That’s wrong. Dec(H′(\Hat{S}_j), ct_j) requires \alpha the server secret to determine the decryption key using Boneh’s notation of the PSI system. Or looking from the other direction, the encryption uses both w (NeuralHash) and L (\alpha G, for server secret \alpha).
I believe the math you outline above refers to this step, located on page 7 of the Technical Summary:
> Next, the client creates a cryptographic safety voucher that has the following properties: If the user image hash matches the entry in the known CSAM hash list, then the NeuralHash of the user image exactly transforms to the blinded hash if it went through the series of transformations done at database setup time. Based on this property, the server will be able to use the cryptographic header (derived from the NeuralHash) and using the server-side secret, can compute the derived encryption key and successfully decrypt the associated payload data.
Is that correct?
If so, then I agree that it is true that in the PSI system the server secret is completely necessary as part of the decryption process in order to decrypt the matching hash in the pointed-to location in the table. That being said, looking only at the information encrypted by the client, I don't think the server secret comes into play, right?
If I'm misunderstanding, and you're confident that an attacker would have to have the server secret to decrypt a photo (even if they already knew that photo's NeuralHash and were able to defeat the internal layer of encryption), then I definitely recommend posting a well-outlined answer to the Cryptography Stack Exchange, as that would be super helpful!
Yes the server secret comes into play looking at the encryption key on the client. This is the value L, which you can think of as a “public key” in usual ECC schemes.
It’s not useful to talk about defeating the “internal” layer of encryption separately from the “outer” layer because vouchers should be stored as generated by clients at rest. The database leak should not include any unwrapped vouchers (that would be like using a password hash but storing plaintext passwords anyways).
