More Balanced Boolean Functions With Optimal Algebraic Immunity and Good Nonlinearity and Resistance to Fast Algebraic Attacks

Zeng, Xiangyong; Carlet, Claude; Shan, Jinyong; Hu, Lei

doi:10.1109/tit.2011.2109935

Cited by 72 publications

(42 citation statements)

References 24 publications

(47 reference statements)

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“…The properties and constructions of Boolean functions with maximum AI are concerned in a large number of works (to name a few [9,[12][13][14][15][16]). The problem of efficiently constructing balanced Boolean functions with optimal algebraic immunity (and/or other cryptographic properties) is thus of great significance.…”

Section: Algebraic Attacksmentioning

confidence: 99%

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1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity

Han

et al. 2017

Security and Communication Networks

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Several factors (e.g., balancedness, good correlation immunity) are considered as important properties of Boolean functions for using in cryptographic primitives. A Boolean function is perfect algebraic immune if it is with perfect immunity against algebraic and fast algebraic attacks. There is an increasing interest in construction of Boolean function that is perfect algebraic immune combined with other characteristics, like resiliency. A resilient function is a balanced correlation-immune function. This paper uses bivariate representation of Boolean function and theory of finite field to construct a generalized and new class of Boolean functions on even variables by extending the Carlet-Feng functions. We show that the functions generated by this construction support cryptographic properties of 1-resiliency and (sub)optimal algebraic immunity and further propose the sufficient condition of achieving optimal algebraic immunity. Compared experimentally with Carlet-Feng functions and the functions constructed by the method of first-order concatenation existing in the literature on even (from 6 to 16) variables, these functions have better immunity against fast algebraic attacks. Implementation results also show that they are almost perfect algebraic immune functions.

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Section: Algebraic Attacksmentioning

confidence: 99%

“…Algebraic immunity defined for a Boolean function measures the resistance of the function against algebraic attacks. The properties and constructions of Boolean functions with high algebraic immunity are concerned in extensive work, for example, [9,[12][13][14][15][16].…”

Section: Algebraic Immunity Of the Proposed Constructionmentioning

confidence: 99%

1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity

Han

et al. 2017

Security and Communication Networks

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“…Up to now, many classes of Boolean functions with high algebraic immunity have been introduced [1,5,6,7,8,13,14,22,23,28,29,30,31,36,37,38,41,42,43]. However, most of them do not satisfy all the necessary criteria and the few classes which do satisfy, are not very efficiently implementable; moreover, none of the papers studying these classes took BDD-based attacks into consideration.…”

Section: Introductionmentioning

confidence: 99%

Cryptographic properties of the hidden weighted bit function

Wang

Carlet

Stănică

et al. 2014

Discrete Applied Mathematics

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The hidden weighted bit function (HWBF), introduced by R. Bryant in IEEE Trans. Comp. 40 and revisited by D. Knuth in Vol. 4 of The Art of Computer Programming, is a function that seems to be the simplest one with exponential Binary Decision Diagram (BDD) size. This property is interesting from a cryptographic viewpoint since BDDbased attacks are receiving more attention in the cryptographic community. But, to be usable in stream ciphers, the functions must also satisfy all the other main criteria. In this paper, we investigate the cryptographic properties of the HWBF and prove that it is balanced, with optimum algebraic degree and satisfies the strict avalanche criterion. We calculate its exact nonlinearity and give a lower bound on its algebraic immunity. Moreover, we investigate its normality and its resistance against fast algebraic attacks. The HWBF is simple, can be implemented efficiently, has a high BDD size and rather good cryptographic properties, if we take into account that its number of variables can be much larger than for other functions with the same implementation efficiency. Therefore, the HWBF is a good candidate for being used in real ciphers. Indeed, contrary to the case of symmetric functions, which allow such fast implementation but also offer to the attacker some specific possibilities due to their symmetry, its structure is not suspected to be related to such dedicated attacks.

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“…It was shown by N. Courtois and W. Meier [7] that maximum AI of n-variable Boolean functions is ⌈ n 2 ⌉. Constructions of Boolean functions with maximum AI were researched in a large number of papers, e.g., [9,15,14,4,21,23]. However, there are few results referring to constructions of Boolean functions with provable good immunity against fast algebraic attacks.…”

Section: Introductionmentioning

confidence: 99%

“…Although preventing classical and fast algebraic attacks is not sufficient for resisting algebraic attacks on the augmented function [11], the resistance against these attacks depends on the update function and tap positions used in a stream cipher and in actual fact it is not a property of the Boolean function. In [17] Several classes of Boolean functions, e.g., [4,23,19,20], are observed through computer experiments to have good behavior against fast algebraic attacks, but in previous literature only Carlet-Feng function (see [10,4]), which is affine equivalent to discrete logarithm function [12], was proven in [17] to be optimal against fast algebraic attacks as well as algebraic attacks. The results of [17] imply that Carlet-Feng function is PAI for n = 2 s + 1 and is almost PAI for n ̸ = 2 s + 1.…”

Section: Introductionmentioning

confidence: 99%

Almost perfect algebraic immune functions with good nonlinearity

Liu

Lin

2014

2014 IEEE International Symposium on Information Theory

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Abstract. In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSR-based stream ciphers. Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been researched in a large number of papers. However, there are few results with respect to Boolean functions with provable good immunity against fast algebraic attacks. In previous literature, only Carlet-Feng function, which is affine equivalent to discrete logarithm function, was proven to be optimal against fast algebraic attacks as well as algebraic attacks. In this paper, it is proven that a family of 2k-variable Boolean functions, including the function recently constructed by Tang et al. [IEEE TIT 59(1): 653-664, 2013], are almost perfect algebraic immune for any integer k ≥ 3. More exactly, they achieve optimal algebraic immunity and almost perfect immunity to fast algebraic attacks. The functions of such family are balanced and have optimal algebraic degree. A lower bound on their nonlinearity is obtained based on the work of Tang et al., which is better than that of Carlet-Feng function. It is also checked for 3 ≤ k ≤ 9 that the exact nonlinearity of such functions is very good, which is slightly smaller than that of Carlet-Feng function, and some functions of this family even have a slightly larger nonlinearity than Tang et al.'s function. To sum up, among the known functions with provable good immunity against fast algebraic attacks, the functions of this family make a trade-off between the exact value and the lower bound of nonlinearity.

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More Balanced Boolean Functions With Optimal Algebraic Immunity and Good Nonlinearity and Resistance to Fast Algebraic Attacks

Abstract: Abstract: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.

Cited by 72 publications

References 24 publications

1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity

1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity

Cryptographic properties of the hidden weighted bit function

Almost perfect algebraic immune functions with good nonlinearity

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