# For an elliptic curve, what is the difference between the base field modulus $Q$ and subgroup $r$

What is the difference between the basefield modulus $$Q$$ and a subgroup of prime order $$r$$?

They are all fields, but what is their relevance to the curve they are defined upon?

How does this relate to a scalar field?

Also: Which field is the curve defined upon?

You are asking about a subset of Elliptic Curve (EC) domain parameters over $$\mathbb{F}_p$$ or $$\mathbb{F}_{2^m}$$

Domain parameters of Elliptic Curves over $$\mathbb{F}_p$$ is the sextuple:

$$T= (p,a,b,G,n,h)$$

• The integer $$p$$ defines the field $$\mathbb{F}_p$$,
• $$a$$ and $$b$$ defines the curve equation in Weierstrass form: $$E: y^2 \equiv x^3 + a \cdot x + b \pmod{p}$$
• $$G=(x_g,y_g)$$ defines the base point
• a prime $$n$$ which is order of $$G$$
• and an integer $$h = \#E(\mathbb{F}_p)/n$$

So, first choose an EC, then we choose a $$G$$ that generates a subgroup of order $$n$$.

They are all fields, but what is their relevance to the curve they are defined upon?

EC is an Algebraic Group but it is defined over the Field $$\mathbb{F}_p$$

How does this relate to a scalar field?

We have scalar (point) multiplication to represent the $$k$$ addition of a point $$P$$ by itself as $$kP$$.

Which field is the curve defined upon?

Above $$\mathbb{F}_p$$, it can be also $$\mathbb{F}_{2^m}$$, but these are for general Cryptographic purposes. EC's can be defined over the Complex, too.