Today I saw this question again which presents an unusual variant of the Diffie-Hellman key exchange. Having thought about it for a while I could find out how functionally works, but have essentially little cue whether it actuappy provides security.
My question is now:
Does the below stated Diffie-Hellman-like key exchange protocol enjoy similar security properties as normal Diffie-Hellman?
First the variables: Let $(\mathbb G,g,q)$ be a multiplicatively-written group where the discrete logarithm problem is hard with prime order $q$ and generator $g$. Let $a,b,x,y \stackrel{$}{\gets} \{0,\ldots,q-1\}$ be sampled uniformly at random. Let $A=g^a, B=g^b$. Let $a,x$ only be known to Alice and $b,y$ only be known to Bob. Let $a,b$ be static and $x,y$ be ephemeral across multiple sessions. Let $K=g^{x+y}$ be the shared secret, which can be computed by both parties after the exchange. Now define the following messages (the letters on the left denote Alice and Bob): \begin{align} A\to B&:B^x=g^{bx}\\ B\to A&:A^y=g^{ay}\\ \end{align}
If you are questioning functional correctness at this point note that Alice can trivially compute $g^x$, knowing $x$, and can compute $g^y$ as $(A^y)^{a^{-1}\bmod q}$, yielding $K=g^x\cdot g^y$. The computation for Bob goes analogous.