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International Conference on Applied Cryptography and Network Security

ACNS 2020: Applied Cryptography and Network Security pp 24–44Cite as

Random Walks and Concurrent Zero-Knowledge

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12146))

Abstract

The established bounds on the round-complexity of (black-box) concurrent zero-knowledge (\(\mathrm {c}\mathcal {ZK}\)) consider adversarial verifiers with complete control over the scheduling of messages of different sessions. Consequently, such bounds only represent a worst case study of concurrent schedules, forcing \(\widetilde{\varOmega }(\log n)\) rounds for all protocol sessions. What happens in “average” cases against random schedules? Must all sessions still suffer large number of rounds?

Rosen and Shelat first considered such possibility, and constructed a \(\mathrm {c}\mathcal {ZK}\) protocol that adjusts its round-complexity based on existing network conditions. While they provide experimental evidence for its average-case performance, no provable guarantees are known.

In general, a proper framework for studying and understanding the average-case schedules for \(\mathrm {c}\mathcal {ZK}\) is missing. We present the first theoretical framework for performing such average-case studies. Our framework models the network as a stochastic process where a new session is opened with probability p or an existing session receives the next message with probability \(1-p\); the existing session can be chosen either in a first-in-first-out (\(\mathsf {FIFO}\)) or last-in-first-out (\(\mathsf {LIFO}\)) order. These two orders are fundamental and serve as good upper and lower bounds for other simple variations. We also develop methods for establishing provable average-case bounds for \(\mathrm {c}\mathcal {ZK}\) in these models. The bounds in these models turn out to be intimately connected to various properties of one-dimensional random walks that reflect at the origin. Consequently, we establish new and tight asymptotic bounds for such random walks, including: expected rate of return-to-origin, changes of direction, and concentration of “positive” movements. These results may be of independent interest.

Our analysis shows that the Rosen-Shelat protocol is highly sensitive to even moderate network conditions, resulting in a large fraction of non-optimal sessions. We construct a more robust protocol by generalizing the “footer-free” condition of Rosen-Shelat which leads to significant improvements for both \(\mathsf {FIFO}\) and \(\mathsf {LIFO}\) models.

Research supported in part by NSF grant 1907908, the MITRE Innovation Program, and a Cisco Research Award. The views expressed are those of the authors and do not reflect the official policy or position of the funding agencies.

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Notes

  1. 1.

    We were not able to find these results, or derive them as simple corollaries of known results, in any standard texts on probability such as [17].

  2. 2.

    For canonical protocols, we can allow an inconsequential first message from the prover (see Sect. 2.4).

  3. 3.

    We provide the derivation in the full version of this work [1].

  4. 4.

    The statement of this definition in [36] actually has (Vk) instead of (p1) as A’s nested message. However, we believe that it is a typo and by (Vk) authors really mean the presence of second stage messages; this is guaranteed by having (p1) in the definition but not by (Vk). Indeed, many nested protocols may terminate without ever reaching (Vk). If (Vk) is used in the definition, the simulator in [36] will run in exponential time even for the simple concurrent schedule described in [13] (and shown in red in Fig. 1 in [36]).

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Authors and Affiliations

  1. Stony Brook University, Stony Brook, USA

    Anand Aiyer, Xiao Liang, Nilu Nalini & Omkant Pandey

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  1. Anand Aiyer
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  4. Omkant Pandey
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Correspondence to Xiao Liang .

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Editors and Affiliations

  1. Department of Mathematics, University of Padua, Padua, Italy

    Mauro Conti

  2. Singapore University of Technology and Design, Singapore, Singapore

    Jianying Zhou

  3. Dipt di Informatica Sistemi e Produ, Università di Roma “Tor Vergata”, Rome, Roma, Italy

    Emiliano Casalicchio

  4. Sapienza University of Rome, Rome, Italy

    Angelo Spognardi

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Aiyer, A., Liang, X., Nalini, N., Pandey, O. (2020). Random Walks and Concurrent Zero-Knowledge. In: Conti, M., Zhou, J., Casalicchio, E., Spognardi, A. (eds) Applied Cryptography and Network Security. ACNS 2020. Lecture Notes in Computer Science(), vol 12146. Springer, Cham. https://doi.org/10.1007/978-3-030-57808-4_2

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