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Cryptographic Primitives with Hinting Property

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Advances in Cryptology – ASIACRYPT 2022 (ASIACRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13791))

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Abstract

A hinting PRG is a (potentially) stronger variant of PRG with a “deterministic” form of circular security with respect to the seed of the PRG (Koppula and Waters, CRYPTO 2019). Hinting PRGs enable many cryptographic applications, most notably CCA-secure public-key encryption and trapdoor functions. In this paper, we study cryptographic primitives with the hinting property, yielding the following results:

  • We present a novel and conceptually simpler approach for designing hinting PRGs from certain decisional assumptions over cyclic groups or isogeny-based group actions, which enables simpler security proofs as compared to the existing approaches for designing such primitives.

  • We introduce hinting weak PRFs, a natural extension of the hinting property to weak PRFs, and show how to realize circular/KDM-secure symmetric-key encryption from any hinting weak PRF. We demonstrate that our simple approach for building hinting PRGs can be extended to realize hinting weak PRFs from the same set of decisional assumptions.

  • We propose a stronger version of the hinting property, which we call the functional hinting property, that guarantees security even in the presence of hints about functions of the secret seed/key. We show how to instantiate functional hinting PRGs and functional hinting weak PRFs for certain (families of) functions by building upon our simple techniques for realizing plain hinting PRGs/weak PRFs. We also demonstrate the applicability of a functional hinting weak PRF with certain algebraic properties in realizing KDM-secure public-key encryption in a black-box manner.

  • Finally, we show the first black-box separation between hinting weak PRFs (and hinting PRGs) from public-key encryption using simple realizations of these primitives given only a random oracle.

S. Patranabis—Work done while at VISA Research.

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Notes

  1. 1.

    The original definition of hinting PRG in [KW19] uses an additional output element \(z_0\in Y\) which has no hint about the seed of the PRG. We omit this element from the definition of hinting PRG here for simplicity of exposition.

  2. 2.

    Specifically, the authors of [KW19] use the set \(\{ 0,1 \}^{\ell }\) for each block (where \(\ell \) is fixed during the setup) whereas we use a sufficiently large (efficiently representable) set Y. Our definition allows defining hinting PRG in a setting where Y does not necessarily have a compact representation, i.e., when each element of Y is represented using more than \({\log {| Y |}}\) bits (which is the case for isogeny-based group actions). One can obtain a hinting PRG with bit-string blocks by using a suitable (statistical) extractor.

  3. 3.

    For any matrix with \(n + 1\) rows, we number rows from 0 to n.

  4. 4.

    Note that given any vector of group elements \([\textbf{v}] \in \mathbb {G}^n\) and any vector \(\textbf{s}\in \{ 0,1 \}^n\), one can efficiently compute \([\langle \textbf{v}, \textbf{s}\rangle ]\) without the need to solve the discrete log problem.

  5. 5.

    This is simply Lemma 2 with \(k = n + 1\), where we wrote the first row separately.

  6. 6.

    As before, we number rows from 0 to n.

  7. 7.

    Note that we cannot use the bit representation of an element of X to generate a group element G without using extractor, because for some EGAs (and in particular for isogeny-based group actions), elements of X do not have compact represenation.

  8. 8.

    \(\textbf{m}_i\) denotes the i row of \(\textbf{M}\).

  9. 9.

    Note that given any matrix of group elements \([\textbf{M}] \in \mathbb {G}^{n \times n}\) and any binary vector \(\textbf{s}\in \{ 0,1 \}^n\), one can efficiently compute \([\textbf{s}^t \textbf{M}\textbf{s}]\).

  10. 10.

    Note that this is simply Lemma 3 with \(n^2 + 1\) samples.

  11. 11.

    Note that we are using Lemma 3 with \(Q \cdot n^2 = {{\,\textrm{poly}\,}}(\lambda )\) samples.

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Authors and Affiliations

  1. VISA Research, Palo Alto, USA

    Navid Alamati

  2. IBM Research India, Bangalore, India

    Sikhar Patranabis

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  1. Navid Alamati
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Correspondence to Navid Alamati .

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Editors and Affiliations

  1. Indian Institute of Technology Madras, Chennai, India

    Shweta Agrawal

  2. Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

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Alamati, N., Patranabis, S. (2022). Cryptographic Primitives with Hinting Property. In: Agrawal, S., Lin, D. (eds) Advances in Cryptology – ASIACRYPT 2022. ASIACRYPT 2022. Lecture Notes in Computer Science, vol 13791. Springer, Cham. https://doi.org/10.1007/978-3-031-22963-3_2

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  • DOI: https://doi.org/10.1007/978-3-031-22963-3_2

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