On the inherent intractability of certain coding problems (Corresp.)

Berlekamp, Elwyn R.; McEliece, Robert J.; Tilborg, H. van

doi:10.1109/tit.1978.1055873

Cited by 1,196 publications

(876 citation statements)

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“…As the second pre-image resistance is strictly weaker than collision resistance, we will only check that the hash function is collision free and resistant to inversion. We show that the inversion and collision finding are related to two problems very close to syndrome decoding, which is a hard problem [3]. We describe them here and show (in appendix) that they are also NP-complete.…”

Section: Theoretical Securitymentioning

confidence: 88%

“…The ISD algorithm consists in picking information sets at random, until a valid one is found. Checking whether the information set is valid or not is done in polynomial time 3 , so the exponential nature of the algorithm originates from the exponentially small probability of finding a valid information set: let V (r) be the cost of checking the validity of an information set, and P w the probability for a random information set to be valid; then the complexity of this algorithm is V (r)/P w .…”

Section: Information Set Decodingmentioning

confidence: 99%

“…To prove that they are NP-complete we will use a reduction similar to the one given by Berlekamp, McEliece and van Tilborg for syndrome decoding [3]. We will use the following known NP-complete problem.…”

Section: A Np-completeness Proofsmentioning

confidence: 99%

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A Family of Fast Syndrome Based Cryptographic Hash Functions

Augot

Finiasz

Sendrier

2005

Lecture Notes in Computer Science

110

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Abstract. Recently, some collisions have been exposed for a variety of cryptographic hash functions [20,21] including some of the most widely used today. Many other hash functions using similar constructions can however still be considered secure. Nevertheless, this has drawn attention on the need for new hash function designs. In this article is presented a family of secure hash functions, whose security is directly related to the syndrome decoding problem from the theory of error-correcting codes. Taking into account the analysis by Coron and Joux [4] based on Wagner's generalized birthday algorithm [19] we study the asymptotical security of our functions. We demonstrate that this attack is always exponential in terms of the length of the hash value. We also study the work-factor of this attack, along with other attacks from coding theory, for non asymptotic range, i.e. for practical values. Accordingly, we propose a few sets of parameters giving a good security and either a faster hashing or a shorter description for the function.

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Section: Theoretical Securitymentioning

confidence: 88%

Section: Information Set Decodingmentioning

confidence: 99%

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A Family of Fast Syndrome Based Cryptographic Hash Functions

Augot

Finiasz

Sendrier

2005

Lecture Notes in Computer Science

110

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“…We reduce from the Syndrome Decoding problem of a linear error correcting code that is NP-complete. The proof for the case q = 2 is to be found in [2], and an extension to the arbitrary field is sketched in [39], page 1764. Let (n, k, d) be an error correcting code.…”

Section: Minrank Is Np-hardmentioning

confidence: 99%

“…Still there are schemes using an NP-hard problem and still practical, for example PKP by Shamir [31], CLE by Stern [35] or PPP by Pointcheval [27]. However the most interesting schemes are in our opinion the schemes related to coding, as the decoding problem(s) are believed intractable even since the 1970s [2]. There were many proposals [34,40,20,16,4] and the best of them is the scheme SD by Stern [34,40].…”

Section: Introductionmentioning

confidence: 99%

Efficient Zero-Knowledge Authentication Based on a Linear Algebra Problem MinRank

Courtois

2001

Advances in Cryptology — ASIACRYPT 2001

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Abstract.A Zero-knowledge protocol provides provably secure entity authentication based on a hard computational problem. Among many schemes proposed since 1984, the most practical rely on factoring and discrete log, but still they are practical schemes based on NP-hard problems. Among them, the problem SD of decoding linear codes is in spite of some 30 years of research effort, still exponential. We study a more general problem called MinRank that generalizes SD and contains also other well known hard problems. MinRank is also used in cryptanalysis of several public key cryptosystems such as birational schemes (Crypto'93), HFE (Crypto'99), GPT cryptosystem (Eurocrypt'91), TTM (Asiacrypt'2000) and Chen's authentication scheme (1996). We propose a new Zero-knowledge scheme based on MinRank. We prove it to be Zero-knowledge by black-box simulation. An adversary able to fraud for a given MinRank instance is either able to solve it, or is able to compute a collision on a given hash function. MinRank is one of the most efficient schemes based on NP-complete problems. It can be used to prove in Zero-knowledge a solution to any problem described by multivariate equations. We also present a version with a public key shared by a few users, that allows anonymous group signatures (a.k.a. ring signatures).

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Quantum Resistant Cryptography

Yan¹

2012

Computational Number Theory and Modern Cryptography

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On the inherent intractability of certain coding problems (Corresp.)

Cited by 1,196 publications

References 3 publications

A Family of Fast Syndrome Based Cryptographic Hash Functions

A Family of Fast Syndrome Based Cryptographic Hash Functions

Efficient Zero-Knowledge Authentication Based on a Linear Algebra Problem MinRank

Quantum Resistant Cryptography

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