In this paper, we propose public key cryptography using recursive block matrices involving generalized Fibonacci numbers over a finite field $\mathbb{Z}_{p}$. For this, we define multinacci block matrices, a kind of upper triangular matrix involving multinacci matrices at diagonal places and give some of its algebraic properties. Moreover, we set up a method for key element agreement at end users, which makes cryptography more efficient. The proposed cryptography comes with a large key space and its security relies on the Discrete Logarithm Problem (DLP).

This study proposes a scale-invariant digital color image encryption method that includes three main steps the pre-substitution, the 3D scale-invariant modular chaotic map, and the post-substitution. 1) The pre-substitution: At the first stage, pixels of plain sub-images are XOR with different key patterns. By starting from one of the plain sub-images, the pixels of the selected plain sub-image is XOR with the initial key, and the result is used as the cipher sub-image and as the next key pattern for performing XOR operations on the next plain sub-image. In other words, the XOR result of each step is used as the next step key pattern, 2) the 3D permutation: At the second stage, first the red, green, and blue components of a color image is divided to m sub-images with size. Then m sub-images are partitioned into windows to with size sub-images. The last two ï؟ ¼ and ï؟ ¼ windows may be overlap in several sub-images. Finally, the three-dimensional modular chaotic maps are performed on the ï؟ ¼ to ï؟ ¼ windows with MIPF keys and selected by LFSR, 3) the post-substitution: At the final stage, the color image is initially divided into a set of color sub-images. Then, the 24-bit pixels of each sub-images are circularly shifted with several bits specified in the secret key. Modular arithmetic is used in the 3D scale-invariant chaotic maps to increase Keyspace and enhance security parameters. With repeat at least one round of main steps, the proposed encryption scheme reaches optimum parameter values and it is highly sensitive to minor differences in both secret key and plain image. The proposed encryption method for images improves the standard parameters of evaluation such as entropy, adjacent pixel correlations, histogram, and expanded Keyspace.

In this paper, we propose a generalized Lucas matrix (a recursive matrix of higher order) obtained from the generalized Fibonacci sequences. We obtain their algebraic properties such as direct inverse calculation, recursive nature, etc. Then, we propose a modified public key cryptography using the generalized Lucas matrices as a key element that optimizes the Keyspace construction complexity. Furthermore, we establish a key agreement for encryption-decryption with a combination of the terms of generalized Lucas sequences under the residue operation.

In this study, we propose a secure communication scheme based on the synchronization of two identical fractional-order chaotic systems. The fractional-order derivative is in Caputo sense, and for synchronization, we use a robust sliding-mode control scheme. The designed sliding surface is taken simply due to using special technic for fractional-order systems. Also, unlike most manuscripts, the fractional-order derivatives of state variables can be chosen differently. The stability of the error system is proved using the Lyapunov stability of fractional-order systems. Numerical simulations illustrate the ability and effectiveness of the proposed method. Moreover, synchronization results are applied to secure communication using the masking method. The security analysis demonstrates that the introduced algorithm has a large Keyspace, high sensitivity to encryption keys, higher security, and the acceptable performance speed.

This paper presents a novel image cryptosystem using a specific feature of boundary elementary cellular automata with permutation-diffusion architecture. The characteristics of some periodic boundary elementary cellular automata (CAs) with the length of 8 have been studied over the past years. The State-transition diagram shows that some elementary cellular automata (ECA) rule lead to some state attractors that are suitable to use to implement an encrypting function in order to transform the pixels’ values while meeting the basic requirements of the encryption scheme. Minimizing computational overhead, the production of these attractors could be done by using an initial global state of the CA solely, and the implementation of which needs no extra hardware cost. Correspondingly, the proposed new image cryptosystem includes an encryption method based on both permutations of the image pixels and the replacement of the pixel values. The permutation method is optional and can be done by every permutation algorithm. Simulation results revealed this fact that the proposed CA-based image cryptosystem shows strong performance in terms of encryption and decryption. The outstanding characteristics of the proposed image encryption method are as follows: lossless and symmetric private key encryption, low data expansion, the possibility of encryption when there is more than one image with the use of just one key image, large Keyspace, and checking data integrity.