btw it's open source

https://github.com/Slokh/anoncast

Does this require a server? Is there a way to build it without a server, by making the farcaster account itself a smart contract that accepts a zk proof as its signature scheme?

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Technically you could use something like @hatsprotocol  to allow someone to create a signer if they have a valid proof, and they could use that signer to post directly to fc from the client. But signers have full write permission for the account so probably doesn’t work for this use case

Creating a signer also requires an onchain transaction so easier to accidentally dox yourself