the lower bound of nonlinearity.

Constructions of involutions with optimal minimum degree

The minimum degree (resp. algebraic degree) of a Boolean permutation is the minimum (resp. maximum) algebraic degree of all the non-zero linear combinations of its coordinate functions. In this study, the authors concentrate on the design of Boolean permutations with optimal minimum degree. First, they present a novel method for optimising the minimum degrees of known Boolean permutations. Second, they show that the Boolean permutations, which are obtained by optimising Boolean permutations without optimal algebraic degree, have optimal minimum degree. At last, it is shown that their method generates an infinite class of involutions with optimal minimum degree.

Three new infinite families of bent functions

Bent functions are maximally nonlinear Boolean functionswith an even number of variables. They are closely related to someinteresting combinatorial objects and also have importantapplications in coding, cryptography and sequence design. In thispaper, we firstly give a necessary and sufficient condition for a type of Boolean functions, which obtained by adding the product of finitely many linear functions to given bent functions, to be bent.In the case that these known bent functions are chosen to be Kasamifunctions, Gold-like functions and functions with Niho exponents,respectively, three new explicit infinite families of bent functionsare obtained. Computer experiments show that the proposed familesalso contain such bent functions attaining optimal algebraic degree.

Constructions with high algebraic degree of differentially 4-uniform (n, n − 1)-functions and differentially 8-uniform (n, n − 2)-functions

Quadratic differentially 4-uniform (n, n − 1)-functions are given in Carlet J. Adv. Math. Commun. 9(4), 541–565 (2015) where a question is raised of whether non-quadratic differentially 4-uniform (n, n − 1)-functions exist. In this paper, we give highly nonlinear differentially 4-uniform (n, n − 1)-functions of optimal algebraic degree for both n even and odd. Using the approach in Carlet J. Adv. Math. Commun. 9(4), 541–565 (2015), we construct these functions using two APN (n − 1, n − 1)-functions which are EA-equivalent Inverse functions satisfying some necessary and sufficient conditions when n is even. We slightly generalize the approach to construct differentially 4-uniform (n, n − 1)-functions from two differentially 4-uniform (n − 1, n − 1)-functions satisfying some necessary conditions. This allows us to derive the differentially 4-uniform (n, n − 1)-functions $(x,x_{n})\mapsto (x_{n}+1)x^{2^{n}-2}+x_{n} \alpha x^{2^{n}-2}$ , $x \in \mathbb {F}_{2^{n-1}}$ , $x_{n}\in \mathbb {F}_{2}$ , and $\alpha \in \mathbb {F}_{2^{n-1}}\setminus \mathbb {F}_{2}$ , where $Tr_{1}^{n-1}(\alpha )=Tr_{1}^{n-1}(\frac {1}{\alpha })=1$ . These (n, n − 1)-functions are balanced whatever the parity of n is and are then better suited for use as S-boxes in a Feistel cipher. We also give some properties of the Walsh spectrum of these functions to prove that they are CCZ-inequivalent to the differentially 4-uniform (n, n − 1)-functions of the form L ∘ F, where F is a known APN (n, n)-function and L is an affine surjective (n, n − 1)-function. Finally, we also give two new constructions of differentially 8-uniform (n, n − 2)-functions from EA-equivalent Cubic functions and from EA-equivalent Inverse functions.

On the construction of differentially 4-uniform involutions

An involution is a permutation whose compositional inverse is itself. Differentially 4-uniform involutions with high algebraic degree and high nonlinearity are important for the design of block ciphers because they possess good cryptographic properties and the same component can be used in both encryption circuit and decryption circuit. A well known differentially 4-uniform involution is the multiplicative inverse function, which is used as the S-boxes in AES and Camellia. Although many differential 4-uniform permutations have been constructed, there are only a few classes of differentially 4-uniform involutions. Recently, Charpin et al. presented an idea of constructing involutions, which is called piece by piece construction. With this method, we construct a family of differentially 4-uniform involutions with optimal algebraic degree and high nonlinearity. It is also shown with the help of computer that such involutions which are CCZ-inequivalent to the known differentially 4-uniform permutations in small number of even dimensions are constructed. Furthermore, the number of CCZ-inequivalent classes of our involutions over F2n increases exponentially when n increases.

Construction of Highly Nonlinear 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Provably High Fast Algebraic Immunity

In 2013, Tang, Carlet, and Tang [IEEE TIT 59(1): 653–664, 2013] presented two classes of Boolean functions. The functions in the first class are unbalanced and the functions in the second one are balanced. Both of those two classes of functions have high nonlinearity, high algebraic degree, optimal algebraic immunity, and high fast algebraic immunity. However, they are not 1-resilient which represents a drawback for their use as filter functions in stream ciphers. In this paper, we first propose a large family of 1-resilient Boolean functions having high lower bound on nonlinearity, optimal algebraic immunity, and optimal algebraic degree, that is, meeting the Siegenthaler bound. Most notably, we can mathematically prove that every function in $n$ variables belonging to this family has fast algebraic immunity no less than $n-6$ , which is the first time that an infinite family of 1-resilient functions with provably high fast algebraic immunity has been invented. Furthermore, we exhibit a subclass of the family which has higher lower bound on nonlinearity than all the known 1-resilient functions with (potentially) optimal algebraic immunity and potentially high fast algebraic immunity.

