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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?

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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.

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