69 research outputs found
Generalized bent Boolean functions and strongly regular Cayley graphs
In this paper we define the (edge-weighted) Cayley graph associated to a
generalized Boolean function, introduce a notion of strong regularity and give
several of its properties. We show some connections between this concept and
generalized bent functions (gbent), that is, functions with flat Walsh-Hadamard
spectrum. In particular, we find a complete characterization of quartic gbent
functions in terms of the strong regularity of their associated Cayley graph.Comment: 13 pages, 2 figure
A complete characterization of plateaued Boolean functions in terms of their Cayley graphs
In this paper we find a complete characterization of plateaued Boolean
functions in terms of the associated Cayley graphs. Precisely, we show that a
Boolean function is -plateaued (of weight ) if and only
if the associated Cayley graph is a complete bipartite graph between the
support of and its complement (hence the graph is strongly regular of
parameters ). Moreover, a Boolean function is
-plateaued (of weight ) if and only if the associated
Cayley graph is strongly -walk-regular (and also strongly
-walk-regular, for all odd ) with some explicitly given
parameters.Comment: 7 pages, 1 figure, Proceedings of Africacrypt 201
Twin bent functions, strongly regular Cayley graphs, and Hurwitz-Radon theory
The real monomial representations of Clifford algebras give rise to two
sequences of bent functions. For each of these sequences, the corresponding
Cayley graphs are strongly regular graphs, and the corresponding sequences of
strongly regular graph parameters coincide. Even so, the corresponding graphs
in the two sequences are not isomorphic, except in the first 3 cases. The proof
of this non-isomorphism is a simple consequence of a theorem of Radon.Comment: 13 pages. Addressed one reviewer's questions in the Discussion
section, including more references. Resubmitted to JACODES Math, with updated
affiliation (I am now an Honorary Fellow of the University of Melbourne
Strongly Regular Graphs Constructed from -ary Bent Functions
In this paper, we generalize the construction of strongly regular graphs in
[Y. Tan et al., Strongly regular graphs associated with ternary bent functions,
J. Combin.Theory Ser. A (2010), 117, 668-682] from ternary bent functions to
-ary bent functions, where is an odd prime. We obtain strongly regular
graphs with three types of parameters. Using certain non-quadratic -ary bent
functions, our constructions can give rise to new strongly regular graphs for
small parameters.Comment: to appear in Journal of Algebraic Combinatoric
Some results on uniform mixing on abelian Cayley graphs
In the past few decades, quantum algorithms have become a popular research
area of both mathematicians and engineers. Among them, uniform mixing provides
a uniform probability distribution of quantum information over time which
attracts a special attention. However, there are only a few known examples of
graphs which admit uniform mixing. In this paper, a characterization of abelian
Cayley graphs having uniform mixing is presented. Some concrete constructions
of such graphs are provided. Specifically, for cubelike graphs, it is shown
that the Cayley graph has uniform mixing if
the characteristic function of is bent. Moreover, a difference-balanced
property of the eigenvalues of an abelian Cayley graph having uniform mixing is
established. Furthermore, it is proved that an integral abelian Cayley graph
exhibits uniform mixing if and only if the underlying group is one of the
groups: , or
for some integers .
Thus the classification of integral abelian Cayley graphs having uniform mixing
is completed.Comment: 33 page
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