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Short Leakage Resilient and Non-malleable Secret Sharing Schemes

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

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Abstract

Leakage resilient secret sharing (LRSS) allows a dealer to share a secret amongst n parties such that any authorized subset of the parties can recover the secret from their shares, while an adversary that obtains shares of any unauthorized subset of parties along with bounded leakage from the other shares learns no information about the secret. Non-malleable secret sharing (NMSS) provides a guarantee that even shares that are tampered by an adversary will reconstruct to either the original message or something independent of it.

The most important parameter of LRSS and NMSS schemes is the size of each share. For LRSS, in the local leakage model (i.e., when the leakage functions on each share are independent of each other and bounded), Srinivasan and Vasudevan (CRYPTO 2019), gave a scheme for threshold access structures with share size of approximately \((3\cdot \mathsf {message \ length} \ + \mu )\), where \(\mu \) is the number of bits of leakage tolerated from every share. For the case of NMSS, the best known result (again due to the above work) has share size of \((11\cdot \mathsf {message \ length})\).

In this work, we build LRSS and NMSS schemes with much improved share size. Additionally, our LRSS scheme obtains optimal share and leakage size. In particular, we get the following results:

  • We build an information-theoretic LRSS scheme for threshold access structures with a share size of \((\mathsf {message \ length} \ + \mu )\).

  • As an application of the above result, we obtain an NMSS with a share size of \((4\cdot \mathsf {message \ length})\). Further, for the special case of sharing random messages, we obtain a share size of \((2\cdot \mathsf {message \ length})\).

B. Kanukurthi—Research supported by Microsoft Research, India.

S. Sekar—This work was done while at Indian Institute of Science, Bangalore.

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Notes

  1. 1.

    In fact, Guruswami and Wooters  [GW16] show that Shamir’s secret sharing scheme over a field of characteristic 2 is completely insecure if the adversary gets \(t-1\) shares and just one-bit of leakage from other shares. Further, Nielsen and Simkin [NS20] show that for larger characteristic fields and large n, Shamir’s secret sharing scheme is not leakage resilient for threshold \(t \le cn/\log n\), for constant \(0<c<1\).

  2. 2.

    Where the underlying field is a large characteristic field, the number of parties n is large, the threshold t is at least \(n-o(\log n)\), and the adversary can only obtain a constant number of full shares.

  3. 3.

    In [BDIR18], under the same restrictions (on n, the field and the number of full shares allowed), Shamir secret sharing is also shown to allow constant bits of leakage per share, under threshold \(t \le \alpha n\), for \(\alpha < 1\). The lower bound of [NS20, Section 4] mentioned in footnote 1 proves that this is the best possible trade-off from Shamir. However, for the LRSS scheme of [SV19] or ours, their lower bound allows for leakage almost as large as the size of a single share.

  4. 4.

    We require the secret sharing to satisfy an additional property of “local uniformity”, which requires every share to individually have (an almost) uniform distribution. We show later that such a property is already satisfied by many natural secret sharing schemes (e.g.: Shamir secret sharing).

  5. 5.

    A similar technique of using linear and invertible extractors to get rate optimality has been used in two prior settings before: information-theoretic privacy of communication data in the wiretap channel setting in [BT12, CDS12] and binary secret sharing schemes in [LCG+19].

  6. 6.

    In literature, invertible (seeded) extractors (see [CDS12] for an exposition on the same) are well-studied which allow efficient sampling of a source w and a seed s such that the extractor output on w and s equals a given value y. Note that our requirement to sample a source w given a seed s and a value y is stronger than the guarantee provided by invertible extractors. Hence we explicitly show that certain extractors allow such sampling.

  7. 7.

    \(\mathcal {A}\) is a monotone access structure if for all A, B such that \(A\subset B \subseteq [n]\) and \(A\in \mathcal {A}\), it holds that \(B\in \mathcal {A}\). Throughout this paper whenever we consider a general access structure, we mean a monotone access structure.

  8. 8.

    This is formally proven in [CKOS21, Claim 2].

  9. 9.

    Here, we let R denote the function that computes the rate to secret share \(l\)-size secrets.

  10. 10.

    Particularly, instantiating the NMSS sharing scheme of [GK18, Theorem 1] with the NMC of [AKO+22, Theorem 3] and our LRSS would give: First encode the secret m using the NMC to get states L and R. For [AKO+22]’s NMC, one of the states, L is larger than R. Secret share L using our LRSS from Theorem 1 for (n, 2)-threshold access structure to get \(L_1,\cdots ,L_n\), and R using an (n, t)-threshold secret sharing scheme to get \(R_1,\cdots ,R_n\). Set the i-th share as \((L_i,R_i)\).

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Authors and Affiliations

  1. Microsoft Research, Bengaluru, India

    Nishanth Chandran & Sai Lakshmi Bhavana Obbattu

  2. Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India

    Bhavana Kanukurthi

  3. UC Berkeley, Berkeley, CA, USA

    Sruthi Sekar

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  1. Nishanth Chandran
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  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

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Chandran, N., Kanukurthi, B., Obbattu, S.L.B., Sekar, S. (2022). Short Leakage Resilient and Non-malleable Secret Sharing Schemes. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13507. Springer, Cham. https://doi.org/10.1007/978-3-031-15802-5_7

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