Boolean Functions with Five Controllable Cryptographic Properties

Information Processing Letters, 1999

The GAC (Global Avalanche Characteristics) were introduced by Zhang and Zheng (1995) as a measure of cryptographic strength of Boolean functions. Two indicators r~f and Af related to GAC are introduced. Son et al. (1998) gave a lower bound on of for a balanced Boolean function. In this paper, we provide an improved lower bound. Moreover, we provide bounds on nonlinearity for a balanced Boolean function satisfying the propagation criterion with respect to r vectors. 0 1999 Elsevier Science B.V. All rights reserved.

Lecture Notes in Computer Science, 2004

By considering a new metric, we generalize cryptographic properties of Boolean functions such as resiliency and propagation char- acteristics. These new definitions result in a better understanding of the properties of Boolean functions and provide a better insight in the space defined by this metric. This approach leads to the construction of "hand- made" Boolean functions, i.e., functions for which the security with re- spect to some specific monotone sets of inputs is considered, instead of the security with respect to all possible monotone sets with the same cardinality, as in the usual definitions. This approach has the advantage that some trade-os between important properties of Boolean functions can be relaxed.

2005

Algebraic attack has recently become an important tool in cryptanalysing different stream and block cipher systems. A Boolean function, when used in some cryptosystem, should be designed properly to resist this kind of attack. The cryptographic property of a Boolean function, that resists algebraic attack, is known as Algebraic Immunity (AI). So far, the attempt in designing Boolean functions with required algebraic immunity was only ad-hoc, i.e., the functions were designed keeping in mind the other cryptographic criteria, and then it has been checked whether it can provide good algebraic immunity too. For the first time, in this paper, we present a construction method to generate Boolean functions on n variables with highest possible algebraic immunity n 2 . Such a function can be used in conjunction with (using direct sum) functions having other cryptographic properties. In a different direction we identify that functions, having low degree subfunctions, are weak in terms of algebraic immunity and analyse some existing constructions from this viewpoint.

Information Sciences, 2014

Cryptographically strong Boolean functions play an imperative role in the design of almost every modern symmetric cipher. In this context, the cryptographic properties of Boolean functions, such as non-linearity, algebraic degree, correlation immunity and propagation criteria, are critically considered in the process of designing these ciphers. More recently, with the emergence of algebraic and fast algebraic attacks, algebraic immunity has also been included as an integral property to be considered. As a result, several constructions of Boolean functions with high non-linearity, maximal algebraic degree and optimal algebraic immunity have been devised since then. This paper focuses on some of these constructions and presents two hybrid classes of Boolean functions. The functions constructed within these classes possess maximal algebraic degree for balanced functions, optimal algebraic immunity, high non-linearity and good resistance to algebraic and fast algebraic attacks. A hybrid class of 1-resilient functions has also been proposed that also possesses high algebraic degree, optimal algebraic immunity, high non-linearity and good resistance to algebraic and fast algebraic attacks.

Information Processing Letters, 1994

This letter presents a simple yet e ective method for transforming Boolean functions that do not satisfy the strict avalanche criterion (SAC) into ones that satisfy the criterion. Such a method has a wide range of applications in designing cryptographically strong functions, including substitution boxes (S-boxes) employed by common key block encryption algorithms.

Information Processing Letters, 1998

The GAC (global avalanche characteristic) was introduced to measure cryptographic behavior, such as, propagation characteristic, in a global manner. The two indicators af and Af related to GAC are introduced. However, an important problem to compute lower bounds of two indicators for balanced Boolean functions was not solved yet. In this paper, we derive lower bounds on the two indicators for balanced Boolean functions: af 2 2'" + 2n+3 and Af 2 8. Moreover, we obtain an upper bound on nonlinearity for balanced Boolean functions: Nf < 2"-' -~2-"iz,/Z7, which improves the previously known results.

