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There are many well known and studied ways of constructing a hash function from a block cipher. A thorough (but reasonably readable for a beginner) treatment of many of the classic approaches, and the security properties of the various constructions, can be found in Black-Box Analysis of the Block-Cipher-Based Hash-Function Constructions from PGV, which is ...


2

You have just to look at the signing/verification relation. Just write it as $$m\cdot s \equiv r\cdot \alpha + k \bmod (p-1)$$ And the verification relation should be $$g^{s\cdot m}\stackrel{?}{\equiv} y^r\cdot r \bmod p$$ where $y=g^\alpha$ is the public key and you eavesdrop a signature $(r,s)$ for $m$. Obseve that you can take any multiplicative ...


1

Actually you have a bug in your step of key generation, it should be $Q_A=d_A \times G$ and you want to have a point $G$ of large prime order $n$ such that the ECDLP is hard on the group. The last check ensures that the public key $Q_A$ is a point of order $n$. If this is the case, then $Q_A\times n = (d_A \times G)\times n = d_A\times(n\times G)=d_A\times ...



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