Superpolynomial Lower Bounds for Monotone Span Programs

Babai, László; Gál, Anna; Wigderson, Avi

doi:10.1007/s004930050058

Cited by 76 publications

(110 citation statements)

References 38 publications

Supporting

Mentioning

105

Contrasting

Order By: Relevance

“…The best known lower bounds for linear secret-sharing schemes is n Ω(log n) [1,16,17]. By modification of the claims in [17], we show that these lower bounds hold also for multi-linear secret-sharing schemes.…”

Section: Lower Bounds For Multi-linear Secret-sharing Schemesmentioning

confidence: 77%

“…Let ψ : G 1 → G 2 be a group homomorphism, and φ : G 2 N a group action. Then ψ induces a group action ψ * (φ): G 1 N , given by composition (ψ * (φ))(x) := φ(ψ(x)). Furthermore, if ψ is surjective and the action φ is non-trivial then so is ψ * (φ).…”

Section: A the Construction Of The Group G Pmentioning

confidence: 99%

“…Brickell [7] considered ideal schemes and constructed ideal schemes for some access structures, e.g., for hierarchical access structures. Brickell and Davenport [8] showed an interesting connection between ideal access structures and matroids, that is, (1) If an access structure is ideal then it is induced by a matroid, (2) If an access structure is induced by a representable matroid, then the access structure is ideal. Following this work, many works have studied ideal access structures and matroids, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to general secret-sharing schemes, super-polynomial lower bounds are known for linear secret-sharing schemes. That is, there exist explicit access structures such that the total share size of any linear secret-sharing scheme realizing them is n Ω(log n) times the size of the secret [1,16,17].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multi-linear Secret-Sharing Schemes

Beimel

Ben-Efraim

Padrö

et al. 2014

Theory of Cryptography

View full text Add to dashboard Cite

Abstract. Multi-linear secret-sharing schemes are the most common secret-sharing schemes. In these schemes the secret is composed of some field elements and the sharing is done by applying some fixed linear mapping on the field elements of the secret and some randomly chosen field elements. If the secret contains one field element, then the scheme is called linear. The importance of multi-linear schemes is that they provide a simple non-interactive mechanism for computing shares of linear combinations of previously shared secrets. Thus, they can be easily used in cryptographic protocols.In this work we study the power of multi-linear secret-sharing schemes. On one hand, we prove that ideal multi-linear secret-sharing schemes in which the secret is composed of p field elements are more powerful than schemes in which the secret is composed of less than p field elements (for every prime p). On the other hand, we prove super-polynomial lower bounds on the share size in multi-linear secret-sharing schemes. Previously, such lower bounds were known only for linear schemes.

show abstract

Section: Lower Bounds For Multi-linear Secret-sharing Schemesmentioning

confidence: 77%

Section: A the Construction Of The Group G Pmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multi-linear Secret-Sharing Schemes

Beimel

Ben-Efraim

Padrö

et al. 2014

Theory of Cryptography

View full text Add to dashboard Cite

show abstract

“…Linear secret-sharing schemes are equivalent to monotone span programs, defined by [46]. Super-polynomial lower bounds for monotone span programs and, therefore, for linear secret-sharing schemes were proved in [5,2,36].…”

Section: Example 1 (Attribute Based Encryption)mentioning

confidence: 99%

Secret-Sharing Schemes: A Survey

Beimel

2011

Lecture Notes in Computer Science

505

305

View full text Add to dashboard Cite

Abstract. A secret-sharing scheme is a method by which a dealer distributes shares to parties such that only authorized subsets of parties can reconstruct the secret. Secret-sharing schemes are important tools in cryptography and they are used as a building box in many secure protocols, e.g., general protocol for multiparty computation, Byzantine agreement, threshold cryptography, access control, attribute-based encryption, and generalized oblivious transfer. In this survey, we will describe the most important constructions of secret-sharing schemes, explaining the connections between secret-sharing schemes and monotone formulae and monotone span programs. The main problem with known secret-sharing schemes is the large share size: it is exponential in the number of parties. We conjecture that this is unavoidable. We will discuss the known lower bounds on the share size. These lower bounds are fairly weak and there is a big gap between the lower and upper bounds. For linear secret-sharing schemes, which is a class of schemes based on linear algebra that contains most known schemes, super-polynomial lower bounds on the share size are known. We will describe the proofs of these lower bounds. We will also present two results connecting secret-sharing schemes for a Hamiltonian access structure to the NP vs. coNP problem and to a major open problem in cryptography -constructing oblivious-transfer protocols from one-way functions.

show abstract

On some simple degree conditions that guarantee the upper bound on the chromatic (choice) number of random graphs

1999

J. Graph Theory

View full text Add to dashboard Cite

It is well known that almost every graph in the random space G(n, p) has chromatic number of order O(np/log(np)), but it is not clear how we can recognize such graphs without eventually computing the chromatic numbers, which is NP‐hard. The first goal of this article is to show that the above‐mentioned upper bound on the chromatic number can be guaranteed by simple degree conditions, which are satisfied by G(n, p) almost surely for most values of p. It turns out that the same conditions imply a similar bound for the choice number of a graph. The proof implies a polynomial coloring algorithm for the case p is not too small. Our result has several applications. It can be used to determine the right order of magnitude of the choice number of random graphs and hypergraphs. It also leads to a general bound on the choice number of locally sparse graphs and to several interesting facts about finite fields. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 201–226, 1999

show abstract

Superpolynomial Lower Bounds for Monotone Span Programs

Cited by 76 publications

References 38 publications

Multi-linear Secret-Sharing Schemes

Multi-linear Secret-Sharing Schemes

Secret-Sharing Schemes: A Survey

On some simple degree conditions that guarantee the upper bound on the chromatic (choice) number of random graphs

Contact Info

Product

Resources

About