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.

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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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