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Collusion Resistant Broadcast and Trace from Positional Witness Encryption

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Public-Key Cryptography – PKC 2019 (PKC 2019)

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Abstract

An emerging trend is for researchers to identify cryptography primitives for which feasibility was first established under obfuscation and then move the realization to a different setting. In this work we explore a new such avenue—to move obfuscation-based cryptography to the assumption of (positional) witness encryption. Our goal is to develop techniques and tools, which we will dub “witness encryption friendly” primitives and use these to develop a methodology for building advanced cryptography from positional witness encryption.

We take a bottom up approach and pursue our general agenda by attacking the specific problem of building collusion-resistant broadcast systems with tracing from positional witness encryption. We achieve a system where the size of ciphertexts, public key and private key are polynomial in the security parameter \(\lambda \) and independent of the number of users N in the broadcast system. Currently, systems with such parameters are only known from indistinguishability obfuscation.

B. Waters—Supported by NSF CNS-1228599 and CNS-1414082, DARPA SafeWare, Microsoft Faculty Fellowship, and Packard Foundation Fellowship.

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Notes

  1. 1.

    This is intended to mirror the term “iO friendly” used elsewhere in the literature.

  2. 2.

    Following prior broadcast encryption literature we will not count a description S of the recipients of a ciphertext toward the ciphertext overhead.

  3. 3.

    Here qualified could alternatively be interpreted as “non-revoked”.

  4. 4.

    Here we only consider BT schemes with public traceability.

  5. 5.

    Here we assume that number of users N is at most \(\mathsf {poly}(\lambda )\).

  6. 6.

    Here comparisons between bit-strings is performed by interpreting each bit-string as non-negative integer.

  7. 7.

    The idea of using Merkle hash tree for efficiently committing to large sets has also been previously used in works such as [3, 60].

  8. 8.

    The proof will involve an exponential number of hybrids. This is because for applying message hiding security property of PWE the index used must be \(2^{\lambda + \ell + k}\) (i.e., the last index), therefore we need to use index hiding security to go from index \((N + 1)\left| \right| 0^\ell \left| \right| 0^k\) to \(2^{\lambda + \ell + k}\) which takes an exponential number of hybrid steps. Here the exact ordering of witness components, i.e. \(i, \sigma , \pi \), is very important for the proof to go through. We can only use the security of PWE scheme if index i is leading term and corresponds to the most significant bits.

  9. 9.

    Although the notion of witness encryption with extractable security has been well studied [28, 36], extractability in the case of positional witness encryption is rather non-trivial to define due to the fact that PWE already requires index hiding to hold for all indices.

  10. 10.

    We would like to point out that our techniques of relaxing extractably-secure assumptions to more standard indistinguishability-based assumptions are in part inspired by analogous results in the regime of moving from differing-inputs obfuscation (diO) to indistinguishability obfuscation (iO) [21, 44, 52].

  11. 11.

    The adversary is not allowed to query the oracle on message \(m^{*}\) to allow trivial distinguishing attacks.

  12. 12.

    Technically one could visualize the proof \(\pi \) as only proving that the \(i^{th}\) bit of pre-image is m[i]. The fact that it also proves that the message hashes to \(H_\mathsf {hk}(m)\) is just due to the structure of the proof.

  13. 13.

    Note that index \(\mathsf {int}(i||0^{k+\ell }) + 2^{k+\ell }\) is same as \(\mathsf {int}(i+1||0^{k+\ell })\).

  14. 14.

    Note that index \(\mathsf {int}(i||0^{k+\ell }) + 2^{k+\ell }\) is same as \(\mathsf {int}(i+1||0^{k+\ell })\).

  15. 15.

    We would like to point out that most existing IBE constructions based on LWE are already verifiable and they can be made anonymous by using the transformation from [39, 59].

  16. 16.

    Using this ABO scheme in our AugBE construction results in an AugBE scheme without perfect correctness.

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Acknowledgement

We thank the anonymous reviewers of PKC 2019 for helpful feedback, especially for pointing out the connection between SPB hashes and (a weakening of) ABO signatures.

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Goyal, R., Vusirikala, S., Waters, B. (2019). Collusion Resistant Broadcast and Trace from Positional Witness Encryption. In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11443. Springer, Cham. https://doi.org/10.1007/978-3-030-17259-6_1

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