Code change:
to the PA protocol: The helper handles each sub-request "PAAggregateSubReq" as follows. <del> It first checks that checks that "PAAggregateSubReq.proto == PAParam.proto".  If not, it aborts and alerts the leader with "incorrect protocol for sub-request".  Otherwise, it computes the HPKE context as where "sk" is its HPKE key and computes the body of the "PAAggregateSubResp".  It then updates its state according to the PA protocol.  After processing all of the sub-requests, the helper encrypts its updated state and constructs its response to the aggregate request. </del> <ins> It first looks up the HPKE config and corresponding secret key associated with "helper_share.config_id".  If not found, then the sub-response consists of an "unrecognized config" alert.  [TODO: We'll want to be more precise about what this means.  See issue#57.] Next, it attempts to decrypt the payload with the following procedure: where "sk" is the HPKE secret key.  If decryption fails, then the sub-response consists of a "decryption error" alert.  [See issue#57.] Otherwise, the helper handles the request for its plaintext input share "input_share" and updates its state as specified by the PA protocol. After processing all of the sub-requests, the helper encrypts its updated state and constructs its response to the aggregate request. </ins> 3.4.2.1.1.

Related discussion:
text: The proof is valid, so the verifier will accept always. To remove interaction, the Fiat-Shamir-like technique in Section 6.2.3 is probably the best way to go. We only sketched the idea in the paper (without formal analysis or proof) but if it's going to be important for your application, I'm happy to sanity-check the protocol you come up with. Another idea—the details of which I have not checked carefully—would be to modify your protocol as follows: The prover sends its data vector x to verifiers, split into k shares x1, ..., xk—one for each of the k verifiers. Each verifier holds a PRF key ki computes ri = PRF(ki, xi) and they return r = r1 + ... + rk to the prover. The verifiers store no state. The prover computes the proof pi and sends (x1, pi1), ..., (xk, pik) to the verifiers—again, one for each of the k verifiers. Each verifier computes ri = PRF(ki, xi), they jointly compute r = r1 + ... + r_k, and check the proof using randomness r. The Fiat-Shamir-like approach seems slightly cleaner to me, since it only requires a single prover-to-verifier message. Either way, please let me know if I misunderstood your question or if anything else here is unclear. [1] URL
Thanks for the nice explanation, Henry. It's perfectly clear and answers my question. It also points to something that I, for one, missed the importance of: the need for the prover to "commit" to the input shares before generating the r-dependent proof. I appreciate you pointing this out. The only requirement for making upload start not idempotent is so that the randomness r can be sent to the client. It would be nice to do away with this requirement. I think the way forward is using the Fiat-Shamir technique. NAME I'll get cracking on this and let you know when I have something for you to look at.
NAME I believe the change is something like this. Let be a hash function with range . Suppose there are verifiers. The prover proceeds as follows: Splits its inputs into shares .