N-dimensional non-degenerate chaos based on singular value estimation with application in dynamic DNA image encryption

Due to the truncation effects inherent in implementing chaotic systems on digital circuits, the limited precision reduces the sensitivity to initial conditions in chaotic systems. This ultimately leads to the overlap of adjacent trajectories, causing the system to transition from a chaotic state to either a periodic state or fixed point state. The Lyapunov exponent is a quantitative measure of chaotic characteristics, quantifying the mean degree of convergence and divergence among the various trajectories of a chaotic system. This paper introduces a construction algorithm for a N-dimensional non-degenerate chaotic system based on singular value estimation (NC-CSVE) from the perspective of controlling the Jacobian matrix of system, achieving indirect control of the Lyapunov exponents (LEs). Experimental analyses using lyapunov spectrum, joint entropy, sample entropy, and NIST tests indicate that compared to existing non-degenerate chaos construction methods, the system constructed by our method demonstrates superior performance in terms of chaotic dynamic characteristics, algorithm complexity, and construction time expenditure, meeting the needs of practical application scenarios. Additionally, to validate the feasibility and reliability of the proposed construction algorithm in practical applications, this paper introduces a new dynamic DNA color image encryption algorithm based on NC-CSVE. The encryption techniques include three-dimensional six-direction bit diffusion (TDSBD), bit-plane global shift and local rotation scrambling (B-GSLR), DNA calculation rule extension (DNA-CRE) and DNA dynamic mutation (DNA-DM), achieving fast encryption of color images. The results of the statistical tests indicate that the algorithm demonstrates higher efficiency and excellent resistance to attacks. The encrypted images approach ideal values in terms of information entropy, histograms and pixel correlation, confirming the effectiveness and reliability of the algorithm.