We have proposed a robust, secure and efficient image encryption algorithm based on chaotic maps and algebraic structure. Nowadays, the chaotic cryptosystems gained more attention due to their efficiency, the assurance of robustness and high sensitivity corresponding to initial conditions. In literature, there are many encryption algorithms that can simply guarantees security while the schemes based on chaotic systems only promises the uncertainty, both of them can not encounter the needs of current scenario. To tackle this issue, this article proposed an image encryption algorithm based on Lorenz chaotic system and primitive irreducible polynomial substitution box. First, we have proposed 16 different S-boxes based on projective general linear group and 16 primitive irreducible polynomials of Galois field of order 256, and then utilized these S-boxes with combination of chaotic map in image encryption scheme. Three chaotic sequences can be produced by the disturbed of Lorenz chaotic system corresponding to variables x, y and z. We have constructed a new pseudo random chaotic sequence k_i based on x, y and z. The plain image is encrypted by the use of chaotic sequence k_i and XOR operation to get a ciphered image. To show the strength of presented image encryption, some renowned analyses are performed.

The Advanced Encryption Standard (AES) is widely used in different kinds of security applications. The substitution box (S-box) is the main component of many modern symmetric encryption ciphers that provides confusion between the secret key and ciphertext. The S-box component that is used in AES is fixed. If we construct this component dynamically, the encryption strength of AES would be greater than before. In this manuscript, we used chaotic logistic map, Mobius transformation and symmetric group $S_{256}$ to construct S-box for AES. The idea behind the proposed work is to make supplementary safe S-box. The presented S-box is analyzed for the following analyses: linear approximation probability (LP), nonlinearity (NL), differential approximation probability (DP), strict avalanche criterion (SAC), and bit independence criterion (BIC). The analyses show that the proposed technique is useful in generating high resistance S-box to known attacks.

There is a need of secure and robust encryption algorithm that can be implemented on each and diverse software and hardware platforms. In this paper, we investigate semifield substitution boxes using permutation of symmetric group on a set of size 8 $$S_8$$S 8 and establish an effective procedure for generating $$S_8$$S 8 semifield substitution boxes having same algebraic properties. Further, the strength analysis of the generated substitution boxes is carried out using the well-known standards, namely bijectivity, non-linearity, strict avalanche criterion, bit independence criterion, XOR table and differential invariant. As application, an encryption algorithm is proposed that can be employed to strengthen any kind of secure communication. The presented algorithm is mainly based on the Shannon idea of substitution–permutation network where the process of substitution is performed by the proposed $$S_8$$S 8 semifield substitution boxes and permutation operation is performed by the binary cyclic shift of substitution box transformed data. In addition, the proposed encryption algorithm utilizes two different chaotic maps. To ensure the appropriate utilization of these chaotic maps, we carry out in-depth analyses of their behavior in the context of secure communication and apply the pseudo-random sequences of chaotic maps in the proposed image encryption algorithm accordingly.

In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities.

In this paper, we study a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. We introduce two convex quadratic relaxations (CQRs) and discuss cases, where the problem is equivalent to exactly one of the CQRs. Particularly, we show that the global optimal solution can be recovered from an optimal solution of the CQRs. Through this equivalence, we introduce new conditions under which the problem enjoys strong Lagrangian duality, generalizing the recent condition in the literature. Finally, under the new conditions, we present necessary and sufficient conditions for global optimality of the problem.