On sharing secrets and Reed-Solomon codes

McEliece, Robert J.; Sarwate, D.V.

doi:10.1145/358746.358762

Cited by 513 publications

(302 citation statements)

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“…In the case of malicious parties, the degree reduction is much more complex and works as long as t < n/3. In order to obtain some intuition as to why it is needed than t < n/3, observe that a Shamir secret sharing can also be viewed as a Reed-Solomon code of the polynomial [23]. With a polynomial of degree t, it is possible to correct up (n − t − 1)/2 errors.…”

Section: The Bgw Protocolmentioning

confidence: 99%

A Full Proof of the BGW Protocol for Perfectly Secure Multiparty Computation

Asharov

Lindell

2015

J Cryptol

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In the setting of secure multiparty computation, a set of n parties with private inputs wish to jointly compute some functionality of their inputs. One of the most fundamental results of information-theoretically secure computation was presented by Ben-Or, Goldwasser and Wigderson (BGW) in 1988. They demonstrated that any n-party functionality can be computed with perfect security, in the private channels model. When the adversary is semi-honest this holds as long as t < n/2 parties are corrupted, and when the adversary is malicious this holds as long as t < n/3 parties are corrupted. Unfortunately, a full detailed proof of these results was never given. In this paper, we remedy this situation and provide a full proof of security of the BGW protocol. We also derive corollaries for security in the presence of adaptive adversaries and under concurrent general composition (equivalently, universal composability). In addition, we give a full specification of the protocol for the malicious setting. This includes one new step for the perfect multiplication protocol in the case of n/4 ≤ t < n/3. *

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Section: The Bgw Protocolmentioning

confidence: 99%

A Full Proof of the BGW Protocol for Perfectly Secure Multiparty Computation

Asharov

Lindell

2015

J Cryptol

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“…It was observed as early as 1981 that an ideal (k, n)-threshold secret-sharing scheme can be interpreted as a Reed-Solomon code [20].…”

Section: (K N)-threshold Schemes and Reed-solomon Codesmentioning

confidence: 99%

“…Therefore, given the vector of shares possessed by the participants in a (k, n) threshold scheme we can recover the corresponding secret, even if up to t of the participants' shares have been corrupted (i.e. replaced by a different element of GF(q)), provided n − k ≥ 2t [20]. Techniques for performing such error correction for threshold secret-sharing schemes have received a certain amount of attention in the literature [23,25,26].…”

Section: (K N)-threshold Schemes and Reed-solomon Codesmentioning

confidence: 99%

“…This approach is taken by information-theoretically secure verifiable secret sharing schemes, as well as many of the secret-sharing schemes with error detection and correction capability. The second approach is to require additional shares, above and beyond what is strictly required to reconstruct the secret, in order to perform error correction [20,23,25,26]. This is also the approach adopted by error decodable secret-sharing schemes, introduced by Kurosawa [16].…”

Section: Introductionmentioning

confidence: 99%

“…Such schemes have previously been considered in the case of threshold adversaries [20,23,25,26]. Kurosawa demonstrates that in fact the question of whether or not a scheme is error decodable depends only on the properties of the adversary structure considered; for appropriate adversaries, the error-decoding ability essentially comes for free.…”

Section: Introductionmentioning

confidence: 99%

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Error decodable secret sharing and one-round perfectly secure message transmission for general adversary structures

2010

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An error decodable secret-sharing scheme is a secret-sharing scheme with the additional property that the secret can be recovered from the set of all shares, even after a coalition of participants corrupts the shares they possess. In this paper we consider schemes that can tolerate corruption by sets of participants belonging to a monotone coalition structure, thus generalising both a related notion studied by Kurosawa, and the well-known error-correction properties of threshold schemes based on Reed-Solomon codes. We deduce a necessary and sufficient condition for the existence of such schemes, and we show how to reduce the storage requirements of a technique of Kurosawa for constructing error-decodable secret-sharing schemes with efficient decoding algorithms.In addition, we explore the connection between one-round perfectly secure message transmission (PSMT) schemes with general adversary structures and secret-sharing schemes, and we exploit this connection to investigate factors affecting the performance of one-round PSMT schemes such as the number of channels required, the communication overhead, and the efficiency of message recovery.

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Fundamentals of Cryptology

Schäfer

2003

Security in Fixed and Wireless Networks

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The protection of sensitive information against unauthorized access or fraudulent changes has been of prime concern throughout the centuries. Modern communication techniques, using computers connected through networks, make all data even more vulnerable for these threats. Also, new issues have come up that were not relevant before, e.g. how to add a (digital) signature to an electronic document in such a way that the signer can not deny later on that the document was signed by him/her. Cryptology addresses the above issues. It is at the foundation of all information security. The techniques employed to this end have become increasingly mathematical of nature. This book serves as an introduction to modern cryptographic methods. After a brief survey of classical cryptosystems, it concentrates on three main areas. First of all, stream ciphers and block ciphers are discussed. These systems have extremely fast implementations, but sender and receiver have to share a secret key. Public key cryptosystems (the second main area) make it possible to protect data without a prearranged key. Their security is based on intractable mathematical problems, like the factorization of large numbers. The remaining chapters cover a variety of topics, such as zero-knowledge proofs, secret sharing schemes and authentication codes. Two appendices explain all mathematical prerequisites in great detail. One is on elementary number theory (Euclid's Algorithm, the Chinese Remainder Theorem, quadratic residues, inversion formulas, and continued fractions). The other appendix gives a thorough introduction to finite fields and their algebraic structure. This book differs from its 1988 version in two ways. That a lot of new material has been added is to be expected in a field that is developing so fast. Apart from a revision of the existing material, there are many new or greatly expanded sections, an entirely new chapter on elliptic curves and also one on authentication codes. The second difference is even more significant. The whole manuscript is electronically available as an interactive Mathematica manuscript. So, there are hyperlinks to other places in the text, but more importantly, it is now possible to work out non-trivial examples. Even a non-expert can easily alter the parameters in the examples and try out new ones. It is our experience, based on teaching at the California Institute of Technology and the Eindhoven University of Technology, that most students truly enjoy the enormous possibilities of a computer algebra notebook. Throughout the book, it has been our intention to make all Mathematica statements as transparent as possible, sometimes sacrificing elegant or smart alternatives that are too dependent on this particular computer algebra package.There are several people that have played a crucial role in the preparation of this manuscript. In alphabetical order of first name, I would like to thank Fred Simons for showing me the full potential of Mathematica for educational purposes and for enhancing many the Mathematica command...

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On sharing secrets and Reed-Solomon codes

Abstract: Shamir's scheme for sharing secrets is closely related to Reed-Solomon coding schemes. Decoding algorithms for Reed-Solomon codes provide extensions and generalizations of Shamir's method.

Cited by 513 publications

References 2 publications

A Full Proof of the BGW Protocol for Perfectly Secure Multiparty Computation

A Full Proof of the BGW Protocol for Perfectly Secure Multiparty Computation

Error decodable secret sharing and one-round perfectly secure message transmission for general adversary structures

Fundamentals of Cryptology

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