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Alice needs to share an already generated numeric secret $s$ with Bob and she doe not want the world to see it. The problem is that she has never contacted Bob before.

I thought Alice and Bob could generate a line à la Shamir Secret Sharing so that $$q(x)=s+\alpha x.$$

To agree with $\alpha$ Alice and Bob could implement the original Diffie-Hellman protocol for key exchange. Finally Alice could just send $q(1)$ to Bob and he could compute $s$ as $$q(1)-\alpha*1=q(1)-\alpha=s.$$

Theoretically even if Eve could see both the values sent in the clear by the Diffie-Hellman protocol and the value $q(1)$, Eve could not compute $s$ from them.

Is this method already used used somewhere? Do you know some literature which prove this to be secure?

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This protocol is insecure because the original Diffie-Hellman method can be exploited by man-in-the-middle attacks.

For this reason, you should never use Diffie-Hellman alone.

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