[Math] Point addition on an elliptic curve

elliptic-curves

I have an elliptic curve $y^2 = x^3 + 2x + 2$ over $Z_{17}$. It has order $19$.

I've been given the equation $6\cdot(5, 1) + 6\cdot(0,6)$ and the answer as $(7, 11)$ and I'm unsure how to derive that answer.

I have $6\cdot(5, 1) = (16,13)$ and $6\cdot(0,6)=(0, 11)$ however when I use point addition to add them together I get $(16,13)+(0, 11)=(14,11)$ which isn't even a point on the curve…

Could someone help me identify where and why I've gone wrong?

For further information here's each of the points:

enter image description here

And here's the curve plotted out: enter image description here

Best Answer

Hints:

  • Your calculation of $6(5,1) = (16,13)$ is correct.
  • Your calculation of $6(0,6) = (0,11)$ is incorrect, you should get $6(0,6) = (3,1)$.

Maybe if you show how you did that calculation, I can spot the issue.

Once you fix that, I verified the author's result is correct, that is:

$$(16,13)+(3,1) = (7,11)$$

Update

Here are some additional hints to help you with intermediate calculations:

  • $P = (0,6)$
  • $2P = (9,1)$
  • $3P = (6,3)$
  • $4P = (7,6)$
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