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Computationally Private Randomizing Polynomials and Their Applications

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Cryptography in Constant Parallel Time

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Notes

  1. 1.

    This similarity is not coincidental as both concepts were raised in the context of secure multiparty computation. Indeed, an information theoretic variant of Yao’s garbled circuit technique was already used in [93] to construct low-degree randomized encoding for NC 1 functions.

  2. 2.

    Previous presentations of Yao’s garbled circuit relied on primitives that seem less likely to allow an NC 0 implementation. Specifically, [25, 115] require linear stretch PRG and [107] requires symmetric encryption that enjoys some additional properties.

  3. 3.

    This condition is redundant in the case of signatures and commitments, whose existence follows from the existence of a PRG. We will later describe a stronger result for such primitives.

  4. 4.

    By symmetric encryption we refer to (probabilistic) stateless encryption for multiple messages, where the parties do not maintain any state information other than the key. If parties are allowed to maintain synchronized states, symmetric encryption can be easily reduced in NC 0 to a PRG.

  5. 5.

    Applying the construction to circuits with a bounded fan-out, even linear length would suffice.

  6. 6.

    Security proofs for variants of this construction were given implicitly in [107, 128, 134] in the context of secure computation. However, they cannot be directly used in our context for different reasons. In particular, the analysis of [107] relies on a special form of symmetric encryption and does not achieve perfect correctness, while that of [128, 134] relies on a linear-stretch PRG.

  7. 7.

    In fact, each application of the encryption scheme will use some additional random bits. To simplify notation, we keep these random inputs implicit.

  8. 8.

    Specifically, the encryption is always invoked on messages whose length is bounded by \(\ell(n)\stackrel {\mathrm {def}}{{=}}O(|C_{n}|\cdot k)\), hence we can use ℓ(n)-one-time symmetric encryption.

  9. 9.

    In some cases, we will need to rely on perfect correctness, which we get “for free” in our main construction. See Table 4.2.

  10. 10.

    Similarly, assuming a linear-stretch PRG in NC 1, we can obtain, for every NC function, a (non-adaptive single-oracle) IHS in which the user is in NC 0 and the oracle is in NC.

  11. 11.

    Assuming that factoring is intractable (or, more generally, that there exists a OWF in SREN) it is provably impossible to obtain an NC 0 reduction from PRFs to sublinear stretch PRGs or OWFs. See Sect. 4.8.

  12. 12.

    For concreteness, we refer here only to the case of symmetric encryption, the case of other primitives which are NC 0-reducible to a PRF (such as identification schemes and MACs) is analogous.

  13. 13.

    Similar examples are the NC 1 transformation of one-to-one OWF to non-interactive commitment scheme (cf. [34]) and of distributionally OWF into standard OWF (cf. [90]).

  14. 14.

    Actually, for the composition theorem to go through, Definition 5.3 should be augmented by providing players and adversaries with auxiliary inputs. We ignore this technicality here, and note that the results in this section apply (with essentially the same proofs) to the augmented model as well.

  15. 15.

    To handle randomized functionalities we use the non-interactive secure reduction mentioned above. Now, we can (m−1)-securely reduce f to a single-output functionality by letting each party mask its output f i with a private randomness. That is, f′((x 1,r 1)…,(x m ,r m ))=((f 1(x 1)⊕r 1)∘⋯∘(f 1(x m )⊕r m )). As both reductions are non-interactive the resulting reduction is also non-interactive. Moreover, the circuit size of f′ is linear in the size of the circuit that computes the original function.

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  1. School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel

    Benny Applebaum

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Applebaum, B. (2014). Computationally Private Randomizing Polynomials and Their Applications. In: Cryptography in Constant Parallel Time. Information Security and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17367-7_5

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  • DOI: https://doi.org/10.1007/978-3-642-17367-7_5

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