8,308 research outputs found
Localized solutions of Lugiato-Lefever equations with focused pump
Lugiato-Lefever (LL) equations in one and two dimensions (1D and 2D)
accurately describe the dynamics of optical fields in pumped lossy cavities
with the intrinsic Kerr nonlinearity. The external pump is usually assumed to
be uniform, but it can be made tightly focused too -- in particular, for
building small pixels. We obtain solutions of the LL equations, with both the
focusing and defocusing intrinsic nonlinearity, for 1D and 2D confined modes
supported by the localized pump. In the 1D setting, we first develop a simple
perturbation theory, based in the sech ansatz, in the case of weak pump and
loss. Then, a family of exact analytical solutions for spatially confined modes
is produced for the pump focused in the form of a delta-function, with a
nonlinear loss (two-photon absorption) added to the LL model. Numerical
findings demonstrate that these exact solutions are stable, both dynamically
and structurally (the latter means that stable numerical solutions close to the
exact ones are found when a specific condition, necessary for the existence of
the analytical solution, does not hold). In 2D, vast families of stable
confined modes are produced by means of a variational approximation and full
numerical simulations.Comment: 26 pages, 9 figures, accepted for publication in Scientific Report
On the Derivative Imbalance and Ambiguity of Functions
In 2007, Carlet and Ding introduced two parameters, denoted by and
, quantifying respectively the balancedness of general functions
between finite Abelian groups and the (global) balancedness of their
derivatives , (providing an
indicator of the nonlinearity of the functions). These authors studied the
properties and cryptographic significance of these two measures. They provided
for S-boxes inequalities relating the nonlinearity to ,
and obtained in particular an upper bound on the nonlinearity which unifies
Sidelnikov-Chabaud-Vaudenay's bound and the covering radius bound. At the
Workshop WCC 2009 and in its postproceedings in 2011, a further study of these
parameters was made; in particular, the first parameter was applied to the
functions where is affine, providing more nonlinearity parameters.
In 2010, motivated by the study of Costas arrays, two parameters called
ambiguity and deficiency were introduced by Panario \emph{et al.} for
permutations over finite Abelian groups to measure the injectivity and
surjectivity of the derivatives respectively. These authors also studied some
fundamental properties and cryptographic significance of these two measures.
Further studies followed without that the second pair of parameters be compared
to the first one.
In the present paper, we observe that ambiguity is the same parameter as
, up to additive and multiplicative constants (i.e. up to rescaling). We
make the necessary work of comparison and unification of the results on ,
respectively on ambiguity, which have been obtained in the five papers devoted
to these parameters. We generalize some known results to any Abelian groups and
we more importantly derive many new results on these parameters
Doubly Perfect Nonlinear Boolean Permutations
Due to implementation constraints the XOR operation is widely used in order
to combine plaintext and key bit-strings in secret-key block ciphers. This
choice directly induces the classical version of the differential attack by the
use of XOR-kind differences. While very natural, there are many alternatives to
the XOR. Each of them inducing a new form for its corresponding differential
attack (using the appropriate notion of difference) and therefore block-ciphers
need to use S-boxes that are resistant against these nonstandard differential
cryptanalysis. In this contribution we study the functions that offer the best
resistance against a differential attack based on a finite field
multiplication. We also show that in some particular cases, there are robust
permutations which offers the best resistant against both multiplication and
exponentiation base differential attacks. We call them doubly perfect nonlinear
permutations
- …