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Title: The mirror effect of public keys Post by: Kostelooscoin on October 21, 2021, 10:11:38 AM Does anyone know the public key mirror ?
Title: Re: The mirror effect of public keys Post by: Kostelooscoin on October 21, 2021, 10:17:54 AM here is the public key 00000000000000000000000000000000000000000000000000000000000000000001
0479be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8 0279be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798 and here is the mirror are fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364140 0479be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798b7c52588d95c3b9aa25b0403f1eef75702e84bb7597aabe663b82f6f04ef2777 0379be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798 how to explain this ? something else: according to this shema: 0=f 1=e 2=d 3=c 4=b 5=a 9=6 8=7 look at the text in red 0479be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08 ffb10d4b8 0479be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798b7c52588d95c3b9aa25b0403f1eef75702e84bb7597aabe663b82f6 f04ef2777 everything fits but the last and 0xb7c52588d95c3b9aa25b0403f1eef75702e84bb7597aabe663b82f6f04ef2777+0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f Title: Re: The mirror effect of public keys Post by: a.a on October 21, 2021, 11:42:14 AM This is the symmetry in secp256k1.
0479be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798483ada7726a3c 4655da4fbfc0e1108a8fd17b448a68554199c47d08 is the uncompressed privatekey of 1. The other one is the uncompressed privatekey of -1. -1 => N - 1 Title: Re: The mirror effect of public keys Post by: Kostelooscoin on October 21, 2021, 11:46:29 AM ok ;)
Title: Re: The mirror effect of public keys Post by: pooya87 on October 21, 2021, 02:22:43 PM If point P is computed like P = k.G then we have -P = -k.G.
For each elliptic curve point P(x,y) we define point negation as -P(x,y) = P(x,-y) and in modular arithmetic we know that -x ≡ N-x (mod N) if 0<= x < N so both -y and -k are in congruence with N-y and N-k respectively. Title: Re: The mirror effect of public keys Post by: Kostelooscoin on October 21, 2021, 04:31:27 PM If point P is computed like P = k.G then we have -P = -k.G. For each elliptic curve point P(x,y) we define point negation as -P(x,y) = P(x,-y) and in modular arithmetic we know that -x ≡ N-x (mod N) if 0<= x < N so both -y and -k are in congruence with N-y and N-k respectively. ok good Title: Re: The mirror effect of public keys Post by: NotATether on October 22, 2021, 06:12:12 AM Also see this:
There are actually 6 combinations so only 2^42.666666...7 unique combos. You have just discovered an endomorphism, where two different Y's will solve the curve equation: y^2 = x^3 + 7 i.e. these two privkeys make two different but opposite Y's (because 115792089237316195423570985008687907852837564279074904382605163141518161494336 is just n-1 or, equivalently [mod n: the cyclic group 0..n, n-1...2n, etc.], it is -1). And -1^2 = 1^2 = 1. Now the other 3 combinations - and why only 3? see the next section - come from the X term. Notice how the X is cubed which means it has three different roots if you consider it as a polynomial. There's obviously X, but there's also 0+Xi and 0-Xi (complex numbers). It follows the pattern [X + Yi], where the Y coord is an imaginary number. This goes to say that if e.g. (7,0) was a valid point, then that, (0,7) and (0,n-7) would all reference similar points. And (x,y), (y,x) and (y, n-x) would similarly reference similar points as well. Now multiply 2*3 combos (endomorphisms) and you get a total of 6 endomorphisms: (x,y), (y,x) (y, n-x) and (x,n-y), (n-y, x), (n-y, n-x). It isn't something like they'd all have the same Y-point, but these points are accessible from the same X-coordinate as well. (See e.g. Roots of x^3+7 example (https://www.wolframalpha.com/input/?i=roots+of+x%5E3%2B7)) There is 6-way symmetry on secp256k1 not just 2-way. Y-coordinate symmetry accounts for 3x the mirror symmetry you're talking about (which is 2). Title: Re: The mirror effect of public keys Post by: a.a on October 23, 2021, 04:20:32 PM So let's say we switch the x and y coordinates and search for the y value, we get e.g. the value 15. Then we could theoretically determine that x can be
225 = x^3 + 7 218 = x^3 3✓218 = x X= 6.01.. ( I am now to lacy to find a solution were we get an integer result for x) |