Extension of elliptic curves on Krasner hyperfields

Derived Hyperstructures from Hyperconics

“…(i) For connecting it to number theory, incidence geometry, and geometry in characteristic one [8][9][10]. (ii) For connecting it to tropical geometry, quadratic forms [11,12] and real algebraic geometry [13,14]. (iii) For relating it to some other objects see [15][16][17][18][19].…”

“…There are different kinds of curves that basically are used in cryptography [34,35]. An elliptical curve is a curve of the form y 2 = p(x), where p(x) is a cubic polynomial with no-repeat roots over the field F. This kind of curves are considered and extended over Krasner's hyperfields in [13]. Now let g(x, y) = ax 2 + bxy + cy 2 + dx + ey + f ∈ F[x, y] and g(x, y) = 0 be the quadratic equation of two variables in field of F, if a = c = 0 and b = 0 then the equation g(x, y) = 0 is called homographic transformation.…”

Superring of Polynomials over a Hyperring

“…Vougiouklis studied the fundamental relation in hyperrings and the general hyperfield in his paper [12]. Extension of elliptic curves on Krasner hyperfields was studied in [13].…”

“…In 1956, Krasner introduced the notion of the hyperfield in order to define a certain approximation of a complete valued field by sequences of such fields [14]. Krasner's hyperfield is based on the generalization of the additive group in a field by the structure of a special hypergroup.…”

“…Later on, this hypergroup was named by Mittas "canonical hypergroup" [15]. The hyperfield that appears in [14] was named by Krasner "residual hyperfield". Krasner also introduced the hyperring, which is related to the hyperfield in the same way as the ring is related to the field.…”