How can the Diffie-Hellman key exchange be extended to three parties?

Question

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Can one generalize the Diffie-Hellman key exchange to three or more parties?

How can Alice, Bob, and Charlie share a common secret key using an extended version of the Diffie-Hellman key exchange protocol?

a=3, b=4, c=5
g=2
p=5

Answer 1

1. Alice computes $A=g^a$, Bob computes $B=g^b$ and Charlie computes $C=g^c$.

2. $A,B$ and $C$ are published.

3. Alice computes $AC=C^a$ and $AB=B^a$ and Bob computes $BC=C^b$.

4. $AB,AC$ and $BC$ are published.

5. Alice computes $ABC=BC^a$, Bob computes $ABC=AC^b$ and Charlie computes $ABC=AB^c$.

6. Everybody shares $ABC=g^{abc}$.

This can be generalized for any number of participants in a tree construction, however it becomes increasingly inefficient, because the number of rounds increases.