In this paper, firstly, we introduce the XOR product and XNOR product of soft sets. In practice, these products lead to simplicity and brevity. We discuss new algebraic structures such as group, ring, isomorphic rings, lattice and MV algebra on the soft sets using the emerging soft products and some present soft operations like intersection, union and complement. Moreover, the axioms for uncertainty measurements among the soft sets are proposed, and then, novel categories of the measures of similarity and distance between two soft sets are presented. The comparison and performance analysis of new measures with some of soft set measures in the literature are provided; thus, it is shown that the proposed ones decide correctly. Finally, we deal with the soft cryptosystem to illustrate the potential application of the new structures.