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On the Complexity of the Rank Syndrome Decoding Problem.

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Titel: On the Complexity of the Rank Syndrome Decoding Problem.
Autoren: Gaborit, Philippe, Ruatta, Olivier, Schrek, Julien
Quelle: IEEE Transactions on Information Theory; Feb2016, Vol. 62 Issue 2, p1006-1019, 14p
Schlagwörter: DECODING algorithms, COMPLEXITY (Philosophy), COMPUTATIONAL complexity, ELECTRONIC linearization, POLYNOMIAL time algorithms
Abstract: In this paper, we propose two new generic attacks on the rank syndrome decoding (RSD) problem. Let C be a random [n,k] rank code over GF(q^m) and let y=x+e be a received word, such that x \in C and rank (e)=r . The first attack, the support attack, is combinatorial and permits to recover an error e of rank weight r operations on GF(q) . This new attack improves the exponent for the best generic attack for the RSD problem in the case n > m , by introducing the ratio m/n in the exponential coefficient of the previously best known attacks. The second attack, the annulator polynomial attack, is an algebraic attack based on the theory of q rather than in GF(q) as it is usually the case. We consider two approaches to solve the problem in this new setting. The linearization technique shows that if n \ge (k+1) (r+1)-1 the RSD problem can be solved in polynomial time. More generally, we prove that if , the RSD problem can be solved with an average complexity of O(r^3k^3q^r\lceil (((r+1)(k+1)-(n+1))/r) \rceil ) operations in the base field GF(q) . We also consider solving with Gröbner bases for which we discuss theoretical complexity, we also consider hybrid solving with Gröbner bases on practical parameters. As an example of application, we use our new attacks on all recent cryptosystems parameters, which repair the GPT cryptosystem, we break all examples of published proposed parameters, and some parameters are broken in less than 1 s in certain cases. [ABSTRACT FROM AUTHOR]
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  Data: On the Complexity of the Rank Syndrome Decoding Problem.
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  Data: <searchLink fieldCode="AR" term="%22Gaborit%2C+Philippe%22">Gaborit, Philippe</searchLink><br /><searchLink fieldCode="AR" term="%22Ruatta%2C+Olivier%22">Ruatta, Olivier</searchLink><br /><searchLink fieldCode="AR" term="%22Schrek%2C+Julien%22">Schrek, Julien</searchLink>
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  Data: IEEE Transactions on Information Theory; Feb2016, Vol. 62 Issue 2, p1006-1019, 14p
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  Data: <searchLink fieldCode="DE" term="%22DECODING+algorithms%22">DECODING algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22COMPLEXITY+%28Philosophy%29%22">COMPLEXITY (Philosophy)</searchLink><br /><searchLink fieldCode="DE" term="%22COMPUTATIONAL+complexity%22">COMPUTATIONAL complexity</searchLink><br /><searchLink fieldCode="DE" term="%22ELECTRONIC+linearization%22">ELECTRONIC linearization</searchLink><br /><searchLink fieldCode="DE" term="%22POLYNOMIAL+time+algorithms%22">POLYNOMIAL time algorithms</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: In this paper, we propose two new generic attacks on the rank syndrome decoding (RSD) problem. Let C be a random [n,k] rank code over GF(q^m) and let y=x+e be a received word, such that x \in C and rank (e)=r . The first attack, the support attack, is combinatorial and permits to recover an error e of rank weight r operations on GF(q) . This new attack improves the exponent for the best generic attack for the RSD problem in the case n > m , by introducing the ratio m/n in the exponential coefficient of the previously best known attacks. The second attack, the annulator polynomial attack, is an algebraic attack based on the theory of q rather than in GF(q) as it is usually the case. We consider two approaches to solve the problem in this new setting. The linearization technique shows that if n \ge (k+1) (r+1)-1 the RSD problem can be solved in polynomial time. More generally, we prove that if , the RSD problem can be solved with an average complexity of O(r^3k^3q^r\lceil (((r+1)(k+1)-(n+1))/r) \rceil ) operations in the base field GF(q) . We also consider solving with Gröbner bases for which we discuss theoretical complexity, we also consider hybrid solving with Gröbner bases on practical parameters. As an example of application, we use our new attacks on all recent cryptosystems parameters, which repair the GPT cryptosystem, we break all examples of published proposed parameters, and some parameters are broken in less than 1 s in certain cases. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label:
  Group: Ab
  Data: <i>Copyright of IEEE Transactions on Information Theory is the property of IEEE and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1109/TIT.2015.2511786
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        Text: English
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        Type: general
      – SubjectFull: COMPLEXITY (Philosophy)
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      – SubjectFull: COMPUTATIONAL complexity
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      – SubjectFull: ELECTRONIC linearization
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      – SubjectFull: POLYNOMIAL time algorithms
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              Text: Feb2016
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