Designing a 1D extended logistic map for a secure image encryption

Abstract

The performance of the keystream is an important index for determining the security of a cryptosystem. However, the parameter range of the classical one-dimensional chaotic map is limited, and the distribution of the generated chaotic sequence is aggregated. Using the keystream generated by such chaotic systems can lead to a small secret key space and low security of the cryptosystem. To address this, this paper proposes a 1D Extended Logistic Map (1D-ELM), which combines the Sin and Logistic maps. The 1D-ELM has the same control parameters as the Logistic map, but the parameter range in which 1D-ELM exhibits chaotic behavior is larger than that of the Logistic map. The performance of 1D-ELM is evaluated using the Lyapunov exponent, bifurcation diagram, and 0-1 test. Additionally, 1D-ELM is applied in image encryption. In the secret key generation stage, the designed cryptosystem employs a combination of public and private keys, ensuring the algorithm's security. An attacker needs to find two secret keys to crack the algorithm. Furthermore, in the diffusion stage, most ciphertext values are related to the three adjacent ciphertext values, reducing the correlation of adjacent ciphertext pixels. The experiment verifies that 1D-ELM has good application in image encryption and plays an important role. The designed cryptographic system has high security.