Testing a Compact Version of secp256k1

Just getting into this stuff, so bear with me. I made a smaller secp256k1 with p=43, n=31. Took point G(2, 31) as the generator and got all valid points. I printed them out and stuck the paper on my wall. When working from home, I often glance at it looking for something interesting.

That's cool! So every Y has 3 corresponding X values, right? And if you check G for those points, you get a lot of connections. Pretty neat how this can apply to the usual secp256k1 that Bitcoin uses.

Yeah, exactly! Using the cube root of unity, each Y indeed has 3 X values (those are endomorphism points), and each X has 2 Y values (symmetry points). It's all interlinked and used in various tools for faster scanning, like VanitySearch.

I’m curious, how did you come up with G = (2, 31) for this curve? Is there some theorem that helps with this kind of setup?

Honestly, I just messed around with some prime numbers and values. I ended up finding that if you add all X values of the points and take mod p, it goes to 0.

Interesting! Like, if you use P=97, you get n=79? And then for Gx points, you can get some cool results too?

Exactly! I used G=(1, 69) and found that if I added the valid points, it cycles back to 0 when you mod it. Kinda funky, right? Maybe there's something useful here.

What’s the smallest example you’ve tried? I’ve been playing around with a curve p=7, n=13, and it’s wild how it rotates, plus that 13th point at infinity.

Smallest I've got is with p=7 and G=(1, 1). Just brute-forced some X values to get valid points. Then I saw that if (n-1) mod 6 = 0, it mirrors the bigger secp256k1.

Nice! So you’re saying the properties can be scaled? How do you ensure the points are all valid for that kind of construction?

I used a site that lets you input random p-prime numbers and find valid points. If you follow the rule for (n-1) mod 6, you get those sets of Y values.

Endomorphism is the key here! You can generate multiple public keys from one point using some constants. Got a script that does it all in one go.

Please share that script! I’d love to see how it works. The way you’ve described it sounds super useful.

Sure, here’s a snippet: from ecdsa.ellipticcurve import Point from ecdsa.curves import SECP256k1 # Secp256k1 parameters curve = SECP256k1.curve p = curve.p() n = SECP256k1.order G = SECP256k1.generator # Endomorphism constants beta =... lmbda =... # Function to apply endomorphism Let me know if you want the full code!

Thanks for sharing! This is great info. I think I’ll play around with it a bit more. Curious to see what else can come from this smaller secp256k1.

idk if scaling down secp256k1 this way really reflects how it works at full scale tho... smaller primes might miss some edge cases or weird behaviors. still cool to experiment but gotta be careful with what conclusions you draw