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April 9th, 2014, 05:04 AM   #1
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Elliptic curve over finite field

Hello guys, I am doing a project about elliptic curve cryptography over finite field. The equation used is : y^2 ≡ x^3+ax+b (mod p). I want to plot the coordinates of that equation in finite field. I found referance which stated that:

Ep (a,b) are the sets of coordinates, with x, y ∈ Zp, such that the equation y^2 = x^3+ax+b with a, b ∈ Zp is satisfied modulo p and such that the condition : 4a^3+27b not equal to zero.

Zp = prime finite field Zp
Ep(a,b) = set of coordinates x,y

1. I want to ask what does it mean by x, y ∈ Zp and a, b ∈ Zp? Does it mean that x,y,a,b must be a prime number?
2. is the following condition necessary : 4a^3+27b != 0 ?
2. Also, what the equation y^2 ≡ x^3+ax+b (mod p) means? is it the same as : (y^2) mod p = (x^3+ax+b) mod p?

I really need explanation for this. Thanks
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April 9th, 2014, 06:03 AM   #2
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Quote:
Originally Posted by Kentut View Post
1. I want to ask what does it mean by x, y ∈ Zp and a, b ∈ Zp? Does it mean that x,y,a,b must be a prime number?
$\mathbb Z_p$ is the field consisting of $0,1,\ldots,p-1$ under addition and multiplication modulo $p$ (a prime). The variables $x,y$ and parameters $a,b$ must belong to this field.


Quote:
Originally Posted by Kentut View Post
2. is the following condition necessary : 4a^3+27b != 0 ?
Yes (that’s what the question explicitly states). By the way $0$ here means $0\pmod p$, in other words $4a^3+27b$ must not be a multiple of $p$.


Quote:
Originally Posted by Kentut View Post
2. Also, what the equation y^2 ≡ x^3+ax+b (mod p) means? is it the same as : (y^2) mod p = (x^3+ax+b) mod p?
Yes.
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