Problem with the signature of message using ECDSA over GF(2^m)
First of all I would like to thank @fgrieu, @kelelaka and the moderator Maarten Bodewes for their support to solve my question.
I had an error to determine K^-1, S, y S^-1, I was also missing an important piece of information which is the order of G (n = 29)
Correcting my mistake we have:
GF(2^7), m = 7
E: y^2 + xy - [x^3 + ax^2 + b]
y^2 + xy - [x^3 + 0x^2 + 1] with a = 0 and b = 1
Irreducible polynomial P(z) = z^7 + z + 1 mod 2
d = 17
G = (124, 68) = (z^6 + z^5 + z^4 + z^3 + z^2, z^6 + z^2), Generator a point that lives on the curve
Q = [d]G = (40, 19), a point that lives on the curve too.
k = 13
P = [k]G = (82, 100)
r = x(P) = 82
e = SHA(m) = 19
n = order of G = 29
k^-1 = 9 (Here was my first mistake when taking modularization with P(z) instead of n)
S = k^-1 (e + d*r) mod n (Here was my second mistake when taking modularization with P(z) instead of n)
S = 9 (19 + 17 * 82) mod 29
S = 15
Finally we obtained the value r and S that is the signature of the message(r,S)=(82,15)
Verifying the signature:
P = [S^-1 * e mod n] G + [(S^-1 * r mod n)] Q
S^1 = 2
P = (2 * 19 mod 29 )G + (2 * 82 mod 29)Q
P = [9]G + [19]Q
P = 9(124, 68) + 19(40, 19)
P = (76, 50) + (98, 104)
P = (82, 100)
x(P) = r
This is how we check that the signature is valid.