Elliptic ElGamal Public Key Cryptosystem doubt I need an example of Elliptic ElGamal Public Key Cryptosystem. I have been trying with some values but I don't get the right solution. 

I have $p=13$, the elliptic curve $E:y^2=x^3+11x+7$ and a point $P=(6,\sqrt{3}) \in E(\mathbb{F}_{13})$.

The message is $M=(0.5,12.625)\in E(\mathbb{F}_{13})$. 

Alice, chooses the value $nA=5$, calculates Q$_A=n_A \cdot P=(2.5,11.125)$ and sends $Q_A$.

Bob chooses the integer $k=2$ randomly and calculates $C_1$ and $C_2$:
$$C_1=k⋅P=(12,\sqrt{23})$$
$$C_2=M+k⋅Q_A=(5.5,8.875)$$

Bob sends $(C_1,C_2)$ to Alice.

Alice calculates M: $$M=C_2−n_A⋅C_1=(−2.5,−8.4455)$$ and the solution should be $ M=(0.5,12.625)$ ... Does someone know what is wrong? I think that my problem is calculating in $\mathbb{F}_{13}$, but I'm not sure.

Thank you :)
In points of elliptic curve instead of $\frac {a}{b} $ you should use $a\cdot b^{-1} \pmod p$. Also: $$\sqrt3=\{i \in [0,12]|i^2=3 \pmod {13}\}=\{4,9\}$$  So $P=(6,4)$ or $P=(6,9)$ which $2\cdot P$ are $(10,8)$ and $(10,5)$ respectively. But your computation show that $2\cdot P=(12,\sqrt {23})$.


Also with $x=12$ we have't a solution for $y$ in your elliptic curve.