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Practical witness encryption for algebraic languages or how to encrypt under Groth–Sahai proofs

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Abstract

Witness encryption (\(\mathsf{WE}\)) is a recent powerful encryption paradigm, which allows to encrypt a message using the description of a hard problem (a word in an \({\mathbf{NP}}\)-language) and someone who knows a solution to this problem (a witness) is able to efficiently decrypt the ciphertext. Recent work thereby focuses on constructing \(\mathsf{WE}\) for \({\mathbf{NP}}\) complete languages (and thus \({\mathbf{NP}}\)). While this rich expressiveness allows flexibility w.r.t. applications, it makes existing instantiations impractical. Thus, it is interesting to study practical variants of \(\mathsf{WE}\) schemes for subsets of \({\mathbf{NP}}\) that are still expressive enough for many cryptographic applications. We show that such \(\mathsf{WE}\) schemes can be generically constructed from smooth projective hash functions (\(\mathsf {SPHF}\)s). In terms of concrete instantiations of \(\mathsf {SPHF}\)s (and thus \(\mathsf{WE}\)), we target languages of statements proven in the popular Groth–Sahai (\(\mathsf {GS}\)) non-interactive witness-indistinguishable/zero-knowledge proof framework. This allows us to provide a novel way to encrypt. In particular, encryption is with respect to a \(\mathsf {GS}\) proof and efficient decryption can only be done by the respective prover. The so obtained constructions are entirely practical. To illustrate our techniques, we apply them in context of privacy-preserving exchange of information.

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Notes

  1. Even if such special-purpose obfuscation exists, this does not rule out that extractable \(\mathsf{WE}\) for a sufficiently large interesting subset of \({\mathbf{NP}}\) exists.

  2. We mention this direction for completeness and stress that we are only interested in non-interactive \(\mathsf{WE}\) schemes.

  3. Our approach is also easily portable to the SXDH setting (and thus relying on DDH instead of DLIN).

  4. Similar to [1] we additionally partition the language \(L_{\mathsf {pp}}\) with respect to some auxiliary information \(\mathsf {aux}\in \mathsf{A}\), where \(\mathsf A\) is determined by \(\mathsf {pp}\). Note that without this partitioning, words \(x \in L_{\mathsf {pp}}\) additionally contain \(\mathsf {aux}\), i.e., so that \(x = (\mathsf {aux}, x')\).

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Acknowledgements

The authors have been supported by EU H2020 Project Prismacloud, Grant Agreement No. 644962. We thank various anonymous referees for their valuable comments.

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Authors and Affiliations

  1. IAIK, Graz University of Technology, Graz, Austria

    David Derler

  2. AIT Austrian Institute of Technology, Vienna, Austria

    Daniel Slamanig

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Correspondence to Daniel Slamanig.

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Communicated by M. Albrecht.

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Derler, D., Slamanig, D. Practical witness encryption for algebraic languages or how to encrypt under Groth–Sahai proofs. Des. Codes Cryptogr. 86, 2525–2547 (2018). https://doi.org/10.1007/s10623-018-0460-y

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