Problem with the signature of message using ECDSA over GF(2^m) I'm trying to set up an ECDSA with Elliptic Curves over $\operatorname{GF}(2^m)$ with an example of toy with the following values:
Using the Weierstrass equation on binary finite fields. $$E: y^2 + x*y - [x^3 + a x^2 + b]$$

Let $E$ be defined over the field $\operatorname{GF}(2^7)$ ($m = 7$) with the equation
$$E: y^2 + x*y - [x^3 + 0*x^2 + 1]$$ where $a = 0$ and $b = 1$, using the irreducible polynomial $P(z) = z^7 + z + 1 \bmod 2$
The points are in decimal which has a polynomial representation. Example; $(z^2 + z; z^2 + z + 1) = (0110; 0111)$ in its binary representation and $(6, 7)$ in its decimal representation.I am using wolframcloud software to validate operations.

$d = 17$
generator $G = (124, 68) = (z^6 + z^5 + z^4 + z^3 + z^2, z^6 + z^2)$, a point that lives on the curve
$Q = [d]G = (40, 19)$, a point that lives on the curve too.
$k = 13$
$P = [k]G = (82, 100)$
$r = x(P) = 82$
$e = SHA(m) = 19$
$k^{-1} = 79$
$S = k^{-1} (e + d*r) \pmod{P(z)}$
$S = 79 (19 + 17*82) \pmod{P(z)}$
$S = 11$

Finally I obtained the value r and S that is the signature of the message: $(r, S) = (82, 11)$

Verification, and then suppose that the second entity know the same parameters over the curve without know d nor k. The second entity will carry out:
$P = [(S^{-1} * e) * G] + [(S^{-1} * r) * Q] \pmod{P(z)}\\
S^{-1} = 74\\
P = (74 * 19) * G + (74 * 82) * Q \pmod{P(z)}\\
P = 102 * G  + 67 * Q \pmod{P(z)}\\
P = (80, 87) + (38, 35) \pmod{P(z)}\\
P = (30, 92) \pmod{P(z)}\\
P.x = 30$
which is different from $r=82$: $P.x$ should be equal to $r$, but, it's NOT.

Now, We suppose that the second entity knows $d$ such that $Q = d * G$ then:
$P = (102 * G)  + (67 * 17 * G) \pmod{P(z)}\\
P = (102 * G)  + (107 * G) \pmod{P(z)}\\
P = (102 + 107) * G \pmod{P(z)}\\
P = 13 * G\pmod{P(z)}\\
P = 13 * (124, 68)\\
P = (82, 100)\\
P.x = 82 = r$
which is correct but the second entity doesn't know $d$.
Could someone help me, please and tell me how I can solve this problem?

Annex code in Wolfram language: To perform scalar multiplication through sum of points and doubling of a point I am using the following code with 
; or just Try It Online!
(* Input example GF(2^7): *)
m=7;
k=&quot;10001&quot;; (*BinaQy representation 10001 = 17 in decimal *)
lim = StringLength [k] + 1 ;
a=0; 
Gx=z^6 + z^5 + z^4 + z^3 + z^2;  (* Gx = 124 *)
Gy=z^6 + z^2; (* Gy = 68 *)
IrreduciblePolynomialCCE= z^7 + z + 1;
Qx=Gx;
Qy=Gy;
For [i=2, i&lt;lim , i++,
    c=StringTake [k ,{i}];
    (*Dubling*)
    {d, {inv, u}}=PolynomialMod[PolynomialExtendedGCD[Qx, IrreduciblePolynomialCCE],2];
    Lamda=PolynomialMod[(Qx + Qy*inv ), {IrreduciblePolynomialCCE, 2}] ;
    X3=PolynomialMod[(Lamda^2 + Lamda + a ), {IrreduciblePolynomialCCE, 2}] ;
    Y3=PolynomialMod [ (Qx^2 + Lamda*X3 + X3) , {IrreduciblePolynomialCCE, 2}] ;
    Qx = X3;
    Qy = Y3;
    If [c==&quot;1&quot;,{
        (*Sum*)
        {d, {inv2, u}}=PolynomialMod[ PolynomialExtendedGCD[Gx + Qx, IrreduciblePolynomialCCE],2];
        Lamda2=PolynomialMod[(Gy + Qy) * inv2 , {IrreduciblePolynomialCCE, 2}] ;
        XX3=PolynomialMod [(Lamda2^2 + Lamda2 + Gx + Qx + a) , {IrreduciblePolynomialCCE, 2}] ;
        YY3=PolynomialMod [Lamda2*(Gx + XX3) + XX3 + Gy, {IrreduciblePolynomialCCE, 2}] ;
        Qx = XX3;
        Qy = YY3;
        },{0}
    ]
]
Print [Qx]
Print [Qy]

The result $Q = (z^5 + z^3, z^4 + z + 1) = (40, 19)$
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.