Algebraic-Geometric Codes

Tsfasman, M. A.; Vlăduţ, Serge

doi:10.1007/978-94-011-3810-9

Cited by 519 publications

(356 citation statements)

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“…For introductions to algebraic geometry, see for instance [20,27], or or textbooks that place special emphasis on curves over F q such as [37,36]. For an accessible, high level overview of the technicalities sketched above see for instance [28].…”

Section: If D Is Rational Then L(d) Admits a Basis Defined Over F Qmentioning

confidence: 99%

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

Chen

Cramer

2006

Lecture Notes in Computer Science

122

188

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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: If D Is Rational Then L(d) Admits a Basis Defined Over F Qmentioning

confidence: 99%

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

Chen

Cramer

2006

Lecture Notes in Computer Science

122

188

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“…For details and for the omitted proofs we refer to [18], [28], [4] for coding theory, and to [29], [26], [20], [24] for geometric Goppa codes.…”

Section: Geometric Goppa Codesmentioning

confidence: 99%

“…Their usefulness for geometric Goppa codes is pointed out in [15]. For an application of Clifford's theorem, see [26,Theorem 3.1.54]. The proposition gives the weight distribution with multiplicities through a generating function.…”

Section: The Coordinate Function B(u T G) ∈ C[u ][[T ]] Is the Genementioning

confidence: 99%

Weight distributions of geometric Goppa codes

Duursma

1999

Trans. Amer. Math. Soc.

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Abstract. The in general hard problem of computing weight distributions of linear codes is considered for the special class of algebraic-geometric codes, defined by Goppa in the early eighties. Known results restrict to codes from elliptic curves. We obtain results for curves of higher genus by expressing the weight distributions in terms of L-series. The results include general properties of weight distributions, a method to describe and compute weight distributions, and worked out examples for curves of genus two and three.

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“…2) There are two nonzero effective divisors D 1 and [11]. According to Proposition 2.4 there are two effective divisors Conversely, if C(P, G) is a decomposable code of length n = 2g + 2 and dimension k = g + 1, it is deg(G) = 2g, so according to 2.5 it is…”

Section: Decomposable Codesmentioning

confidence: 99%

“…For the properties of geometric Goppa codes we refer to the textbook [11]. In order to be able to say something about the dimension and the minimum distance of these codes we restrict to the case 2g − 2 < deg(G) < n, in which C L (X , P, G, F q ) has dimension deg(G)+1−g and at least minimum distance n − deg(G); and C Ω (X , P, G, F q ) has dimension n − deg(G) − 1 + g and at least minimum distance deg(G) − 2g + 2.…”

Section: Introductionmentioning

confidence: 99%