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 2 ⁵¹² 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 χ ² . 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 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.

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 16.6% of the initial values compared to 50% in other EtC systems.

The proposed algorithm called Image encryption based on chaos, elliptic curve, dynamic S-boxes and variable permutations (IECC) is a symmetric cryptosystem for encrypting bmp images, in this case 512 × 512 pixel figures are used for testing. This algorithm consists of 15 rounds with a different S-box for each round and each set of S-boxes is different for each image encryption. Based on the above, a permutation is applied before the first round, additionally, in round 15 the inverse permutation of the first round intervenes. Both the S-boxes and the permutations are built using the E. Lorenz equations, which obtain solutions with two different points of the elliptic curve chosen randomly in each process. To measure the resistance to differential attack of the proposed cryptosystem, the following measurements are carried out: Number of Pixels Change Rate - NPCR, Unified Average Changing Intensity - UACI. Also, taking into account that the generation of the boxes is dynamic, the algebraic attack is avoided; furthermore, with this symmetric cryptosystem it is possible to distribute keys and sign, since an asymmetric system is used in its construction. On the other hand, four types of noise are applied to the encrypted images to evaluate the resistance of the encryption algorithm to this type of attack. Ten instruments are used to measure the quality of encryption; namely: entropy, correlation, discrete Fourier transform, NPCR, UACI, Avalanche Criteria - AC, contrast, energy, homogeneity and a goodness of fit test is proposed using the χ² distribution.

This research proposes an algorithm to sign messages using the ElGamal asymmetric cryptosystem, called: Digital signature algorithm based on ElGamal, Diffie-Hellman protocol and the number π (DSAπ). This proposal has a high level of security like the DSA standard, in addition, execution times are reduced in relation to it. The Diffie-Hellman protocol and the number π were used in the construction of the digital signature algorithm in conjunction with the Hash-Sha-512 function. Some experiments are presented to provide evidence that the bits on the right side of the decimal point of π appear randomly, such as: a color image was constructed of a string of consecutive bits chosen at random, of the number π. Subsequently, parameters such as: entropy, correlation, discrete Fourier transform, goodness-of-fit test χ2 were evaluated. It is mentioned that an S-box 8×8 is involved in the signature. The sensitivity of the signature was measured, that is, when small changes are made to the sender’s private key, or small changes to the message. Finally, an image was constructed using the signatures of random texts, and subsequently the randomness of its pixels was obtained, to verify that there is no relationship between the different signatures.