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I read about elliptic curve cryptography $E$ over $Z_p$ where $p$ is prime number and $G$ is a base point on the curve. I noticed the points resulting from multiplication e.g. $2G$,$3G$,.....,$(N-1)G$ are always be positive numbers for $x , y$ and don't contain any negative values.

Whats the reason for these results not containing any negative values?

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1 Answer 1

Actually, strictly speaking, the $x$ and $y$ values on an elliptic curve point aren't integers; instead, they are field elements. That is, the elliptic curve is defined in a field, which is a group of elements with addition and multiplication operations defined on them (along with a group of identities); the $x$ and $y$ values are members from these elements.

Now, you specify that you're doing a curve over $Z_p$; that field (often called $GF(p)$), which has precisely $p$ element, is most commonly represented by the integers between 0 and $p-1$. With this representation, we perform the addition and multiplication operations by taking the represention, adding or multiplying the two values together (using normal integer arithmetic), and then perform a modulo $p$ operation (which maps the value back into the range of 0 to $p-1$. If $p$ is prime, these operations satisfies all the required identifies of a field.

So, when you say that that points are always have positive numbers for $x, y$, what you're actually saying is that, using this standard representation, negative values never occur. That's trivially true because this representation only contains values between 0 and $p-1$, and never takes on negative values. We can easily define an alternative representation which does contain negative values; we just don't have any reason to.

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