Title: Points where x=y on secp256k1 Post by: garlonicon on July 11, 2023, 09:16:41 PM Are there any such points? For some elliptic curves with less bits they can be found. I wonder if there are any such points for secp256k1, and if so, how to calculate those public keys? Even without private keys (that could be hard to find in such huge space), calculating those public keys alone could be interesting.
For example, if we use "p=14071, n=13933, base=(1,3660)", then we can find those values: Code: n= 5744, x= 2318, y= 2318, y^2=12073 Title: Re: Points where x=y on secp256k1 Post by: digaran on July 11, 2023, 09:41:32 PM Quote so x^2=x^3+7 How exactly x^2 is = x^3+7? Is that a typo? Also have you ever seen x=y in secp256k1? A few months ago as I was playing around with xs and ys, I remember to see something like x=y, but since I had no clue what that could really mean, I just ignored it. Main question, what would be the result of finding such points on secp256k1? Title: Re: Points where x=y on secp256k1 Post by: odolvlobo on July 11, 2023, 11:18:00 PM Are there any such points? For starters, the real solution to y2 = x3+7 when x = y is -1.63109... I know that doesn't help, but I had fun looking it up. Title: Re: Points where x=y on secp256k1 Post by: DaCryptoRaccoon on July 12, 2023, 02:20:12 PM I see what you're getting at! In the equation
Code: y^2 = x^3 + 7, If we assume that x and y are equal, we can substitute that into the equation to simplify it to. Code: x^2 = x^3 + 7. When we're working with elliptic curve cryptography it's important to understand that we're dealing with points on the curve that satisfy both the x and y coordinates. Not every combination of x and y will be a valid point on the curve. For the secp256k1 curve which is the one used in Bitcoin. Code: y^2 = x^3 + 7 This is because the curve is defined over a finite field, and the coordinates (x, y) must satisfy the equation modulo a large prime number. So, while it's true that there may exist y-values corresponding to certain x-values that satisfy the equation, we need to check if both x and y fall within the valid range of values for the finite field defined by the curve's prime modulus. In your example of : Code: (1, sqrt(8)), Quote y^2 = x^3 + 7, It is important to note that it may not correspond to a valid point on the secp256k1 curve. To determine if a point (x, y) is on the curve, we must ensure that both x and y are valid values within the finite field defined by the curve's prime modulus. Title: Re: Points where x=y on secp256k1 Post by: CrunchyF on July 28, 2023, 06:54:38 PM No there is no point in secp256k1 where x==y
but there is one where x==y+1 x=103219894018170979103981239500535823206309202530631329673674059809050911020508 y=103219894018170979103981239500535823206309202530631329673674059809050911020507 (x**3+7)%P==(y**2)%P => True Title: Re: Points where x=y on secp256k1 Post by: interiawp on July 28, 2023, 07:21:58 PM This is wrong. You think x==y as pointy on curve.
Theo real value is y == modinv(x.n) or x == modinv(y,n) I have found only 5 points with this. One is satoshi pubkey Title: Re: Points where x=y on secp256k1 Post by: CrunchyF on July 28, 2023, 07:43:41 PM This is wrong. You think x==y as pointy on curve. Theo real value is y == modinv(x.n) or x == modinv(y,n) I have found only 5 points with this. One is satoshi pubkey Sorry i don't understand your post...and why I'm wrong I speak about coordinate in affine plan as Q->(x,y) Q : (103219894018170979103981239500535823206309202530631329673674059809050911020508,103219894018170979103981239500535823206309202530631329673674059809050911020507) or 04e43463c1a7b06b6e49f555d75238bd140690ee0f689fda75d87623e10acf95dce43463c1a7b06 b6e49f555d75238bd140690ee0f689fda75d87623e10acf95db (uncompresed pubkey) or 03e43463c1a7b06b6e49f555d75238bd140690ee0f689fda75d87623e10acf95dc is a perfect valid bitcoin pubkey Title: Re: Points where x=y on secp256k1 Post by: garlonicon on July 29, 2023, 04:29:33 AM Quote No there is no point in secp256k1 where x==y How do you know that? Is there any simple way to check, if for a given p-value, there is such point or not?Quote but there is one where x==y+1 Nice result! But how it was calculated?Quote This is wrong. You think x==y as pointy on curve. Because it should be a point on curve, exactly as specified in the first post:Code: n= 5744, x= 2318, y= 2318, y^2=12073 Title: Re: Points where x=y on secp256k1 Post by: ripemdhash on July 29, 2023, 09:03:14 AM No there is no point in secp256k1 where x==y but there is one where x==y+1 x=103219894018170979103981239500535823206309202530631329673674059809050911020508 y=103219894018170979103981239500535823206309202530631329673674059809050911020507 (x**3+7)%P==(y**2)%P => True So. if we are talking abouy the curve and we have Fp and N : field Fp is defined by : Fp = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F order: Fn = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141 your point x,y where x==103219894018170979103981239500535823206309202530631329673674059809050911020508 y==103219894018170979103981239500535823206309202530631329673674059809050911020507 it is real point as half_mod Fp/Fn , I got problem to explain my english is not so good to technical explain. Title: Re: Points where x=y on secp256k1 Post by: CrunchyF on July 29, 2023, 12:25:14 PM How do you know that? Is there any simple way to check, if for a given p-value, there is such point or not? x³+7=y² mod(P) or x³+7-y²=0 mod(P) if x=y then x³-x²+7=0 this equation is a polynomial of degree 3 in Finite Field and have no roots (solutions) Quote but there is one where x==y+1 Nice result! But how it was calculated? instead of looking for x=y we can find if roots exists replacing x=y+c in the polynomial equation where c in a constant varying between the range [-10;10] e.g This is my Sage script: Code: P=0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f We test G and -G to see if one corresponding to x==y+c Quote Now I only wonder, what algorithm is needed to get there? see above...Title: Re: Points where x=y on secp256k1 Post by: CrunchyF on July 29, 2023, 12:30:35 PM Main question, what would be the result of finding such points on secp256k1? Absolutely nothing because one Generator in not different from another in term of security Title: Re: Points where x=y on secp256k1 Post by: OneGoLuck on August 04, 2023, 12:51:39 PM Main question, what would be the result of finding such points on secp256k1? Absolutely nothing because one Generator in not different from another in term of security If you could find one "weak" generator, the security of the whole bitcoin would be broken That is why it's assumed there are no weak generators, as none has been found. And to be honest, I cant imagine why one would be weaker than all the others Title: Re: Points where x=y on secp256k1 Post by: digaran on August 04, 2023, 06:07:21 PM Can anyone explain the following?
Code: Lambda : 5363ad4cc05c30e0a5261c028812645a122e22ea20816678df02967c1b23bd72 What is the use of the keys above? Edit, so far I figured if we multiply a point by lambda, it will add our key to it and there won't be any multiplication and it also steals our y coordinate. So what is the use of that? About beta, I can't find anything meaningful in my results, is there any? Title: Re: Points where x=y on secp256k1 Post by: o_e_l_e_o on August 05, 2023, 09:20:54 AM Can anyone explain the following? These are the values for endomorphism on secp256k1. You can read the original post from Hal Finney deriving these values here: https://bitcointalk.org/index.php?topic=3238.msg45565#msg45565Lambda is such that Lambda^3 (mod N) = 1. Beta is such that Beta^3 (mod P) = 1. |