Resilient Boolean Functions Research Articles

Improving the lower bound on the maximum nonlinearity of 1-resilient Boolean functions and designing functions satisfying all cryptographic criteria

In this article we improve the lower bound on the maximum nonlinearity of 1-resilient Boolean functions, for n even, by proposing a method of constructing this class of functions attaining the best nonlinearity currently known. Thus for the first time, at least for small values of n, the upper bound on nonlinearity can be reached in a deterministic manner in difference to some heuristic search methods proposed previously. The nonlinearity of these functions is extremely close to the maximum nonlinearity attained by bent functions and it might be the case that this is the highest possible nonlinearity of 1-resilient functions. Apart from this theoretical contribution, it turns out that the cryptographic properties of these functions are overall good apart from their moderate resistance to fast algebraic attacks (FAA). This weakness is repaired by a suitable modification of the original functions giving a class of balanced functions with almost optimal resistance to FAA whose nonlinearity is better than the nonlinearity of other methods.

Construction of resilient Boolean functions with high nonlinearity and good algebraic degree

AbstractResilient Boolean functions with high nonlinearity and good algebraic properties play an important role in designing certain stream cipher schemes. In this paper, two construction methods are proposed to obtain such functions. It is shown that a class of resilient functions with high algebraic degree and currently best known nonlinearity can be constructed by using our technique. The algebraic immunity of the constructed functions is also analyzed. Copyright © 2015 John Wiley & Sons, Ltd.

Generalized Maiorana–McFarland Construction of Resilient Boolean Functions With High Nonlinearity and Good Algebraic Properties

A new framework concerning the construction of small-order resilient Boolean functions whose nonlinearity is strictly greater than 2 n-1 - 2 [n/2] is given. First, a generalized Maiorana-McFarland construction technique is described, which extends the current approaches by combining the usage of affine and nonlinear functions in a controllable manner. It is shown that for any given m, this technique can be used to construct a large class of n-variable (n both even and odd) m-resilient degree-optimized Boolean functions with currently best known nonlinearity. This class may also provide functions with excellent algebraic properties, measured through the resistance to (fast) algebraic attacks, if the number of n/2-variable affine subfunctions used in the construction is relatively low. Due to a potentially low hardware implementation cost, along with overall good cryptographic properties, this class of functions is an attractive candidate for the use in certain stream cipher schemes.

On the degree of completeness of cryptographic functions

This paper discusses the degree of completeness of cryptographic functions, which is one of the cryptographic criteria should be considered in the design of stream ciphers. We establish the relationships between the degree of completeness and other cryptographic criteria. For resilient Boolean functions, a method to enhance the degree of completeness is proposed, while the nonlinearity and the algebraic degree do not decrease. Moreover, two constructions of resilient functions are provided, which have optimal degree of completeness, high nonlinearity, and high algebraic degree.

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.

Construction of highly nonlinear resilient Boolean functions satisfying strict avalanche criterion

Boolean functions with high nonlinearity, high resiliency and strict avalanche criterion (SAC) play an important role in the designs of conventional cryptographic systems. In this paper, a method is proposed to construct resilient Boolean functions on n variables (n even) satisfying SAC with nonlinearity > 2n−1 − 2n/2. A large class of cryptographic Boolean functions that were not known earlier were obtained.

A Class of 1-Resilient Functions in Odd Variables with High Nonlinearity and Suboptimal Algebraic Immunity

Based on Tu-Deng's conjecture and the Tu-Deng function, in 2010, X. Tang et al. proposed a class of Boolean functions in even variables with optimal algebraic degree, very high nonlinearity and optimal algebraic immunity. In this corresponding, we consider the concatenation of Tang's function and another Boolean function, and study its cryptographic properties. With this idea, we propose a class of 1-resilient Boolean functions in odd variables with optimal algebraic degree, good nonlinearity and suboptimal algebraic immunity based on Tu-Deng's conjecture.

