Constructing a non-degenerate nD hyperchaotic map based on elliptic curve over GF(p)

To overcome the problem of reversibility in chaotic maps over Galois field, a model of non-degenerate nD (n ≥ 2) hyperchaotic map combined with elliptic curve (nD-ECHCM) over Galois field is designed, which embedded with the Elliptic Curve Discrete Logarithm Problem (ECDLP). First, an nD hyperchaotic map (nD-HCM) over GF(p) is designed based on the linear congruential generator (LCG). Due to the reversibility of nD-HCM, a model of irreversible nD-ECHCM is designed, which can construct any non-degenerate nD hyperchaotic map. Furthermore, the global boundedness of the Lyapunov exponents and hyperchaotic behavior of nD-ECHCM are proven, and the irreversibility of nD-ECHCM can be ensured by the embedding of ECDLP. Finally, to verify the effectiveness of nD-ECHCM, two instances are provided to analyze their dynamic behaviors. Experimental results demonstrate that the hyperchaotic maps over Galois field constructed by nD-ECHCM exhibit ergodicity and irreversibility.