In EC starting with a primitive element (x,y) with x,y integers we get all the elements as integers pairs. Is this a general feature of all ellipitic curves or only of the ones we choose to use?
In the realm of Elliptic Curve Cryptography (ECC), the property mentioned, where starting with a primitive element (x,y) with x and y as integers, all subsequent elements are also integer pairs, is not a general feature of all elliptic curves. Instead, it is a characteristic specific to certain types of elliptic curves that are chosen for use in cryptographic applications.
Elliptic curves used in ECC are defined over a finite field, often denoted as Fp, where p is a prime number. The elements of this field are integers modulo p. The primitive element, also known as the generator point, is a special point on the elliptic curve that is chosen in such a way that it generates all other points on the curve when repeatedly added to itself.
When we perform scalar multiplication on the generator point, we obtain a set of points that form a subgroup of the elliptic curve. This subgroup is cyclic, meaning that every element in the subgroup can be expressed as a multiple of the generator point. In ECC, this subgroup is typically denoted as E(Fp), where E represents the elliptic curve and Fp represents the finite field.
The property mentioned, where all elements in the subgroup E(Fp) are integer pairs, arises due to the nature of the finite field Fp. Since the elements of Fp are integers modulo p, the coordinates of the points in the subgroup E(Fp) will also be integers modulo p. Hence, they can be represented as integer pairs.
However, it is important to note that this property does not hold for all elliptic curves. There are elliptic curves defined over other fields, such as binary fields or extension fields, where the elements of the field are not integers. In such cases, the coordinates of the points on the curve will not necessarily be integers, and the property mentioned would not hold.
To illustrate this, let's consider an example. Suppose we have an elliptic curve defined over the field F7, where the elements of the field are integers modulo 7. If we choose a primitive element (x,y) = (2,3) on this curve, and perform scalar multiplication, we will obtain a set of points that are all integer pairs. For instance, 2(2,3) = (5,1), 3(2,3) = (6,6), and so on. In this case, the property holds because the field F7 consists of integers modulo 7.
The property where all elements on an elliptic curve are integer pairs, starting from a primitive element with integer coordinates, is not a general feature of all elliptic curves. It is specific to elliptic curves defined over finite fields where the elements are integers modulo a prime number. Different elliptic curves defined over different fields may exhibit different properties.
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