So I’ve been digging into ECDSA for a bit and found something intriguing. There’s a structural shortcut that seems to be under the radar.
When I looked at the signature equation and thought of it like a 2D rectangle sum, I realized we could break down both axes for a bidirectional approach.
After some math, I got this representation that hasn’t been documented much. Just wanted to share what I found!
Nice find! So you’re saying that knowing the number K doesn’t give away KG since KG is a point on the curve? That makes sense. The modular equations do hold up, but you need K to hit a curve point where the X-Coord is R.
Your three-component breakdown seems to fit, but guessing a private key still looks like a long shot, right? What are the odds?
Thanks for the insights! I see where you're coming from. Totally agree that without the EC relationship (kG leading to R.x = r), just checking scalars doesn’t cut it. Hitting a valid key at random on secp256k1 is practically impossible.
But consider this: your equation h = r(k − d) + (s − r)d + (s − r)(k − d) clearly highlights the links between k and d in multiple terms, rather than just a simple line. It might open new avenues to explore.