On Codes, Matroids and Secure Multi-party Computation from Linear Secret Sharing Schemes

Cramer, Ronald; Daza, Vanesa; Gracia, Ignacio; Urroz, Jorge Jiménez; Leander, Gregor; Martí-Farré, Jaume; Padrö, Carles

doi:10.1007/11535218_20

Cited by 33 publications

(51 citation statements)

References 26 publications

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“…This algorithm is efficient if the secret sharing scheme itself is efficient to begin with. This follows from the results in [10] where it is shown that strong multiplication linearizes this "decoding problem," by means of a generalization of ideas taken from the Berlekamp-Welch algorithm. For completeness we include in this paper a description of the general procedure from [10] as it applies to our algebraic geometric secret sharing schemes, even though in the present case it also follows by efficient decoding algorithms for algebraic geometry codes.…”

Section: It Is Quasi-threshold (With a 2g Gap) This Means That A Tnmentioning

confidence: 91%

“…For completeness we show how strong multiplication linearizes the problem of recovering the secret in the presence of corrupted shared. It is a special case of the more general technique given in [10]. But also note that known techniques for decoding algebraic-geometry codes apply here.…”

Section: Corollary 1 a Tn ⊂ A(m)mentioning

confidence: 99%

“…See Section 2 for the definition. It is also known to linearize in general the problem of recovery of the secret in the presence of corrupted shares [10]. 4.…”

Section: It Is Quasi-threshold (With a 2g Gap) This Means That A Tnmentioning

confidence: 99%

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Algebraic Geometric Secret Sharing Schemes and Secure Multi-Party Computations over Small Fields

Chen

Cramer

2006

Lecture Notes in Computer Science

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Abstract. We introduce algebraic geometric techniques in secret sharing and in secure multi-party computation (MPC) in particular. The main result is a linear secret sharing scheme (LSSS) defined over a finite field Fq, with the following properties. It is ideal. The number of players n can be as large as #C(Fq), whereC is an algebraic curve C of genus g defined over Fq. 2. It is quasi-threshold: it is t-rejecting and t +1+2g-accepting, but not necessarily t + 1-accepting. It is thus in particular a ramp scheme. High information rate can be achieved. 3. It has strong multiplication with respect to the t-threshold adversary structure, if t < 1 3 n − 4 3 g. This is a multi-linear algebraic property on an LSSS facilitating zero-error multi-party multiplication, unconditionally secure against corruption by an active t-adversary. 4. The finite field Fq can be dramatically smaller than n. This is by using algebraic curves with many Fq-rational points. For example, for each small enough , there is a finite field Fq such that for infinitely many n there is an LSSS over Fq with strong multiplication satisfying ( 1 3 − )n ≤ t < 1 3 n. 5. Shamir's scheme, which requires n > q and which has strong multiplication for t < 1 3 n, is a special case by taking g = 0. Now consider the classical ("BGW") scenario of MPC unconditionally secure (with zero error probability) against an active t-adversary with t < 1 3 n, in a synchronous n-player network with secure channels. By known results it now follows that there exist MPC protocols in this scenario, achieving the same communication complexities in terms of the number of field elements exchanged in the network compared with known Shamir-based solutions. However, in return for decreasing corruption tolerance by a small -fraction, q may be dramatically smaller than n. This tolerance decrease is unavoidable due to properties of MDS codes. The techniques extend to other models of MPC. Results on less specialized LSSS can be obtained from more general coding theory arguments.The authors independently submitted similar results to CRYPTO 2006 [5,8]. This paper is the result of merging those two papers.

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Section: It Is Quasi-threshold (With a 2g Gap) This Means That A Tnmentioning

confidence: 91%

Section: Corollary 1 a Tn ⊂ A(m)mentioning

confidence: 99%

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Algebraic Geometric Secret Sharing Schemes and Secure Multi-Party Computations over Small Fields

Chen

Cramer

2006

Lecture Notes in Computer Science

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122

188

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“…These techniques provided a characterization of the hierarchical access structures that admit an ideal perfect secret sharing scheme [12]. Other relevant results on secret sharing have been obtained by using matroid theory as, for instance, the ones in [2,9,13,23].…”

Section: Introductionmentioning

confidence: 99%

Extending Brickell–Davenport theorem to non-perfect secret sharing schemes

Farràs

Padrö

2013

Des. Codes Cryptogr.

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One important result in secret sharing is the Brickell-Davenport Theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. Even though a few attempts have been made, there is no satisfactory definition of ideal secret sharing scheme for the general case, in which non-perfect schemes are considered as well. Without providing another unsatisfactory definition of ideal non-perfect secret sharing scheme, we present a generalization of the Brickell-Davenport Theorem to the general case. After analyzing that result under a new point of view and identifying its combinatorial nature, we present a characterization of the (not necessarily perfect) secret sharing schemes that are associated to matroids. Some optimality properties of such schemes are discussed.

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“…Secure multiplication is a fundamental primitive in its own right, as secure multi-party computation is often based on combinations of secure addition and secure multiplication, the latter typically being demanding and involved while the former is typically much more straightforward. Arithmetic secret sharing allows efficient recovery of the secret in the presence of faulty shares, by a generalization of a result from [12] (see Section 5) and also gives rise to verifiable secret sharing [10].…”

Section: Introductionmentioning

confidence: 99%

The Torsion-Limit for Algebraic Function Fields and Its Application to Arithmetic Secret Sharing

Cascudo

Cramer

Xing

2011

Lecture Notes in Computer Science

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Abstract. An (n, t, d, n−t)-arithmetic secret sharing scheme (with uniformity) for F k q over Fq is an Fq-linear secret sharing scheme where the secret is selected from F k q and each of the n shares is an element of Fq. Moreover, there is t-privacy (in addition, any t shares are uniformly random in F t q ) and, if one considers the d-fold "component-wise" product of any d sharings, then the d-fold component-wise product of the d respective secrets is (n − t)-wise uniquely determined by it. Such schemes are a fundamental primitive in information-theoretically secure multi-party computation. Perhaps counter-intuitively, secure multi-party computation is a very powerful primitive for communication-efficient two-party cryptography, as shown recently in a series of surprising results from 2007 on. Moreover, the existence of asymptotically good arithmetic secret sharing schemes plays a crucial role in their communication-efficiency: for each d ≥ 2, if A(q) > 2d, where A(q) is Ihara's constant, then there exists an infinite family of such schemes over Fq such that n is unbounded, k = Ω(n) and t = Ω(n), as follows from a result at CRYPTO'06. Our main contribution is a novel paradigm for constructing asymptotically good arithmetic secret sharing schemes from towers of algebraic function fields. It is based on a new limit that, for a tower with a given Ihara limit and given positive integer , gives information on the cardinality of the -torsion sub-groups of the associated degree-zero divisor class groups and that we believe is of independent interest. As an application of the bounds we obtain, we relax the condition A(q) > 2d from the CRYPTO'06 result substantially in terms of our torsion-limit. As a consequence, this result now holds over nearly all finite fields Fq. For example, if d = 2, it is sufficient that q = 8, 9 or q ≥ 16.

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On Codes, Matroids and Secure Multi-party Computation from Linear Secret Sharing Schemes

Cited by 33 publications

References 26 publications

Algebraic Geometric Secret Sharing Schemes and Secure Multi-Party Computations over Small Fields

Algebraic Geometric Secret Sharing Schemes and Secure Multi-Party Computations over Small Fields

Extending Brickell–Davenport theorem to non-perfect secret sharing schemes

The Torsion-Limit for Algebraic Function Fields and Its Application to Arithmetic Secret Sharing

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