In this paper, we consider minimal linear codes by a general construction of linear codes from
q
-ary functions. First, we give necessary and sufficient conditions for codewords which are constructed ...by functions to be minimal. Second, as applications, we present three constructions of minimal linear codes. Constructions on minimal linear codes in this paper generalize some recent results in Ding et al. (IEEE Trans. Inf. Theory
64
(10), 6536–6545,
2018
); Heng et al. (Finite Fields Appl.
54
, 176–196,
2018
); Bartoli and Bonini (IEEE Trans. Inf. Theory
65
(7), 4152–4155,
2019
); Mesnager et al. (IEEE Trans. Inf. Theory
66
(9), 5404–5413,
2020
); Bonini and Borello (J. Algebraic Comb.
53
, 327–341,
2021
). In our three constructions, the conditions of functions are much looser than theirs.
Further Results on Niho Bent Functions Budaghyan, L.; Carlet, C.; Helleseth, T. ...
IEEE transactions on information theory,
11/2012, Letnik:
58, Številka:
11
Journal Article
Recenzirano
This paper consists of two main contributions. First, the Niho bent function consisting of 2 r exponents (discovered by Leander and Kholosha) is studied. The dual of the function is found and it is ...shown that this new bent function is not of the Niho type. Second, all known univariate representations of Niho bent functions are analyzed for their relation to the completed Maiorana-McFarland class M . In particular, it is proven that two families do not belong to the completed class M . The latter result gives a positive answer to an open problem whether the class H of bent functions introduced by Dillon in his thesis of 1974 differs from the completed class M .
The S-PUF and S n -PUF designs (proposed in IN-DOCRYPT2019) are one of the contemporary composite strong PUF candidates of the Delay-PUF family that exhibit two distinguishing and notable attributes ...- (i) it is one of the few PUF constructions which is guided by theoretical analysis of the Strict Avalanche Criteria (SAC) property and not by ad-hoc choices; and (ii) though its construction is quite similar to XOR PUFs, it has very good reliability property unlike the former design due to the introduction of Maiorana-McFarland (M-M) Bent Function. These make S n -PUF to be a very good candidate for strong PUF proposals and an interesting target from the point of view of attackers. In this work, we testify that a novel reliability based machine learning attack can be launched in this architecture against the original authors' claim. Though it is challenging to launch a classical or reliability based ML attack directly, we leverage the bias introduced by the AND operation in the M-M bent function due to its non-linearity property. Our proposed novel attack framework, SACReD, is able to break S_{8}, S_{10} and S_{12}-PUF designs, which were originally assumed to be secure, by taking only 400K Challenge-Response Pairs.
Maiorana-McFarland ternary bent functions are analyzed with respect to their relationship with spectral invariant operations and with respect to self-duality. In analogy to the Maiorana-McFarland ...method to generate binary bent functions, an equation is introduced to generate ternary bent functions in an even number of variables. It is shown that Maiorana-McFarland ternary bent functions are strict bent and that their duals may not be Maiorana-McFarland. It is known that spectral invariant operations preserve the bentness of a function. We show that most spectral invariant operations preserve that Maiorana-McFarland bentness of ternary functions. The concept of generalized self-duality is introduced and it is shown how Maiorana-McFarland ternary bent functions exhibit this kind of self-duality. It is shown that Maiorana-McFarland ternary bent functions in two variables may be divided into 3 classes of 54 functions each. A list of these functions for a first class is given in an Appendix.
A bent function is
self-dual
if it is equal to its dual function. We study the metric properties of the self-dual bent functions constructed on using available constructions. We find the full Hamming ...distance spectrum between self-dual Maiorana–McFarland bent functions. Basing on this, we find the minimal Hamming distance between the functions under study.
The Gowers
U
3
norm of a Boolean function is a measure of its resistance to quadratic approximations. It is known that smaller the Gowers
U
3
norm for a Boolean function larger is its resistance to ...quadratic approximations. Here, we compute Gowers
U
3
norms for some classes of Maiorana–McFarland bent functions. In particular, we explicitly determine the value of the Gowers
U
3
norm of Maiorana–McFarland bent functions obtained by using APN permutations. We prove that this value is always smaller than the Gowers
U
3
norms of Maiorana–McFarland bent functions obtained by using differentially
δ
-uniform permutations, for all
δ
≥
4
. We also compute the Gowers
U
3
norms for a class of cubic monomial functions, not necessarily bent, and show that for
n
=
6
, these norm values are less than that of Maiorana–McFarland bent functions. Further, we computationally show that there exist 6-variable functions in this class which are not bent but achieve the maximum second-order nonlinearity for 6 variables.
