39,164 research outputs found
Metrical properties of the set of bent functions in view of duality
In the paper, we give a review of metrical properties of the entire set of bent functions and its significant subclasses of self-dual and anti-self-dual bent functions. We present results for iterative construction of bent functions in n + 2 variables based on the concatenation of four bent functions and consider related open problem proposed by one of the authors. Criterion of self-duality of such functions is discussed. It is explored that the pair of sets of bent functions and affine functions as well as a pair of sets of self-dual and anti-self-dual bent functions in n > 4 variables is a pair of mutually maximally distant sets that implies metrical duality. Groups of automorphisms of the sets of bent functions and (anti-)self-dual bent functions are discussed. The solution to the problem of preserving bentness and the Hamming distance between bent function and its dual within automorphisms of the set of all Boolean functions in n variables is considered
Bent functions, SDP designs and their automorphism groups
PhD ThesisIn a 1976 paper Rothaus coined the term “bent” to describe a function f from a vector space V (n, 2) to F2 with the property that the Fourier coefficients of (−1)f have unit magnitude. Such a function has the maximum possible distance from the set of linear functions, hence the name, and has useful correlation properties. These lead to various applications to coding theory and cryptography, some of which are
described. A standard notion of the equivalence of two bent functions is discussed and related to the coding theory setting.
Two constructions mentioned by Rothaus and generalised by Maiorana are described. A further generalisation of one of these, involving sets of bent functions on direct summands of the original vector space, is described and proved. Various methods including computer searches are used to find appropriate sets of bent functions and hence many new equivalence classes of bent functions of 8 variables.
Equivalence class invariants are used to show that most of these classes cannot be constructed by the earlier methods. Some bounds on numbers of bent functions are discussed.
A 2-design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block — such a design is very close to being a 3-design. All SDP designs are induced by bent functions, and conversely. Work on the automorphism groups of various SDP designs involving computer algebra is described. An SDP design on 256 points with
trivial automorphism group is noted.
Some connections with strongly-regular graphs are discussed. An infinite class of pseudo-geometric strongly-regular graphs induced by bent functions is noted, and bent functions which are their own Fourier transform duals are investigated. Finally, some open problems and ideas for future work are described
On Self-Dual Quantum Codes, Graphs, and Boolean Functions
A short introduction to quantum error correction is given, and it is shown
that zero-dimensional quantum codes can be represented as self-dual additive
codes over GF(4) and also as graphs. We show that graphs representing several
such codes with high minimum distance can be described as nested regular graphs
having minimum regular vertex degree and containing long cycles. Two graphs
correspond to equivalent quantum codes if they are related by a sequence of
local complementations. We use this operation to generate orbits of graphs, and
thus classify all inequivalent self-dual additive codes over GF(4) of length up
to 12, where previously only all codes of length up to 9 were known. We show
that these codes can be interpreted as quadratic Boolean functions, and we
define non-quadratic quantum codes, corresponding to Boolean functions of
higher degree. We look at various cryptographic properties of Boolean
functions, in particular the propagation criteria. The new aperiodic
propagation criterion (APC) and the APC distance are then defined. We show that
the distance of a zero-dimensional quantum code is equal to the APC distance of
the corresponding Boolean function. Orbits of Boolean functions with respect to
the {I,H,N}^n transform set are generated. We also study the peak-to-average
power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove
that PAR_IHN of a quadratic Boolean function is related to the size of the
maximum independent set over the corresponding orbit of graphs. A construction
technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It
is finally shown that both PAR_IHN and APC distance can be interpreted as
partial entanglement measures.Comment: Master's thesis. 105 pages, 33 figure
Non-Boolean almost perfect nonlinear functions on non-Abelian groups
The purpose of this paper is to present the extended definitions and
characterizations of the classical notions of APN and maximum nonlinear Boolean
functions to deal with the case of mappings from a finite group K to another
one N with the possibility that one or both groups are non-Abelian.Comment: 17 page
On metric regularity of Reed-Muller codes
In this work we study metric properties of the well-known family of binary
Reed-Muller codes. Let be an arbitrary subset of the Boolean cube, and
be the metric complement of -- the set of all vectors of the
Boolean cube at the maximal possible distance from . If the metric
complement of coincides with , then the set is called a
{\it metrically regular set}. The problem of investigating metrically regular
sets appeared when studying {\it bent functions}, which have important
applications in cryptography and coding theory and are also one of the earliest
examples of a metrically regular set. In this work we describe metric
complements and establish the metric regularity of the codes
and for .
Additionally, the metric regularity of the codes and
is proved. Combined with previous results by Tokareva N.
(2012) concerning duality of affine and bent functions, this establishes the
metric regularity of most Reed-Muller codes with known covering radius. It is
conjectured that all Reed-Muller codes are metrically regular.Comment: 29 page
On metric complements and metric regularity in finite metric spaces
This review deals with the metric complements and metric regularity in the Boolean cube and in arbitrary finite metric spaces. Let A be an arbitrary subset of a finite metric space M, and A be the metric complement of A — the set of all points of M at the maximal possible distance from A. If the metric complement of the set A coincides with A, then the set A is called a metrically regular set. The problem of investigating metrically regular sets was posed by N. Tokareva in 2012 when studying metric properties of bent functions, which have important applications in cryptography and coding theory and are also one of the earliest examples of a metrically regular set. In this paper, main known problems and results concerning the metric regularity are overviewed, such as the problem of finding the largest and the smallest metrically regular sets, both in the general case and in the case of fixed covering radius, and the problem of obtaining metric complements and establishing metric regularity of linear codes. Results concerning metric regularity of partition sets of functions and Reed — Muller codes are presented
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