Secret sharing is a cryptographic technique to share a secret among participants in such a way that only authorized subsets are able to recover the secret. Ramp secret sharing schemes can achieve better information ratio than perfect schemes while some partial information on a secret which is composed of several sub-secrets leaks out. The notion of strong security has been introduced to control the amount of information on every subset of the sub-secrets unauthorized sets can obtain. In this paper, we reduce the construction of strongly secure ramp secret sharing for general access structures to a linear algebraic problem. As a result, we show that previous results on strongly secure network coding imply two constructions of a linear transformation which makes a given linear ramp scheme strongly secure. They are explicit or provide a deterministic algorithm while the previous method which works for any linear ramp scheme is probabilistic.
Secret sharing is a cryptographic technique to share a secret among participants in such a way that only authorized subsets are able to recover the secret. Ramp secret sharing schemes can achieve better information ratio than perfect schemes while some partial information on a secret which is composed of several sub-secrets leaks out. The notion of strong security has been introduced to control the amount of information on every subset of the sub-secrets unauthorized sets can obtain. In this paper, we reduce the construction of strongly secure ramp secret sharing for general access structures to a linear algebraic problem. As a result, we show that previous results on strongly secure network coding imply two constructions of a linear transformation which makes a given linear ramp scheme strongly secure. They are explicit or provide a deterministic algorithm while the previous method which works for any linear ramp scheme is probabilistic.
Quantum secret sharing is a scheme for encoding a quantum state (the secret) into multiple shares and distributing them among several participants. If a sufficient number of shares are put together, then the secret can be fully reconstructed. If an insufficient number of shares are put together however, no information about the secret can be revealed. In quantum ramp secret sharing, partial information about the secret is allowed to leak to a set of participants, called an unqualified set, that cannot fully reconstruct the secret. By allowing this, the size of a share can be drastically reduced. This paper introduces a quantum analog of classical strong security in ramp secret sharing schemes. While the ramp secret sharing scheme still leaks partial information about the secret to unqualified sets of participants, the strong security condition ensures that qudits with critical information can no longer be leaked.
In this paper, we consider linear secret sharing schemes over a finite field F q , where the secret is a vector in Fℓ q and each of the n shares is a single element of F q . We obtain lower bounds on the so-called threshold gap g of such schemes, defined as the quantity r-t where r is the smallest number such that any subset of r shares uniquely determines the secret and t is the largest number such that any subset of t shares provides no information about the secret. Our main result establishes a family of bounds which are tighter than previously known bounds for ℓ ≳ 2 . Furthermore, we also provide bounds, in terms of n and q , on the partial reconstruction and privacy thresholds, a more fine-grained notion that considers the amount of information about the secret that can be contained in a set of shares of a given size. Finally, we compare our lower bounds with known upper bounds in the asymptotic setting.
Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties in such a way that certain authorized subsets can reconstruct the secret, while unauthorized subsets obtain no information about it. Previous works on QSS for general access structures focused solely on the existence of perfectly secure schemes, and the share size of the known schemes is necessarily exponential even in cases where the access structure is computed by polynomial size monotone circuits. This stands in stark contrast to the classical setting, where polynomial-time computationally-secure secret sharing schemes have been long known for all access structures computed by polynomial-size monotone circuits under standard hardness assumptions, and one can even obtain shares which are much shorter than the secret (which is impossible with perfect security). While QSS was introduced over twenty years ago, previous works only considered information-theoretic privacy. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes, just as in the classical case. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time computationally-secure QSS schemes under standard hardness assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS necessarily required exponential share size. In fact, we can go even further: We construct QSS schemes for which the size of the quantum shares is significantly smaller than the size of the secret. As in the classical setting, this is impossible with perfect security. We also apply our compiler to obtain results beyond computational QSS. In the information-theoretic setting, we improve the share size of perfect QSS schemes for a large class of n-party access structures to 1.5^{n+o(n)}, improving upon best known schemes and matching the best known result for general access structures in the classical setting. Finally, among other things, we study the class of access structures which can be efficiently implemented when the quantum secret sharing scheme has access to a given number of copies of the secret, including all such functions in 𝖯 and NP.
In this paper, we consider linear secret sharing schemes over a finite field Fq, where the secret is a vector in Fℓqand each of the n shares is a single element of Fq. We obtain lower bounds on the so-called threshold gap g of such schemes, defined as the quantity r-t where r is the smallest number such that any subset of r shares uniquely determines the secret and t is the largest number such that any subset of t shares provides no information about the secret. Our main result establishes a family of bounds which are tighter than previously known bounds for ℓ ≳ 2 . Furthermore, we also provide bounds, in terms of n and q , on the partial reconstruction and privacy thresholds, a more fine-grained notion that considers the amount of information about the secret that can be contained in a set of shares of a given size. Finally, we compare our lower bounds with known upper bounds in the asymptotic setting.
The study of probabilistic secret sharing schemes using arbitrary probability spaces and possibly infinite number of participants lets us investigate abstract properties of such schemes. It highlights important properties, explains why certain definitions work better than others, connects this topic to other branches of mathematics, and might yield new design paradigms. A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions for qualified subsets. The scheme is measurable if the recovery functions are measurable. Depending on how much information an unqualified subset might have, we define four scheme types: perfect, almost perfect, ramp, and almost ramp. Our main results characterize the access structures which can be realized by schemes of these types. We show that every access structure can be realized by a non-measurable perfect probabilistic scheme. The construction is based on a paradoxical pair of independent random variables which determine each other. For measurable schemes we have the following complete characterization. An access structure can be realized by a (measurable) perfect, or almost perfect scheme if and only if the access structure, as a subset of the Sierpiński space {0, 1} P, is open, if and only if it can be realized by a span program. The access structure can be realized by a (measurable) ramp or almost ramp scheme if and only if the access structure is a Gδ set (intersection of countably many open sets) in the Sierpiński topology, if and only if it can be realized by a Hilbert-space program.
Abstract We improve the flexibility in designing access structures of quantum stabilizer-based secret sharing schemes for classical secrets, by introducing message randomization in their encoding procedures. We generalize the Gilbert–Varshamov bound for deterministic encoding to randomized encoding of classical secrets. We also provide an explicit example of a ramp secret sharing scheme with which multiple symbols in its classical secret are revealed to an intermediate set, and justify the necessity of incorporating strong security criterion of conventional secret sharing. Finally, we propose an explicit construction of strongly secure ramp secret sharing scheme by quantum stabilizers, which can support twice as large classical secrets as the McEliece–Sarwate strongly secure ramp secret sharing scheme of the same share size and the access structure.