In the literature, few
n
-variable rotation symmetric bent functions have been constructed. In this paper, we present two infinite classes of rotation symmetric bent functions on
F
2
n
of the two ...forms:
f
(
x
)
=
∑
i
=
0
m
-
1
x
i
x
i
+
m
+
γ
(
x
0
+
x
m
,
…
,
x
m
-
1
+
x
2
m
-
1
)
,
f
t
(
x
)
=
∑
i
=
0
n
-
1
(
x
i
x
i
+
t
x
i
+
m
+
x
i
x
i
+
t
)
+
∑
i
=
0
m
-
1
x
i
x
i
+
m
+
γ
(
x
0
+
x
m
,
…
,
x
m
-
1
+
x
2
m
-
1
)
,
where
n
=
2
m
,
γ
(
X
0
,
X
1
,
…
,
X
m
-
1
)
is any rotation symmetric polynomial, and
m
/
gcd
(
m
,
t
)
is odd. The class (i) of rotation symmetric bent functions has algebraic degree ranging from 2 to
m
and the other class (ii) has algebraic degree ranging from 3 to
m
. Moreover, the two classes of rotation symmetric bent functions are disjoint.
This work extends the idea introduced by Hou and Langevin (J. Combin. Theory, Ser. A,
80
:232–246,
1997
) of applying nonlinear permutations to (a portion of) the input variable space of a given ...Boolean function so that the resulting function is bent. Applying such a permutation to a bent function that can be represented in a suitable form then gives an affine inequivalent bent function which potentially does not belong to the same class as the original one. While Hou and Langevin only provided two sporadic examples of bent functions that can be turned into affine inequivalent ones, in this article we identify two generic families of bent functions suitable for generating such affine inequivalent counterparts. The same method when applied to the Marioana-McFarland class of bent functions, depending on the subset of inputs to which a nonlinear action is applied, either lead to bent functions that are provably within the same class or to bent functions that are potentially outside this class. The problem of finding suitable permutations that act nonlinearly on more than two input variables of the initial function and ensure the bentness of the resulting function appears to be generally hard. In this direction, we only slightly extend the approach of Hou and Langevin by identifying suitable permutations that act nonlinearly on three input variabl es. Most notably, the existence of nonlinear permutations that act without strict separation of the input space in terms of linear and nonlinear action is also confirmed. Finally, we show a direct correspondence between (some classes of) bent functions and permutations by providing an efficient method to define permutations using the derivatives of a given bent function. This not only gives a relationship between two seemingly different algebraic objects, but also provides us with a new infinite family of permutations over finite fields.
We consider negabent Boolean functions that have Trace representation. To the best of our knowledge, this is the first ever work on negabent functions with such representation. We completely ...characterize negabent quadratic monomial functions. We also present necessary and sufficient condition for a Maiorana-McFarland bent function to be a negabent function. As a consequence of that result we present a nice characterization of a bent-negabent Maiorana-McFarland function which is based on the permutation \documentclass12pt{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$x \mapsto x^{2^i}$\end{document}.
Some results on q-ary bent functions Singh, Deep; Bhaintwal, Maheshanand; Singh, Brajesh Kumar
International journal of computer mathematics,
09/2013, Letnik:
90, Številka:
9
Journal Article
Recenzirano
Odprti dostop
Kumar et al. Generalized bent functions and their properties, J. Comb. Theory Ser. A 40 (1985), pp. 90-107 have extended the notion of classical bent Boolean functions in the generalized setup on
. ...They have provided an analogue of classical Maiorana-McFarland type bent functions. In this paper, we study the cross-correlation of a subclass of such generalized Maiorana-McFarland type bent functions. We provide a characterization of quaternary (q=4) bent functions on n+1 variables in terms of their subfunctions on n-variables. Analogues of sum-of-squares' indicator and absolute indicator of cross-correlation of Boolean functions are defined in the generalized setup. Further, q-ary functions are studied in terms of these indicators and some upper bounds of these indicators are obtained. Finally, we provide some constructions of balanced quaternary functions with high nonlinearity under Lee metric.
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