Construction of 1-resilient boolean functions with optimal algebraic immunity and good nonlinearity

This paper presents a construction for a class of 1-resilient functions with optimal algebraic immunity on an even number of variables. The construction is based on the concatenation of two balanced functions in associative classes. For some n, a part of 1-resilient functions with maximum algebraic immunity constructed in the paper can achieve almost optimal nonlinearity. Apart from their high nonlinearity, the functions reach Siegenthaler's upper bound of algebraic degree. Also a class of 1-resilient functions on any number n > 2 of variables with at least sub-optimal algebraic immunity is provided.

Construction of 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Good Nonlinearity

This paper presents a construction for a class of 1-resilient functions with optimal algebraic immunity on an even number of variables. The construction is based on the concatenation of two balanced functions in associative classes. For some n, a part of 1-resilient functions with maximum algebraic immunity constructed in the paper can achieve almost optimal nonlinearity. Apart from their high nonlinearity, the functions reach Siegenthaler's upper bound of algebraic degree. Also a class of 1-resilient functions on any number n > 2 of variables with at least sub-optimal algebraic immunity is provided.

Construction of Odd-Variable Resilient Boolean Functions with Optimal Degree

Constructing degree-optimized resilient Boolean functions with high nonlinearity is a significant study area in Boolean functions. In this letter, we provide a construction of degree-optimized n-variable (n odd and n ≥ 35) resilient Boolean functions, and it is shown that the resultant functions achieve the currently best known nonlinearity.

Vectorial resilient PC(l) of order k Boolean functions from AG-codes

Propagation criteria and resiliency of vectorial Boolean functions are important for cryptographic purpose (see [1–4, 7, 8, 1, 11, 16]). Kurosawa, Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying PC(l) of order k from binary linear or nonlinear codes. In this paper, the algebraic-geometric codes over GF(2m) are used to modify the Carlet and Kurosawa-Satoh’s construction for giving vectorial resilient Boolean functions satisfying PC(l) of order k criterion. This new construction is compared with previously known results.

Construction of 1-resilient Boolean functions with optimum algebraic immunity

In this paper, for an integer n≥10, two classes of n-variable Boolean functions with optimum algebraic immunity (AI) are constructed, and their nonlinearities are also determined. Based on non-degenerate linear transforms to the proposed functions, we can obtain 1-resilient n-variable Boolean functions with optimum AI and high nonlinearity if n−1 is never equal to any power of 2.

On the Security of Blockwise Secure Modes of Operation Beyond the Birthday Bound

This paper presents a complete characterization of the first order correlation immune Boolean functions that includes the functions that are 1-resilient. The approach consists in defining an equivalence relation on the full set of Boolean functions with a fixed number of variables. An equivalence class in this relation, called a first-order correlation class, provides a measure of the distance between the Boolean functions it contains and the correlation-immune Boolean functions. The key idea consists on manipulating only the equivalence classes instead of the set of Boolean functions. To achieve this goal, a class operator is introduced to construct a class with n variables from two classes of n - 1 variables. In particular, the class of 1-resilient functions on n variables is considered. An original and efficient method to enumerate all the Boolean functions in this class is proposed by performing a recursive decomposition of classes with less variables. A bottom up algorithm provides the exact number of 1-resilient Boolean functions with seven variables which is 23478015754788854439497622689296. A tight estimation of the number of 1-resilient functions with eight variables is obtained by performing a partial enumeration. It is conjectured that the exact complete enumeration for general n is intractable.

Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any m, one can construct infinitely many re-variable (n even), m-resilient functions with nonlinearity > 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-1</sup> - 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-2</sup> . A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.

