Some Cryptographic Properties of H Boolean Functions

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.

Algebraic immunity is an important cryptographic property for Boolean functions against algebraic attacks. Constructions of Boolean functions with the maximum algebraic immunity (MAI Boolean functions) by using univariate polynomial representation of Boolean functions over finite fields have received more and more attention. In this paper, how to obtain more MAI Boolean functions from a known MAI Boolean function under univariate polynomial representation is further investigated. The sufficient condition of Boolean functions having the maximum algebraic immunity obtained by changing a known MAI Boolean function under univariate polynomial representation is given. With this condition, more balanced MAI Boolean functions under univariate polynomial representation can be obtained. The algebraic degree and the nonlinearity of these Boolean functions are analyzed.

Boolean functions with a variety of secure cipher properties are the key factors to design cryptosystem with the ability to resist multiple cipher attacks and good safety performance. In this paper, using the derivative of the Boolean functions and the e-derivative defined by ourselves as the main research tools, we study algebraic immunity, correlation immunity and other cryptographic properties of the quadratic rotation symmetric Boolean functions. We determine the quadratic rotation symmetric Boolean functions which are H Boolean functions, and the range of weight distribution of the quadratic rotation symmetry H Boolean functions. Besides, we get the compatibility among propagation, balance, correlation immunity and algebraic immunity of the quadratic rotation symmetry H Boolean functions, and also focus on the relationship of balance, correlation immunity and dimension. Furthermore, we check the existence of the cubic rotation symmetry H Boolean functions, and obtain the relationship between existence and dimension of the cubic rotation symmetry H Boolean functions. Moreover, we obtain a more convenient method for solving annihilator. Such researches are important in cryptographic primitive designs, and have significance and role in the theory and application range of cryptosystems.

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

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

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.

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

The cryptographic primitives such as Boolean function and S-Box are used as a building block to design stream cipher and block cipher, respectively. However, it is important to evaluate the cryptographic strength to measure the security of such crytposystem resistance against important cryptanalytic attacks. We develop a new tool called Boolean function and S-Box analysis tool (BSAT) to evaluate the cryptographic strength of the Boolean function and S-Box. As the size of S-Box increases, it takes a lot of computational effort to evaluate all the required cryptographic properties. In this paper, we develop a tool to evaluate important cryptographic properties of both Boolean function and S-Box to minimize the computational time. The computational time of BSAT is lesser than the SET tool in terms of calculating the algebraic normal form and algebraic degree. Our BSAT takes only 0.404 s to evaluate the AES properties faster than the SET, which takes 650 millisecond (0.65 s).

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.

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

We propose a general approach to construct cryptographic significant Boolean functions of (r + 1)m variables based on the additive decomposition F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2rm</sub> × F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sub> of the finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2(r+1)m</sub> , where r ≥ 1 is odd and m ≥ 3. A class of unbalanced functions is constructed first via this approach, which coincides with a variant of the unbalanced class of generalized Tu-Deng functions in the case r = 1. Functions belonging to this class have high algebraic degree, but their algebraic immunity does not exceed m, which is impossible to be optimal when r > 1. By modifying these unbalanced functions, we obtain a class of balanced functions which have optimal algebraic degree and high nonlinearity (shown by a lower bound we prove). These functions have optimal algebraic immunity provided a combinatorial conjecture on binary strings which generalizes the Tu-Deng conjecture is true. Computer investigations show that, at least for small values of number of variables, functions from this class also behave well against fast algebraic attacks.

