A novel chaotic Umbrella map and its application to image encryption

In this paper a new nonlinear discrete chaotic Umbrella map is introduced. Chaotic behavior of the Umbrella map is investigated using the bifurcation diagrams, Lyapunov exponents, Chaos Decision Tree Algorithms, and sensitivity of parameters. The dynamical behavior of the proposed Umbrella shaped chaotic map is also investigated along with its fixed points and their stability. An image encryption algorithm is also proposed to show the potential application of the proposed map. The principle of proposed encryption algorithm is based on the double random phase encoding scheme. The Umbrella map is used for the pixel scrambling of the image and its parameters act as encryption keys of the cryptosystem. The proposed chaotic map is sensitive to initial values of its parameters. The proposed cryptosystem is simulated for various grayscale images and results of the "Baboon" image are presented in this paper. The efficacy of the proposed cryptosystem is also analyzed by various statistical measurements, such as, information entropy, correlation coefficient, and mean squared error. A detailed security analysis in terms of the robustness against various cryptographic attacks, such as, noise, occlusion, and brute-force attacks is also performed. The numerical simulation results indicate the robustness and effectiveness of the proposed cryptosystem.