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Would increasing the number of introduction points help distribute the load among more relays? Would it have any security implications?

There doesn't seem to be a limit on the number of introduction points a hidden service can choose, so it seems viable to increase it, I'm just not sure if it'd be any help.

[The specification is vague about how many introduction points a hidden service can have ("a small number", "arbitrary"), which I'm assuming leaves it to the implementation.]

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Introduction points carry very little load in the hidden service protocol. They're asked once by the hidden service to accept introduction requests, and whenever one comes in, they forward it to the hidden service. That happens each time a client attempts to connect to a hidden service, and each introduction point only has to handle 1/3 of those introductions. Unlikely that reducing that to 1/4, 1/5, 1/100 (not sure what you had in mind) would matter for most hidden services.

However, there's a downside of raising the number of introduction points: whenever that set changes, the hidden service publishes a new descriptor to the hidden service directories. Now, it's going to happen much more frequently that 1 out of 4 (5, 100) introduction points fails and needs to be replaced than 1 out of 3. That hidden service would basically publish new descriptors all the time. And that's going to put a lot of load on relays. (It's also possibly anonymity-related, because that hidden service would sure stand out.)

tl;dr: not helpful, probably harmful, don't do this.

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