In this work we introduce a new method of cryptography based on the matrices over a finite field Fq, were q is a power of a prime number p. The first time we construct the matrix M = A 1 A 2 0 A 3 were A 1 , A 2 and A 3 are matrices of order n with coefficients in Fq and 0 is the zero matrix of order n. We prove thatA 2 A k 3 for all l ∈ N * . After we will make a cryptographic scheme between the two traditional entities Alice and Bob.

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Let N be a prime left near-ring, and I be a nonzero semigroup ideal of N. We prove that if N admits a derivation d satisfying any one of the following properties: (i) d([x, y]) = [x, y], (ii) d([x, y]) = [d(x), y], (iii) [d(x), y] = [x, y], (iv) d(x•y) = x•y, (v) d(x)•y = x•y and (vi) d(x)•y = x•y for all x, y ∈ I, then N is a commutative ring. Moreover, example proving the necessity of the primeness condition is given.

In this paper, we investigate generalized derivations satisfying certain differential identities on semigroup ideals of right near-rings and discuss related results. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

In this paper it is shown that zero symmetric prime left near-rings satisfying certain identities involving semiderivations are commutative rings.