In this paper, we construct a new substitution box (S-box) structure based on the elements of the maximal cyclic subgroup of the multiplicative group of units in a finite Galois ring instead of Galois field. We analyze the potency of the proposed S-box by using the majority logic criterion. Moreover, we illustrate the utility of the projected S-box in watermarking.

The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer n ≥ 2 , the nth-commutativity degree of a finite algebraic structure S, denoted by P n (S) , is the probability that for chosen randomly two elements x and y of S, the relator x n y = yx n holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the nth-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for nth-commutativity degree of these loops, we will obtain best upper bounds for this probability.

ResumoUm código BCH C (respectivamente, um código BCH C ) de comprimento n sobre o anel local Z p k (respectivamente, sobre o corpo Z p )é um ideal no anel(X n −1) (respectivamente, no anel Zp[X](X n −1) ), queé gerado por um polinômio mônico que divide X n − 1. Shankar [1] mostrou que as raízes de X n − 1 são as unidades do anel de. Neste estudo, assumimos que para s i = b i , onde bé um primo e ié um inteiro não negativo tal que 0 ≤ i ≤ t, existem extensões de anéis de Galois correspondentes GR(p k , s i ) (respectivamente, extensões do corpo de Ga- Um código BCH C (respectivamente, um código BCH C ) de comprimento n sobre um anel local Z p k (respectivamente, sobre o corpo Z p ), onde p e n são primos entre si,é um ideal no anel(X n −1) (respectivamente, no anel Zp[X](X n −1) ), queé gerado por um polinômio mônico que divide X n − 1 no anel(X n −1) (respectivamente, no * Os autores agradecem o apoio da Fapesp 2007/56052-8 e 2011/03441-2 e também da Propg -Unesp.

For a non negative integer t, let A0 ⊂ A1 ⊂ · · · ⊂ At−1 ⊂ At be a chain of unitary commutative rings, where each Ai is constructed by the direct product of suitable Galois rings with multiplicative group A * i of units, and K0 ⊂ K1 ⊂ · · · ⊂ Kt−1 ⊂ Kt be the corresponding chain of unitary commutative rings, where each Ki is constructed by the direct product of corresponding residue fields of given Galois rings, with multiplicative groups K * i of units. This correspondence presents four different type of construction techniques of generator polynomials of sequences of BCH codes having entries from A * i and K * i for each i, where 0 ≤ i ≤ t. The BCH codes constructed in [1] are limited to given code rate and error correction capability, however, proposed work offers a choice for picking a suitable BCH code concerning code rate and error correction capability.