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I'm attempting to send a message to a group of recipients, through an untrusted intermediary.

Each of the recipients will have been invited to the group by someone who already has a trusted public key from them.

The best solution I've come up with so far is that a shared secret is created by the first member of the group, and that is sent encrypted to each of the future recipients by whoever invited them. All messages addressed to the group will be encrypted with the shared secret.

Are there any major flaws with this approach? Is there a better, more well known approach?

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One of the major flaw in this approach is that you wont have the option to remove a member from the group. If you want to you will have to send a new shared secret to each of the other members which is not very scalable if the group is large or the number of people leaving the group is large.

There are a lot of ways to accomplish broadcast key distribution, One of the methods is Logical Key Hierarchy (LKH). The basic idea of this scheme is arrange members at leaves of a key tree, each member receives keys on its path. In this scheme both entry and exit of members has low computation and communication overhead.

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