Dynamics analysis and cryptographic application of fractional logistic map

Abstract

Based on Lyapunov exponent, Schwarzian derivative, Shannon entropy and Kolmogorov entropy, we will firstly study chaos and bifurcation of fractional (semi-) logistic map (FLM). It is derived from fractional integration (not fractional derivative) of the classical logistic map. Then, this paper put forward a new accumulated coupled fractional (semi-) logistic map lattice (ACFLML) whose lattice function is the FLM. Local stability, pattern information, high-dimensional chaos and bifurcation of the ACFLML are analyzed based on stability theory, pattern simulation, 0–1 test for chaos, Lyapunov exponent spectrum and Kolmogorov entropy. Finally, the chaotic ACFLML is successfully applied to encryption and decryption of digital image. To evaluate security, histogram analysis, correlation analysis, differential attack, key space, key sensitivity, encryption time, computational complexity and chosen/known-plaintext attacks, analysis is performed. Simulation analysis shows that this encryption scheme is effective and has good statistical effect.