The aim of the work is to evaluate the cryptographic properties of the pCollapser family of pseudo-random functions (PRF) based on the study of the properties of its mini_pCollapser_12x12 miniversionusing fixed substitutions with extremely low cryptographic properties. As a comparison element,we used a mini-version of a typical function based on an SP-net, containing a similar number of fixedsubstitutions, and having a similar input/output dimension equal to 12 bits. To achieve this goal, thefollowing tasks were solved: – determination of the structure of the studied functions and the number ofrounds; – definition of a model for the formation of fixed substitutions with extremely low cryptographicproperties; – generation of sets of 6-bit fixed substitutions with extremely low cryptographic properties; – inclusion of the substitutions obtained into the functions under study and determination of the maincryptographic properties of functions – the maximum dominance value for individual key values and themaximum dominance value averaged over the entire set of keys, the maximum and averaged over theentire set of keys value in the difference distribution table, algebraic degree and algebraic immunity;– analysis of the obtained results. The paper presents two models for the formation of fixed substitutionswith extremely low cryptographic properties – based on the mixing of cell values in a pre-filled tableand based on the simplest ARX function (consisting of modulo addition, cyclic shift and XOR). The useof fixed substitutions with extremely low non-linearity makes it possible to estimate how complex (nonlinear)the function under study is and what minimum level of non-linearity is necessary to effectivelydestroy the statistical dependencies between input/output data. In addition, it becomes clear that ARXfunctions can be used as non-linear elements, which often have controversial and clearly low cryptographicproperties, but allow creating high-speed software and hardware implementations. It has beendetermined that the PRF pCollapser mini-version, in contrast to the typical function based on the SPnetwork, makes it possible to obtain a high-quality non-linear function from the set of ARX-functionswith extremely low cryptographic properties, given that no other non-linear elements are presented inpCollapser. The obtained results reflect the existence of a fundamental difference between thepCollapser PRF and a typical SP-network based PRF and confirm the correctness of the concept ofPD-sbox pseudo-dynamic substitutions and the pCollapser function consisting of them as a whole.

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.

Cryptography plays an important role in the network environment and wireless applications. Most of the time, the information in wireless and network environment is encrypted by stream ciphers which must resist many kinds of attacks. Algebraic immunity (AI) is one of the most important characteristics which evaluate the resistance against algebraic attacks. Constructing a Boolean function which has both AI and other crypto characteristics is an important studying field. In this paper, we study the AI and other properties, and then propose a construction method based on genetic algorithm that can obtain Boolean functions with good cryptographic appearance.

Block of nonlinear replacements (complication node, substitution block, S-box) is an important element of most modern symmetric ciphers. This is an elementary cryptographic primitive designed to mix input data and introduce non-linearity. By replacing small blocks with other blocks, mixing and scattering of data is achieved, and repeated cyclic repetition of such a procedure makes it possible to achieve the required cryptographic properties of the cipher. Many different criteria are put forward to substitution blocks (balance, high nonlinearity, low autocorrelation, correlation immunity, required avalanche properties, and many others). Each criterion expresses a formalized requirement of resistance to certain types of cryptographic analysis (differential, linear, etc.), i.e. when designing ciphers, they use an integrated approach, choosing S-blocks according to the totality of individual indicators. With the development of algebraic cryptanalysis, a new indicator of the effectiveness of substitution nodes has appeared – algebraic immunity, which is calculated as the minimum degree of the simplest (in a sense) algebraic equation describing the S-block. To search for such an equation, special methods are used, based on the construction of Gröbner bases. If you specify a knot of substitutions through a Boolean function, then to calculate algebraic immunity it is enough to find a function of the least degree from the annihilator space. This article discusses various methods of calculating algebraic immunity, analyzes their computational efficiency, discusses implementation details, substantiates methods for optimizing calculations in terms of time (in terms of number of operations) and capacitive (in terms of memory costs) of complexity. An advanced algorithm for calculating algebraic immunity is proposed, optimized for computing resources, including the necessary RAM sizes. The results of experimental studies are presented, in particular, the results of calculations of the algebraic immunity of 8x8 S-blocks for some well-known modern ciphers (AES, Kalina, Grasshopper, BelT), as well as the results obtained for random 8x8, 9x9 and 10x10 S-blocks.