Found an Arithmetic Relationship in secp256k1's G.x

Hey all, Did some digging into secp256k1 and stumbled upon this cool arithmetic relationship for the x-coordinate of the generator G: G.x = 27·(k₀·d' + r') where - D = 2³² + 977 - d' = D/27 = 159072899 - r' = (G.x % D)/27 = 15460270 - k₀ = 346169984758229267385003896202133930596503452506876071803407489043052 Seems like the odds of this being random are less than 1 in 10 billion. I checked out the first 2000 multiples of G and guess what? Only for k=1 does `(x//27)%d' == r'` hold.

I went all out testing, from 1 to 10 million. Range: 1 to 10000000. Hits: 371005. That's around 3.71%.

Not sure why you think this is groundbreaking. It just shows how the math works with the curve. You could swap in different numbers and it’d be similar.

Nah, that’s overthinking it. OP just found that G.x is a multiple of 27, which means it shows up about 3.71% of the time. No need to complicate things with advanced math.

Appreciate your thoughts but the fact that G.x is divisible by 27 doesn’t imply it’s trivial. There’s more context. Let me break it down: - D is chosen to be divisible by 27. - d' is a legit integer. - r' is derived straightforwardly from G.x.

Nope, your ‘discovery’ is basically saying G.x is divisible by 27 because you picked 27 deliberately since D is also divisible by it. Your r' is just a byproduct, so it’s not surprising that G.x satisfies the equation.

Let’s clarify. We know: - `D = 2³² + 977` - `d' = D/27` - `r' = (G.x % D) / 27` Now, G.x follows this integer equation: G.x = 27 * (k₀·d' + r') This isn’t just a coincidence; it’s a specific integer equation tied directly to G.

First, D is a constant and divisible by 27, that’s 100%. Also, if two numbers share a factor, their mod will always divide. This math has been known forever.

True, but let’s not sidestep the fact that 15460270 is actually a 24-bit number, not 27-bit as you said. Good catch. Also, you are right. It's misleading to imply D’s divisibility by 27 relates to a 1/27 probability without context.

Nah, it’s still 1 in 27, always. Just pick any integer and you’ll see that any of them will show this rule. This is true regardless of the curve.

Wait, so if x is a multiple of 27, then that x % D scenario automatically applies. So both conditions aren’t independent. That means finding integers fitting both just depends on the first.

I still don’t see your point. Just select any point where x % 27 == 0, compute an r', and claim it's special because it fits this equation. What's stopping you from doing that for any point?

Come on, this is just nonsense. You could create similar claims for any random point. It loses its meaning here. G does have unique properties, but not in the way OP is pushing.

I've scanned up to 100 billion points, looking for patterns or anomalies. The results were just random. Nothing stood out, all seemed typical to me.

+1 for hitting 100 billion, but it seems like just randomness in the curve.

Got to agree, it’s not groundbreaking stuff, just math as it is.

Any chance you can share more details on your findings? Sounds interesting.

Can you guys back up your claims with math? Or is it just a feeling?