The aim of this work is to synthesize 8*8 substitution boxes (S-boxes) for block ciphers. The confusion creating potential of an S-box depends on its construction technique. In the first step, we have applied the algebraic action of the projective general linear group PGL(2,GF(28)) on Galois field GF(28). In step 2 we have used the permutations of the symmetric group S256 to construct new kind of S-boxes. To explain the proposed extension scheme, we have given an example and constructed one new S-box. The strength of the extended S-box is computed, and an insight is given to calculate the confusion-creating potency. To analyze the security of the S-box some popular algebraic and statistical attacks are performed as well. The proposed S-box has been analyzed by bit independent criterion, linear approximation probability test, non-linearity test, strict avalanche criterion, differential approximation probability test, and majority logic criterion. A comparison of the proposed S-box with existing S-boxes shows that the analyses of the extended S-box are comparatively better.

This paper deals with the study of generalizations of fuzzy subalgebras and fuzzy ideals in BCK/BCI-algebras. In fact, the notions of ∈ , ∈ ∨ κ ~ ∗ , q κ ~ -fuzzy subalgebras, ∈ , ∈ ∨ κ ~ ∗ , q κ ~ -fuzzy ideals, and ∈ ∨ κ ~ ∗ , q κ ~ , ∈ ∨ κ ~ ∗ , q κ ~ -fuzzy ideals in BCK/BCI-algebras are introduced. Some examples are provided to demonstrate the logic of the concepts used in this paper. Moreover, some characterizations of these notions are discussed. In addition, the concept of ∈ , ∈ ∨ κ ~ ∗ , q κ ~ -fuzzy commutative ideals is introduced, and several significant characteristics are discussed. It is shown that for a BCK-algebra A , every ∈ , ∈ ∨ κ ~ ∗ , q κ ~ -commutative ideal of a BCK-algebra is an ∈ , ∈ ∨ κ ~ ∗ , q κ ~ -fuzzy ideal, but the converse does not hold in general; a counter example is constructed.

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In this paper, the notions of ∈ , ∈ ∨ q -fuzzy soft BCK / BCI -algebras and ∈ , ∈ ∨ q -fuzzy soft sub- BCK / BCI -algebras are introduced, and related properties are investigated. Furthermore, relations between fuzzy soft BCK / BCI -algebras and ∈ , ∈ ∨ q -fuzzy soft BCK / BCI -algebras are displayed. Moreover, conditions for an ∈ , ∈ ∨ q -fuzzy soft BCK / BCI -algebra to be a fuzzy soft BCK / BCI -algebra are provided. Also, the union, the extended intersection, and the “AND”-operation of two ∈ , ∈ ∨ q -fuzzy soft (sub-) BCK / BCI -algebras are discussed, and a characterization of an ∈ , ∈ ∨ q -fuzzy soft BCK / BCI -algebra is established.