Watermarking based on discrete wavelet transform and q -deformed chaotic map

Abstract

Hierarchy of one-dimensional ergodic chaotic maps with Tsallis type of q-deformation are studied. We find that in the chaotic region, these maps with q-deformation are ergodic as the Birkhoff ergodic theorem predicts. q-deformed maps are defined as ratios of polynomials of degree N. Hence, by using the Stieltjes transform approach (STA), invariant measure is proposed. In addition, considering Sinai-Ruelle-Bowen (SRB) measure, Kolmogorov-Sinai (KS) entropy for q-deformed maps is calculated analytically. The new q-deformed scheme have ability to keep previous significant properties such as ergodicity, sensitivity to initial condition. By adding q-parameter to the hierarchy in order increase the randomness and oneway computation, we present a new scheme for watermarking. The introduced algorithm tries to improve the problem of failure of encryption such as small key space, encryption speed and level of security. To illustrate the effectiveness of the proposed scheme, some security analyses are presented. By considering the obtained results, it can be concluded that, this scheme have a high potential to be adopted for watermarking. It can be concluded that, the proposed novel watermarking scheme for image authentication can be applied for practical applications.

On <i>q</i>-deformed logistic maps

“…28-39] that the equality, jointly with the fact that the map has negative Schwarzian derivative, implies that x + 0 is LAS. We point out that equation (5) are the conditions for a period doubling bifurcation, where a LAS fixed point becomes unstable and a LAS periodic point of period two arises (see [30]).…”

“…In addition, q-deformations have been also studied with fractional systems (see [21,32]). Among the applications of q-deformations we can mention [5,27].…”

q-Análogos em Corpos Finitos: Definição, Propriedades Algébricas e Aplicação em Geradores de Números Pseudo-Aleatórios

“…A partir de (6), segue que [n] q , n ≥ 1 pode ser alternativamente definido pela relação de recorrência…”

“…Em outro contexto, alguns trabalhos vêm considerando a introdução da q-deformação em mapas caóticos reais, com o objetivo de investigar o efeito da q-deformação na dinâmica caótica [5], [6], [7], [8]. A introdução da q-deformação, portanto, incorpora um grau de liberdade adicional na dinâmica complexa gerada por mapas caóticos, sendo as q-deformações potenciais generalizações para agregar uma camada adicional de complexidade à dinâmica caótica.…”

Dynamics of observables in a q-deformed harmonic oscillator

“…An important application of q-deformed algebras in quantum information theory is to construct quantum logic gates, for e.g., using q-deformed harmonic oscillator algebras [12]. Digital watermarking using one-dimensional ergodic chaotic maps with Tsallis type of q-deformation has been developed recently [13]. In nanoscience, low temperature behaviour of deformed fermion gas models have been used to study the interactions of quasiparticles in nanomaterials [14].…”