Alice uses an ElGamal signature with base the group $Z^*_{107}$ and parameter $g=3$ of order $q=53$.The private key of Alice is some $x \in \{0,1,.....,52\}$ and the public key of her is $y=10$. To sign the message m, she calculates $r=g^k \bmod107$ for $k \in \{0,1,......,52\}$ and $s=(k \cdot h(m)+r\cdot x) \bmod 53$. For signing the first message, Alice chooses a random $k_1 \in \{0,1,......,52\}$. To sign the second message she uses $k_2=(2 \cdot k_1 +1) \bmod 53$ and generally if for the signature of $i$-th message she has used the $k_i$ for the $(i+1)$-th message she uses the $k_{i+1}=(2\cdot k_i +1)\bmod 53$. You know two successive signatures of Alice: $(r,s)=(79,7)$ of message $m$ with $h(m)=2$ and the signature $(r',s')=(105,41)$ of message $m'$ with $h(m')=3$. Find the private key of Alice (of course withουt calculating directly any discrete logarithm in group $Z^*_{107}$)
I am trying to solve this. I tried to apply the ElGamal Algorithm but I do not know how to use the $h(m)$ hash function. Can anyone help to solve it and help me how to use correct the ElGamal signature?
(source: mathematical competition 2008,France)