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In this paper, a method for image encryption is proposed based on a novel approach using chaos and transcendental numbers for generating the substitution box and the ciphering keys. The Chaotic Pi Ciphering (CPC) is a robust symmetric cryptosystem of the substitution-permutation-network type, that ciphers images in ten rounds. The objective of CPC is to avoid some general weakness points in cryptosystems, such as the vulnerability to certain types of attacks or the encryption with data loss. The number of rounds can be increased up to 14, with an even further increment of the ciphering security. An 8×8 substitution box is constructed using chaos, and its value of the Differential Power Analysis is lower than the corresponding to the Advanced Encryption Standard (AES). It makes CPC more resistant to differential and linear attacks than AES. CPC has approximately 2512 keys, presenting a high resistance to brute-force attacks. Chaos is produced by a nonlinear differential equation, and the Walsh function is applied to measure the nonlinearity of the box. The transcendental number pi and a positive integer are employed to generate the schedule keys, and the key length is equal to the image size. The randomness of the encrypted images is evaluated with four tests: entropy, correlation, Discrete Fourier Transform, and a goodness-of-fit test using the chi square distribution χ2. CPC was tested using well-known images from the related literature, and its performance was compared to the corresponding to AES. The results show a good entropy and a high randomness, indicating a high-quality ciphering.

In this work, a proposal to quantify the security gap between Encryption-then-Compression (EtC) systems and the Substitution-Permutation Network (SPN) is presented. A new image transmission system for JPEG files, EtC with Quantization before Transformation (EtCQT) was developed to test the proposed comparison. SPN is a well-known model used for security comparisons but is incompatible with the lossy compression required for saving storage while transmitting images. EtC systems have been designed considering compression performance rather than security, and in most developments in this field, only compression parameters are evaluated. The proposed evaluation was carried out with the metrics used for SPN, and the results showed the weaknesses of the EtC systems by quantifying them. This could help to improve security while keeping compatibility in future designs. EtCQT is based on the EtC scheme combined with block scrambling, with a quantization step instead of a permutation in the color transformation stage. The impact of this modification is an important reduction of the original pixel values in the color distribution that increases the information security, preserving only $16.\overline {6}\%$ of the initial values compared to 50% in other EtC systems.

In this work, a novel symmetric cryptosystem for image encryption is presented. The Symmetric Cryptosystem of the Elliptic Curve (SCEC) can resist damage to encrypted figures, up to 40% of the original figure. SCEC uses chaos to generate an 8×8 S-box with high nonlinearity to avoid the linear attack. A random permutation is used before starting encryption, making the cryptosystem more robust. For testing, damage according to four types of noise was applied to the encrypted images: additive, multiplicative, Gaussian, or occlusion. The median filter was applied to correct the damage in encrypted images, improving its sharpness, and a new measure, the Similarity Parameter (SP), is proposed to evaluate the difference between the original image and the decrypted image with damage. Several parameters and tests were applied to evaluate the performance of SCEC, from the encryption quality to the resistance to the differential attack. Experimental results indicate that SCEC has high-quality cryptographic properties, very much similar to the corresponding values of AES but with the addition of a high protection to noise damage on the encrypted images.