61 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
Rayleigh quotients of Dillon's functions
The Walsh--Hadamard spectrum of a bent function uniquely determines a dual
function. The dual of a bent function is also bent. A bent function that is
equal to its dual is called a self-dual function. The Hamming distance between
a bent function and its dual is related to its Rayleigh quotient. Carlet,
Danielsen, Parker, and Sole studied Rayleigh quotients of bent functions in
, and obtained an expression in terms of a character sum.
We use another approach comprising of Desarguesian spreads to obtain the
complete spectrum of Rayleigh quotients of bent functions in
The Good lower bound of Second-order nonlinearity of a class of Boolean function
In this paper we find the lower bound of second-order nonlinearity of Boolean function with , and . It is also demonstrated that the lower bound obtained in this paper is much better than the lower bound obtained by Iwata-Kurosawa \cite{c14}, and Gangopadhyay et al. (Theorem 1, \cite{c12})
On construction of non-normal Boolean functions
Given two non-weakly -normal Boolean functions on variables a method is proposed to construct a non-weakly -normal Boolean function on variables
A New Class of Bent--Negabent Boolean Functions
In this paper we develop a technique of constructing bent--negabent
Boolean functions by using complete mapping polynomials. Using this
technique we demonstrate that for each there exits
bent--negabent functions on variables with algebraic degree
. It is also demonstrated that there exist
bent--negabent functions on variables with algebraic degrees
, and
On Kasami Bent Functions
It is proved that no non-quadratic Kasami bent is affine equivalent to Maiorana-MacFarland type bent functions
On lower bounds of second-order nonlinearities of cubic bent functions constructed by concatenating Gold functions
In this paper we consider cubic bent functions obtained by Leander and McGuire
(J. Comb. Th. Series A, 116 (2009) 960-970) which are
concatenations of quadratic Gold functions.
A lower bound of second-order nonlinearities of these
functions is obtained. This bound is compared with the lower
bounds of second-order nonlinearities obtained for functions
belonging to some other classes of functions which are recently
studied
On lower bounds on second--order nonliearities of bent functions obtained by using Niho power functions
In this paper we find a lower bound of the second-order nonlinearities
of Boolean bent functions of the form ,where and are Niho exponents. A lower bound of the second-order nonlinearities of these Boolean functions can also be obtained by using a result proved by Li, Hu and Gao (eprint.iacr.org/2010 /009.pdf). It is demonstrated that for large values of the lower bound obtained in this paper are better than the lower bound obtained by Li, Hu and Gao
Generalized Boolean Functions and Quantum Circuits on IBM-Q
We explicitly derive a connection between quantum circuits utilising IBM's
quantum gate set and multivariate quadratic polynomials over integers modulo 8.
We demonstrate that the action of a quantum circuit over input qubits can be
written as generalized Walsh-Hadamard transform. Here, we derive the
polynomials corresponding to implementations of the Swap gate and Toffoli gate
using IBM-Q gate set.Comment: 7 pages, 8 figures, Accepted to Publish in: 10th International
Conference on Computing, Communication and Networking Technologies and IEEE
Xplor
On Learning with LAD
The logical analysis of data, LAD, is a technique that yields two-class
classifiers based on Boolean functions having disjunctive normal form (DNF)
representation. Although LAD algorithms employ optimization techniques, the
resulting binary classifiers or binary rules do not lead to overfitting. We
propose a theoretical justification for the absence of overfitting by
estimating the Vapnik-Chervonenkis dimension (VC dimension) for LAD models
where hypothesis sets consist of DNFs with a small number of cubic monomials.
We illustrate and confirm our observations empirically
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