Constructing Boolean functions with potentially optimal algebraic immunity based on additive decompositions of finite fields (extended abstract)

After the improvement by Courtois and Meier of the algebraic attacks on stream ciphers and the introduction of the related notion of algebraic immunity, several constructions of infinite classes of Boolean functions with optimum algebraic immunity have been proposed. All of them gave functions whose algebraic degrees are high enough for resisting the Berlekamp-Massey attack and the recent Ronjom-Helleseth attack, but whose nonlinearities either achieve the worst possible value (given by Lobanov's bound) or are slightly superior to it. Hence, these functions do not allow resistance to fast correlation attacks. Moreover, they do not behave well with respect to fast algebraic attacks. In this paper, we study an infinite class of functions which achieve an optimum algebraic immunity. We prove that they have an optimum algebraic degree and a much better nonlinearity than all the previously obtained infinite classes of functions. We check that, at least for small values of the number of variables, the functions of this class have in fact a very good nonlinearity and also a good behavior against fast algebraic attacks.

Results on highly nonlinear Boolean functions with provably good immunity to 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.

Balanced [formula omitted]-variable rotation symmetric Boolean functions with optimal algebraic immunity, good nonlinearity, and good algebraic degree

Boolean functions with maximum algebraic immunity have been considered as one class of cryptographically significant functions. It is known that Boolean functions on odd variables have maximum algebraic immunity if and only if a correlative matrix has column full rank, and Boolean functions on even variables have maximum algebraic immunity if and only if two correlative matrices have column full rank. Recently, a smaller matrix was used in the odd case. We find that one or two smaller matrices can be used in the even case and consequently present several sufficient and necessary conditions for Boolean functions with maximum algebraic immunity. This result advances the ability to identify whether Boolean functions on even variables achieve maximum algebraic immunity. We also present a construction algorithm for n-variable Boolean functions with maximum algebraic immunity, specially with the Hamming weights of \( \sum {_{i = 0}^{\left\lceil {\frac{n} {2}} \right\rceil - 1} } \left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right) \). It is easily realized for not too large n and helps construct balanced Boolean functions with maximum algebraic immunity on even variables. Furthermore, we present a sufficient and necessary condition for balanced Boolean functions to achieve maximum algebraic immunity and optimum algebraic degree, and modify the construction algorithm to construct Boolean functions on odd variables with maximum algebraic immunity, optimum algebraic degree and high nonlinearity.

In this paper, we study the algebraic immunity of Boolean functions and consider in particular the problem of constructing Boolean functions with a good algebraic immunity. We first give heuristic arguments which seem to indicate that the algebraic immunity of a random Boolean function on n variables is at least lfloorn/2rfloor with a very high probability (while the upper bound is lceiln/2rceil, the "ceiling" of n/2). We give an upper bound, under a reasonable assumption, on the algebraic immunity of Boolean functions constructed through Maiorana-MacFarland construction. At last we give examples of balanced functions with optimal algebraic immunity and a good nonlinearity and of balanced functions with a good algebraic immunity, a good nonlinearity and a good correlation immunity, which can be used for cryptographic purposes

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure a Boolean function's resistance to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. Some of the binary LCD codes presented in this paper are optimal. These binary LCD codes have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems.

We explore three applications of geometric sequences in constructing cryptographic Boolean functions. First, we construct 1-resilient functions of n Boolean variables with nonlinearity 2n-1-2(n-1)/2, n odd. The Hadamard transform of these functions is 3-valued, which limits the efficiency of certain stream cipher attacks. From the case for n odd, we construct highly nonlinear 1-resilient functions which disprove a conjecture of Pasalic and Johansson for n even. Our constructions do not have a potential weakness shared by resilient functions which are formed from concatenation of linear functions. Second, we give a new construction for balanced Boolean functions with high nonlinearity, exceeding 2n-1-2(n-1)/2, which is not based on the direct sum construction. Moreover, these functions have high algebraic degree and large linear span. Third, we construct balanced vectorial Boolean functions with nonlinearity 2n-1-2(n-1)/2 and low maximum correlation. They can be used as nonlinear combiners for stream cipher systems with high throughput.

Given a positive integer n, let and . The generator matrix of the kth-order Reed–Muller code RM is an important tool in the study of Boolean functions' algebraic immunity. In this paper, choosing the last s column vectors in as a basis of the vector space , we study the values of the coefficients in the linear expressions of 's column vectors over this basis. As an application, we present a new construction of balanced even-variable Boolean functions on with optimal algebraic immunity by modifying the outputs of majority function. The nonlinearities of these constructed functions are also determined.

We describe the algebraic immunity and calculation of algebraic immunity of boolean functions. We propose a new balanced n-variable boolean function with high algebraic immunity. We check the nonlinearity of the new function for n < 20 and the algebraic immunity of the function in case of n < 18. We conjecture it has optimum algebraic immunity.

Boolean functions play an important role in symmetric ciphers. One of important open problems on Boolean functions is determining the maximum possible resiliency order of n-variable Boolean functions with optimal algebraic immunity. In this letter, we search Boolean functions in the rotation symmetric class, and determine the maximum possible resiliency order of 9-variable Boolean functions with optimal algebraic immunity. Moreover, the maximum possible nonlinearity of 9-variable rotation symmetric Boolean functions with optimal algebraic immunity-resiliency trade-off is determined to be 224.

Algebraic immunity is a new cryptographic criterion proposed to against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in cryptosystem should have high algebraic immunity. This paper presents a construction for even-variable Boolean function with optimum algebraic immunity.

International audience

Since 1970, Boolean functions have been the focus of a lot of attention in cryptography. An important topic in symmetric ciphers concerns the cryptographic properties of Boolean functions and constructions of Boolean functions with good cryptographic properties, that is, good resistance to known attacks. An important progress in cryptanalysis areas made in 2003 was the introduction by Courtois and Meier of algebraic attacks and fast algebraic attacks which are very powerful analysis concepts and can be applied to almost all cryptographic algorithms. To study the resistance against algebraic attacks, the notion of algebraic immunity has been introduced. In this paper, we use a parameter introduced by Liu and al., called fast algebraic immunity, as a tool to measure the resistance of a cryptosystem (involving Boolean functions) to fast algebraic attacks. We prove an upper bound on the fast algebraic immunity. Using our upper bound, we establish the weakness of trace inverse functions against fast algebraic attacks confirming a recent result of Feng and Gong.

We go deep into the internal structure of the Boolean functions values, and study the relationship of algebraic immunity and algebraic degree of Boolean functions with the Hamming weight with the diffusion included. Then we get some theorems which relevance together algebraic immunity, annihilators and algebraic degree of H Boolean functions by the e-derivative which is a part of the H Boolean function. Besides, we also get the results that algebraic immunity and algebraic degree of Boolean functions can be linked together by the e-derivative of H Boolean functions and so on.