# Understanding elliptic curve point addition over a finite field

I am new to elliptic curve cryptography as well as finite field theory. I am trying to understand point addition in affine coordinates.

I understand, that for an elliptic curve $y^{2}=x^{3}+ax+b$ over $\mathbb R$ the sum of two points $P=(x_{p},y_{p})$ and $Q=(x_{q},y_{q})$ is $R=(x_{r},y_{r})$: $$x_{r}=\lambda^{2}-x_{p}-x_{q}$$ $$y_{r}=\lambda(x_{p}-x_{r})-y_{p}$$

with the slope $$\lambda=\frac{y_{q}-y_{p}}{x_{q}-x_{p}}$$

Excluding the cases: $P=Q$ (e.g. tangent slope), $P=0$ and $Q=0$ (e.g. $R=0$). If however the elliptic curve is defined over a finite field with prime size $n$: $$y^{2}=x^{3}+ax+b\pmod n$$

Can I just compute the slope for the "standard case" as follows (source: Slide 6)?

$$\lambda=\frac{y_{q}-y_{p}}{x_{q}-x_{p}} \pmod n$$

I understand that for an element in a finite field (f.e. point $P=(x_{p},y_{p})$) amongst other things an multiplicative inverse has to exist. However the formula for the slope $\lambda$ does only include coordinates of the point, not the element itself.

• The formula for $\lambda$ w/o the $\bmod n$ is exactly the same, the modular reduction is just implicit (if this was your question), assuming you're operating over a prime field. – SEJPM Mar 1 '17 at 14:39
• Your formula for $\lambda$ has numerator and denominator swapped (in both instances). – yyyyyyy Mar 1 '17 at 14:54

Your coordinates are elements of the finite field. For a point $P = (x_p, y_p)$ with $x_p,y_p \in \mathbb{F}_n$, where $\mathbf{F}_n$ is the finite field of order $n$ over which the elliptic curve is defined.
Since $n$ is prime, your slope will also be an element of the field $\mathbb{F}_n$. Even if $n$ was composite and $\gcd(x_q - x_p, n) = 1$, $\lambda$ is still an element of your field since an inverse of $(x_q - x_p)$ would still exist in this case. To compute the slope, you must find the multiplicative inverse of $(x_q-x_p)$ in your field $\mathbb{F}_n$. Then $$\lambda \equiv (y_q - y_p)\cdot(x_q - x_p)^{-1} \pmod n \text.$$
• Thank you. So if all coordinates, such as $x_p,y_p$ are $\in \mathbf{F}_n$, then every calculation has to be done in the field $\mathbf{F}_n$? Since however $(x_q - x_p)$ are not points I can use "normal" subtraction / addition here, right? Also including the modulo operation already here is possible to calculate with smaller numbers, but not necessary? Like this: $(x_q - x_p) \pmod n$ – floyd Mar 1 '17 at 17:22
• @floyd Be sure every operation is done modular $n$. – shaun1010 Mar 1 '17 at 20:53
• As $n$ is a prime larger than both $x_p,x_q$ this won't happen unless $x_p=x_q$ and in that case you're trying to do a point-doubling and there's a different formula for that. – SEJPM Jun 2 '19 at 14:38