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)
Topic: The mirror effect of public keys (Read 226 times)
Also see this:
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).
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)
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)
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).
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.
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
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.
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 
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
0479be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798483ada7726a3c 4655da4fbfc0e1108a8fd17b448a68554199c47d08 is the uncompressed privatekey of 1. The other one is the uncompressed privatekey of -1.
-1 => N - 1
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
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
Does anyone know the public key mirror ?
Jump to: