Proofs of partial knowledge demonstrate the possession of certain subsets of witnesses for a given collection of statements $x_1,dots,x_n$.
Cramer, Damg{aa}rd, and Schoenmakers (CDS), built proofs of partial knowledge, given “atomic” protocols for individual statements $x_i$, by having the prover randomly secret share the verifier’s challenge and using the shares as challenges for the atomic protocols. This simple and highly-influential transformation has been used in numerous applications, ranging from anonymous credentials to ring signatures.

We consider what happens if, instead of using the shares directly as challenges, the prover first hashes them. We show that this elementary enhancement can result in significant benefits:
begin{itemize}
item the proof contains a {em single} atomic transcript per statement $x_i$,
item it suffices that the atomic protocols are $kappa$-special sound for $kappa geq 2$,
item when compiled to a signature scheme using the Fiat-Shamir heuristic, its unforgeability can be proved in the {em non-programmable} random oracle model.
end{itemize}
None of the above features is satisfied by the CDS transformation.

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