To achieve additive homomorphism, the decryption structure with respect to adding [c.sub.1] and [c.sub.2] is required to keep the structure as x([m.sub.1]+[m.sub.2]) + [e.sup.+], where [e.sup.+] is the noise in the sum and x is an unknown variable.

Let f : [G.sub.1] [right arrow] [G.sub.2] be a homomorphism. Since [G.sub.2] is subgraph of [G.sub.1][bar.V][G.sub.2], then there exists a homomorphism r : [G.sub.1][bar.V][G.sub.2] [right arrow] [G.sub.2] with r(x) = x, for any vertex x of [G.sub.2] and so [G.sub.2] is a retract of [G.sub.1][bar.V][G.sub.2].

A set-valued mapping H : [Q.sub.t] [right arrow] [P.sup.*]([Q'.sub.t]), where [P.sup.*]([Q.sub.t]) represents the collection of all nonempty subsets of [Q'.sub.t], is called a set-valued homomorphism if, for all [a.sub.i], a, b [member of] [Q.sub.t] (i [member of] I),

Our proposed algorithm consists of four components: image encryption with homomorphism, data embedding with additive homomorphism, data extraction, and image recovery.

Let f: R [right arrow] S be a ring homomorphism and let [mu] be a fuzzy ideal of R such that [mu] is constant on Ker f and let [xi] be a fuzzy ideal of S.

In [3], Eshaghi Gordji introduced the concept of an n-Jordan homomorphism. A linear map [phi] between Banach algebras A and B is called an n-Jordan homomorphism if