Cubic bent functions outside the completed Maiorana-McFarland class

Polujan, Alexandr; Pott, Alexander

doi:10.1007/s10623-019-00712-y

Cited by 16 publications

(13 citation statements)

References 31 publications

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“…Indeed, the first interesting remark of our computer search is that the algebraic degree of the generated functions is always n/2, also in the PS + case. It is known that up to n = 6 variables, all bent functions of degree n/2 belong to the completed Maiorana-McFarland class [16]. Therefore, the smallest interesting case to consider concerning EA-equivalence is n = 8 variables.…”

Section: Computational Results On Ranks and Ea-equivalence For N =mentioning

confidence: 99%

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Bent Functions in the Partial Spread Class Generated by Linear Recurring Sequences

Gadouleau¹,

Mariot²,

Picek³

2021

Preprint

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We present a construction of partial spread bent functions using subspaces generated by linear recurring sequences (LRS). We first show that the kernels of the linear mappings defined by two LRS have a trivial intersection if and only if their feedback polynomials are relatively prime. Then, we characterize the appropriate parameters for a family of pairwise coprime polynomials to generate a partial spread required for the support of a bent function, showing that such families exist if and only if the degrees of the underlying polynomials is either 1 or 2. We then count the resulting sets of polynomials and prove that for degree 1, our LRS construction coincides with the Desarguesian partial spread. Finally, we perform a computer search of all PS − and PS + bent functions of n = 8 variables generated by our construction and compute their 2-ranks. The results show that many of these functions defined by polynomials of degree b = 2 are not EA-equivalent to any Maiorana-McFarland or Desarguesian partial spread function.

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Section: Computational Results On Ranks and Ea-equivalence For N =mentioning

confidence: 99%

“…Besides computing the 2-rank, employing more discriminating invariants would also be interesting. These include, for instance, the Smith normal form of the development of the graph G f of a Boolean function f , which is used by Polujan and Pott in [16] to classify homogeneous cubic bent functions.…”

Section: Discussionmentioning

confidence: 99%

Bent Functions in the Partial Spread Class Generated by Linear Recurring Sequences

Gadouleau¹,

Mariot²,

Picek³

2021

Preprint

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“…A classic and well-known way of building Boolean functions is the Maiorana-McFarland (MM) class, which was created to obtain Bent functions. [8,9] and has also been extended to construct resistant functions [10,11]. For n ≥ 2 an integer, and F n 2 = E ⊕ F, a decomposition into two complementary vector subspaces: E of dimension p and F of dimension q = n − p. For any pair of maps π : E −→ F n 2 and h : E −→ F 2 the Maiorana-McFarland (MM) construction defines a Boolean function f as follows:…”

Section: Maiorana-mcfarland Constructionmentioning

confidence: 99%

Construction of Boolean Functions from Hermitian Codes

et al. 2022

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In 2005, Guillot published a method for the construction of Boolean functions using linear codes through the Maiorana–McFarland construction of Boolean functions. In this work, we present a construction using Hermitian codes, starting from the classic Maiorana–McFarland construction. This new construction describes how the set of variables is divided into two complementary subspaces, one of these subspaces being a Hermitian Code. The ideal theoretical parameters of the Hermitian code are proposed to reach desirable values of the cryptographic properties of the constructed Boolean functions such as nonlinearity, resiliency order, and order of propagation. An extension of Guillot’s work is also made regarding parameters selection using algebraic geometric tools, including explicit examples.

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“…First, we denote the complement of a Boolean function f by f := f ⊕ 1 and by M f an incidence matrix of the translation design dev(G f ), which can be computed as follows M f := (f (x ⊕ y)) x,y∈F n 2 , see [28]. With the use of incidence matrices M f and M f of translation designs dev(D f ) and dev(D f ), respectively, one can decompose the incidence matrix M (dev(G f )) of a Boolean function f : F n 2 → F 2 in the following way [21]:…”

Section: Translation Designs Of Boolean Bent Functionsmentioning

confidence: 99%

“…For instance, Weng et al in [28,Theorem 5.11] used the 2-rank of the translation design dev(D f ) to prove, that almost every Desarguesian partial spread bent function is not extended-affine equivalent to a Maiorana-McFarland bent function. Recently, the authors of this paper in [21] used algebraic invariants of dev(D f ) and dev(G f ) to show inequivalence of certain homogeneous cubic bent functions. Bending in [1, Corollary 10.6] proved that extendedaffine equivalence of bent functions coincides with isomorphism of addition designs.…”

Section: Introductionmentioning

confidence: 99%

On design-theoretic aspects of Boolean and vectorial bent functions

Polujan,

Pott

2020

Preprint

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There are two construction methods of designs from Boolean and vectorial bent functions, known as translation and addition designs. In this paper we analyze, which equivalence relation for Boolean and vectorial bent functions is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial bent functions and isomorphism of addition designs are the same concepts. Further, we show that extended-affine inequivalent Boolean bent functions in n variables, whose translation designs are isomorphic exist for all n ≥ 6. This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence. However, we do not observe the same phenomena for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and show, that in contrast to the Boolean case, one cannot exhibit isomorphic translation designs from extended-affine inequivalent vectorial bent functions in six variables.

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Cubic bent functions outside the completed Maiorana-McFarland class

Cited by 16 publications

References 31 publications

Bent Functions in the Partial Spread Class Generated by Linear Recurring Sequences

Bent Functions in the Partial Spread Class Generated by Linear Recurring Sequences

Construction of Boolean Functions from Hermitian Codes

On design-theoretic aspects of Boolean and vectorial bent functions

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