Lecture Notes in Computer Science, 1995

This paper studies the properties and constructions of nonlinear functions, which are a core component of cryptographic primitives including data encryption algorithms and one-way hash functions. A main contribution of this paper is to reveal the relationship between nonlinearity and propagation characteristic, two critical indicators of the cryptographic strength of a Boolean function. In particular, we prove t h a t (i) if f, a Boolean function on Vn, satis es the propagation criterion with respect to all but a subset < of vectors in Vn, then the nonlinearity o f f satis es N f 2 n;1 ; 2 1 2 (n+t);1 , where t is the rank of <, a n d (ii) When j<j > 2, the nonzero vectors in < are linearly dependent.

IEEE Transactions on Information Theory, 2011

In this paper, three constructions of balanced Boolean functions with optimum algebraic immunity are proposed. The cryptographical properties such as algebraic degree and nonlinearity of the constructed functions are also analyzed.

IEEE Transactions on Information Theory, 2006

Recently, algebraic attacks have received a lot of attention in the cryptographic literature. It has been observed that a Boolean function used as a cryptographic primitive, and interpreted as a multivariate polynomial over 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 variables with highest possible algebraic immunity 2 (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.

2013

Boolean functions play important role in cryptography, since in convention a symmetric encryption algorithm can be designed by composing Boolean functions satisfying good cryptographic criteria. In this paper; state of the art in mathematical and practical study of the most important cryptographic criteria of Boolean functions and how to implement algorithms that fulfill these criteria are introduced. Also; the most known constructions for generating Boolean functions that satisfy good cryptographic criteria are summarized.

Information Sciences, 2017

In this paper, we improve the lower bound on the maximum nonlinearity of 1resilient 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.

Walailak Journal of Science and Technology (WJST)

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.

Discrete Mathematics, 2002

For a Boolean function f , define ∆ f (α) = xf (x)f (x ⊕ α),f (x) = (−1) f (x) , the absolute indicator ∆ f = max α =0 |∆ f (α)|, and the sum-of-squares indicator σ f = α ∆ 2 f (α). We construct a class of functions with good local avalanche characteristics, but bad global avalanche characteristics, namely we show that 2 2n (1 + p) ≤ σ f ≤ 2 3n−2 , ∆ f = 2 n , where p is the number of linear structures (with even Hamming weight) of the first half of an SAC balanced Boolean function f. We also derive some bounds for the nonlinearity of such functions. It improves upon the results of Son et al. [5] and Sung et al. [7]. In our second result we construct a class of highly nonlinear balanced functions with good local and global avalanche characteristics. We show that for these functions, 2 2n+2 ≤ σ f ≤ 2 2n+2+ǫ (ǫ = 0 for n even and ǫ = 1 for n odd).

Cryptography and Communications, 2013

The algebraic immunity of cryptographic Boolean functions with odd number of variables is studied in this paper. Proper modifications of functions with maximum algebraic immunity are proved that yield new functions whose algebraic immunity is also maximum. Several results are provided for both the multivariate and univariate representation, and their applicability is shown on known classes of Boolean functions. Moreover, new efficient algorithms to produce functions of guaranteed maximum algebraic immunity are developed, which further extend and generalize well-known constructions in this area. It is shown that high nonlinearity as well as good behavior against fast algebraic attacks are also achievable in several cases.

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 2019

Boolean functions and vectorial Boolean functions are the most important components of stream ciphers. Their cryptographic properties are crucial to the security of the underlying ciphers. And how to construct such functions with good cryptographic properties is a nice problem that worth to be investigated. In this paper, using two small nonlinear functions with t − 1 resiliency, we provide a method on constructing t-resilient n variables Boolean functions with strictly almost optimal nonlinearity > 2 n−1 − 2 n/2 and optimal algebraic degree n − t − 1. Based on the method, we give another construction so that a large class of resilient vectorial Boolean functions can be obtained. It is shown that the vectorial Boolean functions also have strictly almost optimal nonlinearity and optimal algebraic degree.