The aim of the article is to initiate a new study in the framework of algebraic hyperfields, with an applicative impact in cryptography. First we define the notions of generalized Weierstrass equation and elliptic hypercurve on Krasner hyperfields, as a generalization of the notion of elliptic curve on fields. Then we investigate properties of the hypergroups derived from elliptic hypercurves and of the associated H_v-groups. Finally, we present a class of canonical hypergroups, which can be used as an alphabet in a special cryptographic system.

In this paper, we introduce generalized homographic transformations as hyperhomographies over Krasner hyperfields.These particular algebraic hyperstructues are quotient structures of classical fields modulo normal groups. Besides, we define some hyperoperations and investigate the properties of the derived hypergroups and $H_v$ -groups associated with the considered hyperhomographies. They are equipped hyperhomographies obtained as quotient sets of nondegenerate hyperhomographies modulo a special equivalence. Thus the symmetrical property of the equivalence relations plays a fundamental role in this constructions.

In this paper, we introduce generalized quadratic forms and hyperconics over quotient hyperfields as a generalization of the notion of conics on fields. Conic curves utilized in cryptosystems; in fact the public key cryptosystem is based on the digital signature schemes (DLP) in conic curve groups. We associate some hyperoperations to hyperconics and investigate their properties. At the end, a collection of canonical hypergroups connected to hyperconics is proposed.

In this article, we introduce and analyze a binary relation δ ⁿ on a hypergroup, which generalizes two important equivalence relations, denoted by β and γ. These two equivalence relations are important since the quotient structure is a group and an abelian group respectively. Moreover, we introduce and study the notion of a multisemi-direct hyperproduct of hypergroups, in terms of δ ^n*-equivalence classes.