A new family of differentially 4-uniform permutations over $$\mathbb{F}_{2^{2k} }$$ for odd k

We study the differential uniformity of a class of permutations over $\mathbb{F}_{2^{n}}$ with $n$ even. These permutations are different from the inverse function as the values $x^{-1}$ are modified to be $(\gamma x)^{-1}$ on some cosets of a fixed subgroup $\langle\gamma\rangle$ of $\mathbb{F}_{2^{n}}^*$. We obtain some sufficient conditions for this kind of permutations to be differentially 4-uniform, which enable us to construct a new family of differentially 4-uniform permutations that contains many new Carlet-Charpin-Zinoviev equivalent (CCZ-equivalent) classes as checked by Magma for small numbers $n$. Moreover, all of the newly constructed functions are proved to possess optimal algebraic degree and relatively high nonlinearity.

New differentially 4-uniform permutations by modifying the inverse function on subfields

Permutations over ??22k$\mathbb {F}_{2^{2k}}$ with low differential uniformity, high algebraic degree and high nonlinearity are of great cryptographic importance since they can be chosen as the substitution boxes (S-boxes) for many block ciphers with SPN (Substitution Permutation Network) structure. A well known example is that the S-box of the famous Advanced Encryption Standard (AES) is derived from the inverse function on ??28$\mathbb {F}_{2^{8}}$, which has been proved to be a differentially 4-uniform permutation with the optimal algebraic degree and known best nonlinearity. Recently, Zha et al. proposed two constructions of differentially 4-uniform permutations over ??22k$\mathbb {F}_{2^{2k}}$, say Gt and Gs, t with Tr(s?1) = 1, by applying affine transformations to the inverse function on some subfields of ??22k$\mathbb {F}_{2^{2k}}$ (Zha et al. Finite Fields Appl. 25, 64---78, 2014). In this paper, we generalize their method by applying other types of EA (extended affine) equivalent transformations to the inverse function on some subfields of ??22k$\mathbb {F}_{2^{2k}}$ and present two new constructions of differentially 4-uniform permutations, say F? and Fs, ? with Tr(s?1) = 1. Furthermore, we prove that all the functions Gt with different t are CCZ (Carlet-Charpin-Zinoviev) equivalent to our subclass F0, while all the functions Gs, t with different t are CCZ-equivalent to our subclass Fs,0. In addition, both our two constructions give many new CCZ-inequivalent classes of such functions, as checked by computer in small numbers of variables. Moreover, all these newly constructed permutations are proved to have the optimal algebraic degree and high nonlinearity.

A CLASS OF 1-RESILIENT BOOLEAN FUNCTIONS WITH OPTIMAL ALGEBRAIC IMMUNITY AND GOOD BEHAVIOR AGAINST FAST ALGEBRAIC ATTACKS

Recently, Tang, Carlet and Tang presented a combinatorial conjecture about binary strings, allowing proving that all balanced functions in some infinite class they introduced have optimal algebraic immunity. Later, Cohen and Flori completely proved that the conjecture is true. These functions have good (provable or at least observable) cryptographic properties but they are not 1-resilient, which represents a drawback for their use as filter functions in stream ciphers. We propose a construction of an infinite class of 1-resilient Boolean functions with optimal algebraic immunity by modifying the functions in this class. The constructed functions have optimal algebraic degree, that is, meet the Siegenthaler bound, and high nonlinearity. We prove a lower bound on their nonlinearity, but as for the Carlet-Feng functions and for the functions mentioned above, this bound is not enough for ensuring a nonlinearity sufficient 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. Our computations show that the functions in this class have very good nonlinearity and also good immunity to fast algebraic attacks. This is the first time that an infinite class of functions gathers all of the main criteria allowing these functions to be used as filters in stream ciphers.

Generating highly nonlinear resilient Boolean functions resistance against algebraic and fast algebraic attacks

ABSTRACTBoolean functions play an important role in the design of stream ciphers. In this paper, a simulated annealing algorithm is designed to obtain Boolean functions satisfying all the needed criteria: high nonlinearity, 1‐resiliency, optimal algebraic degree, optimal (or suboptimal) algebraic immunity, and good immunity to fast algebraic attacks. These functions provide a good trade‐off among the criteria to resist the known cryptanalytic techniques. Copyright © 2014 John Wiley & Sons, Ltd.