In the literature, different algebraic techniques have been applied on Galois field (2 8 ) to construct substitution boxes. In this paper, instead of Galois field (2 8 ), we use a cyclic group 255 in the formation of proposed substitution box. The construction proposed S-box involves three simple steps. In the first step, we introduce a special type of transformation of order 255 to generate 255 . Next, we adjoin 0 to 255 and write the elements of 255 ∪ {0} in 16 × 16 matrix to destroy the initial sequence 0, 1, 2, . . . , 255.In the 2 nd step, the randomness in the data is increased by applying certain permutations of the symmetric group 16 on rows and columns of the matrix. In the last step we consider the symmetric group 256 , and positions of the elements of the matrix obtained in step 2 are changed by its certain permutations to construct the suggested S-box. The strength of our S-box to work against cryptanalysis is checked through various tests. The results are then compared with the famous S-boxes. The comparison shows that the ability of our S-box to create confusion is better than most of the famous S-boxes.

This article aims to model two-dimensional, incompressible asymmetric peristaltic propulsion coated with Synovial fluid (“non-Newtonian model”) with mass transport. Due to the coating of the same base-fluid at the surface of the channel, the boundaries become non-porous and exert no slip on the fluid particles. Two illustrative models for the viscosity, namely, shear-thinning (Model 1) and shear-thickening (Model 2), are considered, which reveal the presence and integrity of coating. The perturbation method has been applied to linearize the complicated differential equations. Model 1 predicted higher viscosity values and more significant non-Newtonian behavior than Model 2. It is also observed that the shear-thinning model behaved in quite the opposite manner for the shear thickening model. The converse behavior of Model 1 and Model 2 occurs due to a curvature of the flow domain. Moreover, Model 1 is not able to capture the correct exponential viscosity dependence on concentration for the whole range of shear rates. On the other hand, the second model shows a strong relationship with accurate power. Solutions are attained for velocity field, concentration profile, and pressure gradient. The novelty of all the essential parameters is analyzed through graphical results. Furthermore, streamlines are also drawn to determine the trapping mechanism. The present analysis is beneficial in the study of intrauterine fluid dynamics; furthermore, it is applicable in vivo diagnostic; drug delivery; food diagnostics; protein chips; and cell chips and packaging, i.e., smart sensors.

Abstract.A ring A is called a chain ring if it is a local, both sided artinian, principal ideal ring. Let R be a commutative chain ring. Let A be a faithful R-algebra which is a chain ring such that A = A/J(A) is a separable field extension of R = R/J(R). It follows from a recent result by Alkhamees and Singh that A has a commutative R-subalgebra R 0 which is a chain ring such thatThe structure of A in terms of a skew polynomial ring over R 0 is determined. Let R be a commutative chain ring, and A be a local ring that is a faithful R-algebra. Then J(R) = R ∩ J(A). Let A = A/J(A) be a separable, algebraic field extension of R, and let A be either a locally finite Ralgebra or an artinian duo ring. As proved in [1], A has a commutative local R-subalgebra R 0 such thatThis subalgebra R 0 is also called a coefficient subring of A; such a subring is a commutative chain ring, and is a faithful R-algebra. The group of R-automorphisms of R 0 is investigated in Section 2. Wirt [8] introduced the concept of a distinguished basis of a bimodule over a Galois ring. In Section 3 an analogous concept for bimodules over R 0 is investigated.The main purpose of this paper is to prove a representation theorem for A, in case A is a chain ring, in terms of an appropriate homomorphic image of a skew polynomial ring over its coefficient subring. Sections 4 and 5 are devoted to proving the main theorem (Theorem 5.5). By Cohen [5], any