Generation of Boolean functions using Genetic Algorithm for cryptographic applications

Boolean functions are of great importance for constructing symmetric key ciphers namely stream ciphers and block ciphers. The system can be made secure and immune to cryptanalytic attacks by using strong Boolean functions. In this paper, we find Boolean functions that satisfy balancedness, correlation immunity, autocorrelation and nonlinearity properties using simulated annealing (SA) techniques. Hill climbing technique is also adopted for further improvement.

Boolean functions play a central role in the security of stream and block ciphers (in symmetric cryptography). They are functions mapping binary vectors of some length to either bits (in which case we have Boolean functions strictly speaking) or to binary vectors (of the same length or of another length, we are then speaking of vectorial Boolean functions). In the first case, they play roles in stream ciphers and in the second case, they are mainly used as substitution boxes in block ciphers. Being in each case the only nonlinear components of the ciphers, they are essential to their security. They are also the main components of the ciphers whose choice induces the speed. A tradeoff must then be found between the security and the speed of the cipher. The AES winner (used nowadays as a NIST standard for civil encryption, in particular between all banks) has been chosen because it allowed the best tradeoff. We shall describe what are the main results on them and what are the main open problems they pose currently.

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were introduced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (Reed-Muller codes) and applications in cryptography (design of stream ciphers), they have attracted a lot of research, specially in the last 15 years. The class of bent functions contains a subclass of functions, introduced by Youssef and Gong in 2001, the so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyper-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. This paper is devoted to the constructions of bent and hyper-bent Boolean functions in polynomial forms. We survey and present an overview of the constructions discovered recently. We extensively investigate the link between the bentness property of such functions and some exponential sums (involving Dickson polynomials).

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.

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).

In recent years, almost all systems using linear feedback shift storage device (LFSR) design of stream ciphers have been cracked by Algebraic attack. In light of this, finding a modern cipher that avoids old security issues with design of both security and performance becomes an important issue. In 2004, European Union initiated the selection of a new streaming secret (eSTREAM), which is analogous to United States’ formulation of AES block density standards. Forty eight candidates were chosen after public screening, review and final selection process. Details were published in May 2008 results report. This article is to discuss eSTREAM stream cipher. eSTREAM was part of group of eight surviving third phase of software selection process focusing on HC stream cipher for research. In recent years, stream cipher attack studies have been very popular, especially employing Distinguishing Attack. This article also performed security analysis for HC stream cipher, and proposes a design for a reduced version of HC-256. The reduced version of HC-256 is comparable to original HC-256 in terms of resistance to distinguishing attacks. This new generation of core stream cipher algorithm design can help understand implications of theory and application of mathematical practices towards research and analysis of stream ciphers.

A new package called boolfun is avail-able for R users. The package provides tools tohandle Boolean functions, in particular for cryp-tographic purposes. This document guides theuser through some (code) examples and gives afeel of what can be done with the package.A Boolean function is a mapping f0,1g n !f0,1g.Those mappings are of much interest in the design ofcryptographic algorithms such as secure pseudoran-dom number generators (for the design of stream ci-phers among other applications), hash functions andblock ciphers. The lack of open source software toassess cryptographic properties of Boolean functionsand the increasing interest for statistical testing ofproperties related to random Boolean functions (Fil-iol,2002;Saarinen,2006;Englund et al.,2007;Au-masson et al.,2009) are the main motivations for thedevelopment of this package.The number of Boolean functions with n vari-ables is 2 2 n , i.e. 2 possible output values for each ofthe 2 n input assignments. In this search space ofsize 2

Boolean functions are used as nonlinear filter functions and combiner functions in several stream ciphers. The security of these stream ciphers largely depends upon cryptographic properties of Boolean functions. Finding a balanced Boolean function with optimal cryptographic properties is an open research problem in the cryptographic community. Since the number of n-variable Boolean functions is $$2^{2^n}$$ , it is not computationally feasible to search the entire space of such functions for cryptographically significant functions when $$n \ge 6$$ . In general, the construction of Boolean functions with optimal or near optimal cryptographically significant properties is formulated as combinatorial optimization problems. In this paper, we apply the Genetic algorithm with the integration of a local search procedure called hybrid GA (HGA) searching for the Boolean functions with high nonlinearity and low autocorrelation. The performance of the Genetic algorithm depends upon the tuning parameters such as crossover mechanism, mutation probability, selection criteria, and the choice of fitness/cost function. To achieve an optimal trade-off among two cryptographic properties, the choice of the cost function plays an important role for finding an optimal solution. So our main focus is to construct balanced Boolean functions using HGA with a new cost function and compare it with existing cost functions. Our experimental results shows that the HGA is more efficient than GA and the new cost function is more efficient than existing cost functions for reaching the optimal solutions.

We develop an improved cost function to be used in simulated annealing followed by hill-climbing to find Boolean functions satisfying multiple desirable criteria such as high nonlinearity, low autocorrelation, balancedness, and high algebraic degree. Using this cost function that does not necessitate experimental search for parameter tuning, the annealing-based algorithm reaches the desired function profiles more rapidly. Some Boolean functions of eight and nine variables have been found, which are unattained in the computer search based literature, in terms of joint optimization of nonlinearity and autocorrelation. Global characteristics of eight-variable Boolean functions generated by algebraic construction or computer search are compared with respect to the sum-of-squared-errors in their squared spectra, which is also proportional to the sum-of-squared-errors in their autocorrelation function, the term ‘error’ denoting the deviation from bent function characteristics. Preliminary results consisting of cryptographically strong Boolean functions of nine, ten and eleven variables obtained using a three-stage optimization technique are also presented.

New Constructions of Resilient and Correlation Immune Boolean Functions Achieving Upper Bound on Nonlinearity

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.

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.

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

Welch-Gong (WG) transformation sequences are binary sequences of period 2n. 1 with 2-level auto correlation. These sequences were discovered by Golomb, Gong and Gaal in 1998 and verified for 5 ≤ n ≤ 20. Later on, No, Chung and Yun found another way to construct the WG sequences and verified their result for 5 ≤ n ≤ 23. Dillon first proved this result for odd n in 1998, and finally, Dobbertin and Dillon proved it for even n in 1999. In this paper, we investigate a two-faced property of the WG transformation sequences for application in stream ciphers and pseudo-random number generators. One is to present randomness or unpredictability of the WG transformation sequences. The other is to exhibit the security property of the WG transformations regarded as Boolean functions. It is shown that the WG transformation sequences, in addition to the known 2-level auto correlation, have threelevel cross correlation with m-sequences, large linear ! span increasing exponentially with n and efficient implementation. Thus this is the first type of pseudo-random sequences with good correlation and statistic properties, large linear span and efficient implementation. When the WG transformation are regarded as Boolean functions, it is proved that they have high nonlinearity. A criterion for whether the WG transformations regarded as Boolean functions are r-resilient is derived. It is shown that the WG transformations regarded as Boolean functions have large linear span (this concept will be defined in this paper) and high degree.

Feedback with Carry Shift Registers (FCSRs) are a promising alternative to LFSRs in the design of stream ciphers. The previous constructions based on FCSRs were dedicated to hardware applications [3]. In this paper, we will describe X-FCSR a family of software oriented stream ciphers using FCSRs. The core of the system is composed of two 256-bits FCSRs. We propose two versions: X-FCSR-128 and X-FCSR-256 which output respectively 128 and 256 bits at each iteration. We study the resistance of our design against several cryptanalyses. These stream ciphers achieve a high throughput and are suitable for software applications (6.3 cycles/byte).Keywordsstream cipherFCSRssoftware designcryptanalysis