Question about polynomials

AARRRGH

Hello

I have a question if you don't mind.

There is an equation:

y2 = x3 + ax + b (that's y squared and x cubed)

x and y can be real numbers. This makes perfect sense to me.

But x and y can also be polynomials.

I don't understand this. Does this mean that something like:

if (x = x3 + x2 + 1) and (y = x2 + 6)

The equation can be rewritten as:

(x2 + 6)2 = (x3 + x2 + 1)3 +a(x3 + x2 + 1) + b

Or does it mean something completely different?

Thanks!

Fremen

Yep, that sounds right to me. Do you have a web link I could take a look at?

I'm not 100% certain, because sometimes the notation (f squared) is used to denote composition, rather than multiplication, i.e. f2 = f(f(x)) rather than f(x)*f(x)

Most of the maths you'll cover when studying elliptic curves is more general than what you would have seen with real numbers. My guess is that here, you're working with fields.

In very vague terms, a field is a collection of "things", where you can take two "things" and add, multiply or divide them, to produce a new "thing". The definition of addition and multiplication mightn't be what you would see with the real numbers.

Michael Collins

x and y can be real numbers (polynomials of degree 0), polynomials of any degree > 0, or pretty much anything, once it satisfies the equality. If there's a particular example you're thinking about, post it up - your question is kinda mysterious!

AARRRGH

Thanks for the replies.

Yes, this is for elliptic curve cryptography and it is finite field maths (I think).

From what I understand:

y2 = x3 + ax + b is an equation for an elliptic curve.

y and x can be any finite field, for example, a real number modulo a prime.

They can also be polynomials or normal basis. I don't really get this.

It's easy to visuallise an elliptic curve when I am dealing with numbers, but if I am dealing with a polynomial, my brain hurts.

Does it sound like I am misunderstanding?

I always thought a polynomial was an equation like x + 2, or something like x + 2 = 0. If I am trying to find out what x is, replacing x with a polynomial just seems weird...

Fremen

Don't try to picture it, you really just have to trust the maths here.