Near-Optimal Secret Sharing and Error Correcting Codes in $$\mathsf {AC}^0$$

Cheng, Kuan; Ishai, Yuval; Li, Xin

doi:10.1007/978-3-319-70503-3_14

Cited by 6 publications

(3 citation statements)

References 29 publications

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“…As another example, our recent papers [12,18] built on the same line of work to resolve or nearly resolve a number of longstanding open questions in quantum query complexity. Finally, large-error and threshold degree lower bounds on AC 0 functions have recently proved instrumental in the development of cryptographic secret-sharing schemes with reconstruction procedures in AC 0 [7,8,19]. We thus believe that the new techniques developed in this article will find further applications, perhaps in unexpected areas.…”

Section: Introductionmentioning

confidence: 81%

“…The best previous results could only guarantee security against subsets of shares of size d = Θ(n 1/2 ), or could only guarantee reconstruction advantage bounded away from 1 [7,18]. Cheng et al [19] recently considered a relaxed notion of security, where even very small subsets of shares are allowed to provide (a bounded amount of) information about the secret bit b. Under this relaxed notion of security, they achieved perfect reconstruction and security against subsets of size Ω(n).…”

Section: Authorsmentioning

confidence: 99%

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Bun¹,

Thaler²

2021

Theory of Comput.

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We prove two new results about the inability of low-degree polynomials to uniformly approximate constant-depth circuits, even to slightly-better-than-trivial error. First, we prove a tightΩ(n 1/2 ) lower bound on the threshold degree of the SURJECTIVITY function on n variables. This matches the best known threshold degree bound for any AC 0 function, previously exhibited by a much more complicated circuit of larger depth (Sherstov, FOCS'15). Our result also extends to a 2Ω (n 1/2 ) lower bound on the sign-rank of an AC 0 function, improving on the previous best bound of 2 Ω(n 2/5 ) (Bun and Thaler, ICALP'16).Second, for any δ > 0, we exhibit a function f : {−1, 1} n → {−1, 1} that is computed by a circuit of depth O(1/δ ) and is hard to approximate by polynomials in the following sense: f cannot be uniformly approximated to error ε = 1 − 2 −Ω(n 1−δ ) , even by polynomials of degree n 1−δ . In our FOCS'17 paper we proved a similar lower bound, but which held only for error ε = 1/3.Our result implies 2 Ω(n 1−δ ) lower bounds on the complexity of AC 0 under a variety of measures, including discrepancy, margin complexity, and threshold weight, that are central

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Section: Introductionmentioning

confidence: 81%

Section: Authorsmentioning

confidence: 99%

Untitled

Bun¹,

Thaler²

2021

Theory of Comput.

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“…In [5,6], the secret is one bit. In [9], secrets of length equal to a fraction of N (number of players) is considered. This time binary secret sharing with adaptive and non-adaptive adversaries similar to the model we consider in this work is defined.…”

Section: Related Workmentioning

confidence: 99%

Secret Sharing with Binary Shares

Lin¹,

Cheraghchi²,

Guruswami³

et al. 2018

Preprint

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Shamir's celebrated secret sharing scheme provides an efficient method for encoding a secret of arbitrary length ℓ among any N ≤ 2 ℓ players such that for a threshold parameter t, (i) the knowledge of any t shares does not reveal any information about the secret and, (ii) any choice of t + 1 shares fully reveals the secret. It is known that any such threshold secret sharing scheme necessarily requires shares of length ℓ, and in this sense Shamir's scheme is optimal. The more general notion of ramp schemes requires the reconstruction of secret from any t + g shares, for a positive integer gap parameter g. Ramp secret sharing scheme necessarily requires shares of length ℓ/g. Other than the bound related to secret length ℓ, the share lengths of ramp schemes can not go below a quantity that depends only on the gap ratio g/N .In this work, we study secret sharing in the extremal case of bit-long shares and arbitrarily small gap ratio g/N , where standard ramp secret sharing becomes impossible. We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminate the impossibility. Moreover, we provide explicit constructions of such schemes. One of the consequences of our relaxation is that, unlike standard ramp schemes with perfect secrecy, adaptive and non-adaptive adversaries need different analysis and construction. For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any τ fraction of observed shares, and reconstruction from any ρ fraction of shares, for any choices of 0 ≤ τ < ρ ≤ 1. Our construction achieves secret length N (ρ − τ − o(1)), which we show to be optimal. For adaptive adversaries, we construct explicit schemes attaining a secret length Ω(N (ρ − τ )). We discuss our results and open questions.

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Non-malleable Codes, Extractors and Secret Sharing for Interleaved Tampering and Composition of Tampering

Chattopadhyay

2020

Lecture Notes in Computer Science

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Near-Optimal Secret Sharing and Error Correcting Codes in $$\mathsf {AC}^0$$

Cited by 6 publications

References 29 publications

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Secret Sharing with Binary Shares

Non-malleable Codes, Extractors and Secret Sharing for Interleaved Tampering and Composition of Tampering

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