Maiorana–McFarland Class: Degree Optimization and Algebraic Properties

In this paper, we consider a subclass of the Maiorana-McFarland class used in the design of resilient nonlinear Boolean functions. We show that these functions allow a simple modification so that resilient Boolean functions of maximum algebraic degree may be generated instead of suboptimized degree in the original class. Preserving a high-nonlinearity value immanent to the original construction method, together with the degree optimization gives in many cases functions with cryptographic properties superior to all previously known construction methods. This approach is then used to increase the algebraic degree of functions in the extended Maiorana-McFarland (MM) class (nonlinear resilient functions F:GF(2)n |rarrGF(2)m derived from linear codes). We also show that in the Boolean case, the same subclass seems not to have an optimized algebraic immunity, hence not providing a maximum resistance against algebraic attacks. A theoretical analysis of the algebraic properties of extended Maiorana-McFarland class indicates that this class of functions should be avoided as a filtering function in nonlinear combining generators

Algebraic immunity for cryptographically significant Boolean functions: analysis and construction

Recently, algebraic attacks have received a lot of attention in the cryptographic literature. It has been observed that a Boolean function f used as a cryptographic primitive, and interpreted as a multivariate polynomial over F/sub 2/, should not have low degree multiples obtained by multiplication with low degree nonzero functions. In this paper, we show that a Boolean function having low nonlinearity is (also) weak against algebraic attacks, and we extend this result to higher order nonlinearities. Next, we present enumeration results on linearly independent annihilators. We also study certain classes of highly nonlinear resilient Boolean functions for their algebraic immunity. We identify that functions having low-degree subfunctions are weak in terms of algebraic immunity, and we analyze some existing constructions from this viewpoint. Further, we present a construction method to generate Boolean functions on n variables with highest possible algebraic immunity /spl lceil/n/2/spl rceil/ (this construction, first presented at the 2005 Workshop on Fast Software Encryption (FSE 2005), has been the first one producing such functions). These functions are obtained through a doubly indexed recursive relation. We calculate their Hamming weights and deduce their nonlinearities; we show that they have very high algebraic degrees. We express them as the sums of two functions which can be obtained from simple symmetric functions by a transformation which can be implemented with an algorithm whose complexity is linear in the number of variables. We deduce a very fast way of computing the output to these functions, given their input.

Classification of cubic (n-4)-resilient Boolean functions

Carlet and Charpin classified the set of cubic (n-4)-resilient Boolean functions into four different types with respect to the Walsh spectrum and the dimension of the linear space. Based on the classification of RM(3,6)/RM(1,6), we have completed this classification of cubic (n-4)-resilient Boolean functions by deriving the corresponding algebraic normal form (ANF) and autocorrelation spectrum for each of the four types. At the same time, we have solved an open problem by proving that all plateaued cubic (n-4)-resilient Boolean functions have dimension of the linear space equal either to n-5 or n-6.

On the Covering Radii of Binary Reed–Muller Codes in the Set of Resilient Boolean Functions

Let R/sub t,n/ be the set of t-resilient Boolean functions in n variables, and let /spl rho//spl circ/(t,r,n) be the maximum distance between t-resilient functions and the rth-order Reed-Muller code RM(r,n). We prove that /spl rho//spl circ/(t,2,6)=16 for t=0,1,2 and /spl rho//spl circ/(3,2,7)=32, from which we derive the lower bound /spl rho//spl circ/(t,2,n) /spl ges/ 2/sup n-2/ with t /spl les/ n-4. Using a result from coding theory on the covering radius of (n-3)th- and (n-4)th-order Reed-Muller codes, we establish exact values of the covering radius of RM(n-3,n) in the set of 1-resilient Boolean functions in n variables, when /spl lfloor/n/2/spl rfloor/=1 mod 2 and lower bounds of RM(n-4,n) in the set of 2-resilient Boolean functions in n variables. This result leads again to different lower bounds for general dimensions n and r=0 or 3 mod 4.

Further constructions of resilient Boolean functions with very high nonlinearity

One well-known method of generating key stream sequences for stream ciphers is to combine the outputs of several linear-feedback shift registers (LFSR) using a combining Boolean function. Here we concentrate on the design of good combining Boolean functions. We provide resilient Boolean functions with currently best known nonlinearity. These functions were not known earlier and the issues related to their existence were posed as open questions in the literature. Some of the functions we construct here achieve the provable upper bound on nonlinearity for resilient Boolean functions. Our technique interlinks mathematical results with classical computer search.