A new class of monomial bent functions

Canteaut, Anne; Charpin, Pascale; Kyureghyan, Gohar M.

doi:10.1016/j.ffa.2007.02.004

Cited by 110 publications

(33 citation statements)

References 20 publications

(37 reference statements)

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“…We point out that Theorem 1 can be generalized into the following whose proof is the same as that of Theorem 1. and e the multiplicity of 0 in the following multiset of (9). Then C D f is a binary linear code with length n f and dimension m − log 2 e, and the weight distribution of the code is given by…”

Section: Binary Codes From the Preimage F −1 (B) Of Boolean Functions Fmentioning

confidence: 99%

A construction of binary linear codes from Boolean functions

Ding

2016

Discrete Mathematics

140

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Boolean functions have important applications in cryptography and coding theory. Two famous classes of binary codes derived from Boolean functions are the Reed-Muller codes and Kerdock codes. In the past two decades, a lot of progress on the study of applications of Boolean functions in coding theory has been made. Two generic constructions of binary linear codes with Boolean functions have been well investigated in the literature. The objective of this paper is twofold. The first is to provide a survey on recent results, and the other is to propose open problems on one of the two generic constructions of binary linear codes with Boolean functions. These open problems are expected to stimulate further research on binary linear codes from Boolean functions.Its length is q − 1, and its dimension is at most 2m and is equal to 2m in many cases. The dual of C * ( f ) has dimension at least q − 1 − 2m. This is a generic construction of linear codes, which has a long history and its importance is supported by Delsarte's Theorem [24]. It gives a coding-theory characterisation of APN monomials, almost bent functions, and semibent functions (see, for examples, [13], [8] and [44]) when q = 2. We will not deal with this construction in this paper. The second generic construction of linear codes from functionsIn this section, we present the second generic construction of linear codes over GF(p) with any subset D of GF(p m ), and introduce basic results about the linear codes. In Section 5, we will consider specific families of binary linear codes from Boolean functions obtained with this generic construction.3

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Section: Binary Codes From the Preimage F −1 (B) Of Boolean Functions Fmentioning

confidence: 99%

A construction of binary linear codes from Boolean functions

Ding

2016

Discrete Mathematics

140

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“…From the theory of quadratic Boolean functions (see for instance [9])Ŝ a is constant for every a ∈ V (S) where V (S) ⊆ F 4 is a vector subspace, called the set of linear structures of S. It is well-known that V (S) has dimension 0 if and only if S is bent, it has dimension 4 if and only if S is linear (affine), and it has dimension 2 otherwise. Since V (S) is a vector space, S 3 = S 1 + S 2 .…”

Section: Definitionmentioning

confidence: 99%

On weak differential uniformity of vectorial Boolean functions as a cryptographic criterion

Aragona

Calderini

Maccauro

et al. 2016

AAECC

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We study the relation among some security parameters for vectorial Boolean functions which prevent attacks on the related block cipher. We focus our study on a recently-introduced security criterion, called weak differential uniformity, which prevents the existence of an undetectable trapdoor based on imprimitive group action. We present some properties of functions with low weak differential uniformity, especially for the case of power functions and 4-bit S-Boxes.

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“…The first and the second claims hold, since the statements (4) and (5) were proven in [37,38] for 2-ranks, and by Theorem 1 we know, that 2-ranks and Γ-ranks coincide for all non-constant Boolean functions. Finally, the third claim follows from (5) and the definition of the primary construction. Now we proof the existence of homogeneous cubic bent functions, different from the primary construction.…”

Section: Homogeneous Cubic Bent Functions Different From the Primarymentioning

confidence: 91%