On Constructions and Nonlinearity of Correlation Immune Functions

Seberry, Jennifer; Zhang, Xian-Mo; Zheng, Yuliang

doi:10.1007/3-540-48285-7_16

Cited by 65 publications

(64 citation statements)

References 6 publications

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“…Construction of resilient functions by concatenating the truth tables of small affine functions was first described in [1] and revisited in greater details in [33], [6]. The concatenation simply means that the truth tables of the functions are merged.…”

Section: The MM Class Revisitedmentioning

confidence: 99%

Maiorana–McFarland Class: Degree Optimization and Algebraic Properties

Pašalić

2006

IEEE Trans. Inform. Theory

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Abstract-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 7 ! GF (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.

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Section: The MM Class Revisitedmentioning

confidence: 99%

Maiorana–McFarland Class: Degree Optimization and Algebraic Properties

Pašalić

2006

IEEE Trans. Inform. Theory

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“…Some examples include [2,3,11,21]. A common construction belong to the Maiorana-McFarland class, see [3] for a summary.…”

Section: Comparison With Known Resilient Preferred Functionsmentioning

confidence: 99%

Additive Autocorrelation of Resilient Boolean Functions

Gong

Khoo

2004

Selected Areas in Cryptography

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Abstract. In this paper, we introduce a new notion called the dual function for studying Boolean functions. First, we discuss general properties of the dual function that are related to resiliency and additive autocorrelation. Second, we look at preferred functions which are Boolean functions with the lowest 3-valued spectrum. We prove that if a balanced preferred function has a dual function which is also preferred, then it is resilient, has high nonlinearity and optimal additive autocorrelation. We demonstrate four such constructions of optimal Boolean functions using the Kasami, Dillon-Dobbertin, Segre hyperoval and Welch-Gong Transformation functions. Third, we compute the additive autocorrelation of some known resilient preferred functions in the literature by using the dual function. We conclude that our construction yields highly nonlinear resilient functions with better additive autocorrelation than the Maiorana-McFarland functions. We also analysed the saturated functions, which are resilient functions with optimized algebraic degree and nonlinearity. We show that their additive autocorrelation have high peak values, and they become linear when we fix very few bits. These potential weaknesses have to be considered before we deploy them in applications.

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“…Then h , a function on V,, satisfies Nh = 2"-"Ng. A proof for this special case can be found in, for instance, [18].…”

Section: Corollary 14 Given a ( N M T)-resilient Function Theorementioning

confidence: 99%

On Nonlinear Resilient Functions

Zhang

Zheng

1995

Advances in Cryptology — EUROCRYPT ’95

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Abstract. This paper studies resilient functions which have applications in fault-tolerant distributed computing, quantum cryptographic key distribution and random sequence generation for stream ciphers.We present a number of methods for synthesizing resilient functions. An interestiiig aspect of these methods is that they are applicable both to linear and to nonlinear resilient functions. Our second major contribution is t o show that every linear resilient function can be transformed into a large number of nonlinear resilient functions with the same parameters.As a result, we obtain resilient functions that are highly nonlinear and have a high algebraic degree.

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On Constructions and Nonlinearity of Correlation Immune Functions

Cited by 65 publications

References 6 publications

Maiorana–McFarland Class: Degree Optimization and Algebraic Properties

Maiorana–McFarland Class: Degree Optimization and Algebraic Properties

Additive Autocorrelation of Resilient Boolean Functions

On Nonlinear Resilient Functions

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