Deriving of the y-coordinate on secp256k1 elliptic curve

Question

slope (s) = 75359724952757581356787150982223137069297282952438962440225639443485944634025
Qx = 112711660439710606056748659173929673102114977341539408544630613555209775888121
Qy = 25583027980570883691656905877401976406448868254816295069919888960541586679410

Derived x and y-coordinate from performing point doubling using the above values

Using the below formula;

s = (3 * Qx**2) * pow(2 * Qy, -1, p) % p 
Rx = (s**2 - 2 * Qx) % p
Ry = (s * (Qx - Rx) - Qy) % p

Rx = 115780575977492633039504758427830329241728645270042306223540962614150928364886
Ry = 78735063515800386211891312544505775871260717697865196436804966483607426560663

Secp256k1 elliptic curve perameters

Gx = 55066263022277343669578718895168534326250603453777594175500187360389116729240
Gy = 32670510020758816978083085130507043184471273380659243275938904335757337482424
p = 115792089237316195423570985008687907853269984665640564039457584007908834671663

The above Rx and Ry vairables is derived from slope (s), Qx and Qy by performing point doubling on elliptic curve and all the vairables provided above are valid points on the curve.

Is there any formula that can be used to derive Ry base solely on the slope (s), Gx, Gy, p and Rx?

Answer 1

Given $Q = (Q_x, Q_y)$ the doubling formula under the condition that $Q_y \neq 0$

\begin{align} s & =\frac{3 Q_x^2+a}{2Q_y} &(1)\\ R_x & = s^2 -2 Q_x \mod p & (2)\\ R_y & = s\cdot (Q_x-R_x) -Q_y \mod p & (3) \end{align}

So, basically asking;

• given $s, G_x, G_y, p$, and $R_x$ find $R_y$;

Then, can we find the $R_y$? let see..

1. Find $Q_x$ using (2)

$$ \color{red}{Q_x} = -\frac{R_x - s^2}{2} \mod p $$

2. Use this in (1)

   $$s = \frac{3 \color{red}{Q_x}^2+a}{2Q_y}$$

   to get $Q_y$

   $$\color{blue}{Q_y} = \frac{3 \color{red}{Q_x}^2+a}{2s}$$

3. Use (3) to get $R_y$

   $$R_y = s \cdot (\color{red}{Q_x}-R_x) -\color{blue}{Q_y} \mod p,$$

   where we know $p, s, R_x, Q_x$, and $Q_y$ so that we can find $R_y$