1
$\begingroup$

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?

$\endgroup$
1

1 Answer 1

2
$\begingroup$

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.