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

Courtois, Nicolas T.

doi:10.1007/3-540-45682-1_24

Cited by 60 publications

(50 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It has been proposed by Patarin [29] after his cryptanalysis [28] of the historical multivariate scheme C * [27]. In [26] Kipnis and Shamir proposed a key recovery attack on HFE, which reduces to the so-called MinRank [12] problem. Although the attack is not practical for the proposed parameters, it was conjectured to be sub-exponential.…”

Section: Introductionmentioning

confidence: 99%

Cryptanalysis of Multivariate and Odd-Characteristic HFE Variants

Bettale

Faugère

Perret

2011

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We investigate the security of a generalization of HFE (multivariate and odd-characteristic variants). First, we propose an improved version of the basic Kipnis-Shamir key recovery attack against HFE. Second, we generalize the Kipnis-Shamir attack to Multi-HFE. The attack reduces to solve a MinRank problem directly on the public key. This leads to an improvement of a factor corresponding to the square of the degree of the extension field. We used recent results on MinRank to show that our attack is polynomial in the degree of the extension field. It appears that multi-HFE is less secure than original HFE for equal-sized keys. Finally, adaptations of our attack overcome several variants (i.e. minus modifier and embedding). As a proof of concept, we have practically broken the most conservative parameters given by Chen, Chen, Ding, Werner and Yang in 9 days for 256 bits security. All in all, our results give a more precise picture on the (in)security of several variants of HFE proposed these last years.

show abstract

Section: Introductionmentioning

confidence: 99%

Cryptanalysis of Multivariate and Odd-Characteristic HFE Variants

Bettale

Faugère

Perret

2011

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…The problem of finding a low rank linear combination of matrices is a basic linear algebra problem [20] known as MinRank in cryptography [21]. The MinRank problem over a finite field F q is as follows.…”

Section: The Minrank Problemmentioning

confidence: 99%

“…The attacks on TTM [12], STS [13,14], Rainbow [15], HFE and MultiHFE [16,17,18,19] are all in essence based on the problem of finding a low rank linear combination of matrices, known as MinRank in cryptography [20]. This problem is NP-hard [20] and was used to design a zero-knowledge authentication scheme [21]. Although NP-hard, the instances of MinRank arising from MQ schemes are often easy, thus providing a powerful tool for finding equivalent keys in canonical form.…”

Section: Introductionmentioning

confidence: 99%

A Polynomial-Time Key-Recovery Attack on MQQ Cryptosystems

Faugère

Gligoroski²,

Perret

et al. 2015

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We investigate the security of the family of MQQ public key cryptosystems using multivariate quadratic quasigroups (MQQ). These cryptosystems show especially good performance properties. In particular, the MQQ-SIG signature scheme is the fastest scheme in the ECRYPT benchmarking of cryptographic systems (eBACS). We show that both the signature scheme MQQ-SIG and the encryption scheme MQQ-ENC, although using different types of MQQs, share a common algebraic structure that introduces a weakness in both schemes. We use this weakness to mount a successful polynomial time key-recovery attack. Our key-recovery attack finds an equivalent key using the idea of so-called good keys that reveals the structure gradually. In the process we need to solve a MinRank problem that, because of the structure, can be solved in polynomial-time assuming some mild algebraic assumptions. We highlight that our theoretical results work in characteristic 2 which is known to be the most difficult case to address in theory for MinRank attacks. Also, we emphasize that our attack works without any restriction on the number of polynomials removed from the public-key, that is, using the minus modifier. This was not the case for previous MinRank like-attacks against MQ schemes. From a practical point of view, we are able to break an MQQ-SIG instance of 80 bits security in less than 2 days, and one of the more conservative MQQ-ENC instances of 128 bits security in little bit over 9 days. Altogether, our attack shows that it is very hard to design a secure public key scheme based on an easily invertible MQQ structure.

show abstract

“…Goldwasser and Kalai [7] showed the possibility that the signature depended on (Fiege) FiatShamir can be forged. Courtois has introduced in [8] a new Zero-knowledge proof which is depended on the NP-complete problem that is named MinRank. Wolf has presented in [9] the zero knowledge protocols which are used to fix authentication problems.…”

Section: Introductionmentioning

confidence: 99%

Weak Armendariz Zero Knowledge Cryptosystem

Abduldaim¹

2017

JQCM

View full text Add to dashboard Cite

Innovative idea using ring theory is raised to build a new algorithm for zero knowledge (ZK) cryptosystem. In this paper we introduce an algorithm for zero knowledge protocol based on a specific kind of rings named weak Armendariz. On the other hand, the aim of this paper focuses on the category of noncommutative algebraic structures to describe a new algebraic scheme of zero knowledge proof using weak Armendariz rings. As a result, we employ for the first time weak Armendariz rings in the science of cryptographic which regards as a new application of this class of rings. Finally, we present a novel idea combining between abstract algebra and cryptography.

show abstract

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

Cited by 60 publications

References 31 publications

Cryptanalysis of Multivariate and Odd-Characteristic HFE Variants

Cryptanalysis of Multivariate and Odd-Characteristic HFE Variants

A Polynomial-Time Key-Recovery Attack on MQQ Cryptosystems

Weak Armendariz Zero Knowledge Cryptosystem

Contact Info

Product

Resources

About