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On Tightly Secure Primitives in the Multi-instance Setting

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Public-Key Cryptography – PKC 2019 (PKC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11442))

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Abstract

We initiate the study of general tight reductions in cryptography. There already exist a variety of works that offer tight reductions for a number of cryptographic tasks, ranging from encryption and signature schemes to proof systems. However, our work is the first to provide a universal definition of a tight reduction (for arbitrary primitives), along with several observations and results concerning primitives for which tight reductions have not been known.

Technically, we start from the general notion of reductions due to Reingold, Trevisan, and Vadhan (TCC 2004), and equip it with a quantification of the respective reduction loss, and a canonical multi-instance extension to primitives. We then revisit several standard reductions whose tight security has not yet been considered. For instance, we revisit a generic construction of signature schemes from one-way functions, and show how to tighten the corresponding reduction by assuming collision-resistance from the used one-way function. We also obtain tightly secure pseudorandom generators (by using suitable rerandomisable hard-core predicates), and tightly secure lossy trapdoor functions.

N. K. Nguyen—This work was carried out when the author was doing an internship at the KIT.

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Notes

  1. 1.

    Of course, there are other interesting properties (such as memory usage [4]) of a given adversary \(A\) one would want a reduction to preserve.

  2. 2.

    We also remark that other formalisations of cryptographic assumptions exist, e.g., [14, 38].

  3. 3.

    This notion is not related to the notion of adaptive one-way functions from [40] in which an adversary gets access to a full inversion oracle for the function.

  4. 4.

    For simplicity, we only consider a PRG that stretches its seed \(x\) by one bit.

  5. 5.

    Slightly simplifying, we ignore the unlikely case \(x=0\) and treat \((\frac{x}{p})\) as a bit.

  6. 6.

    We note that the Legendre symbol has already been considered as a hard-core predicate by Damgård [13], although, to the best of our knowledge, its rerandomisability has not been investigated before.

  7. 7.

    Recall that \(\mathsf {queries}(S^{\mathcal {A}},\mathcal {A})\) denotes the worst-case number of queries/messages from \(\mathcal {A}\) to S.

  8. 8.

    Runtime of \(\mathcal {A}\) is included in \(\mathbf T (S^\mathcal {A})\).

  9. 9.

    As long as it is similar to the runtime of \(\mathcal {A}\).

  10. 10.

    If \(\mathcal {A}\) has not requested a signature before, then we just set \(i=1\).

  11. 11.

    On the other hand, one can actually derive a simple tight reduction from \(\forall \textsf {MI}_3(\textsf {LGR})\) to DDH.

  12. 12.

    This occurs with an overwhelming probability.

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Authors and Affiliations

  1. Karlsruhe Institute of Technology, Karlsruhe, Germany

    Dennis Hofheinz

  2. IBM Research Zurich, Rüschlikon, Switzerland

    Ngoc Khanh Nguyen

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  1. Dennis Hofheinz
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Correspondence to Dennis Hofheinz .

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  1. SKLOIS, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

  2. Security Research Laboratories, NEC Corporation, Kawasaki, Japan

    Kazue Sako

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Hofheinz, D., Nguyen, N.K. (2019). On Tightly Secure Primitives in the Multi-instance Setting. In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11442. Springer, Cham. https://doi.org/10.1007/978-3-030-17253-4_20

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