Weak Keys for the Quasi-Cyclic MDPC Public Key Encryption Scheme

  • Magali Bardet
  • Vlad Dragoi
  • Jean-Gabriel Luque
  • Ayoub Otmani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9646)

Abstract

We analyze a new key recovery attack against the Quasi-Cyclic MDPC McEliece scheme. Retrieving the secret key from the public data is usually tackled down using exponential time algorithms aiming to recover minimum weight codewords and thus constructing an equivalent code. We use here a different approach and give under certain hypothesis an algorithm that is able to solve a key equation relating the public key to the private key. We relate this equation to a well known problem the Rational Reconstruction Problem and therefore propose a natural solution based on the extended Euclidean algorithm. All private keys satisfying the hypothesis are declared weak keys. In the same time we give a precise number of weak keys and extend our analysis by considering all possible cyclic shifts on the private keys. This task is accomplished using combinatorial objects like Lyndon words. We improve our approach by using a generalization of the Frobenius action which enables to increase the proportion of weak keys. Lastly, we implement the attack and give the probability to draw a weak key for all the security parameters proposed by the designers of the scheme.

Keywords

Quasi-cyclic MDPC codes McEliece scheme Rational reconstruction problem Extended euclidean algorithm 

Notes

Acknowledgement

We would like to thank the anonymous referees for their careful reading and helpful comments.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Magali Bardet
    • 1
  • Vlad Dragoi
    • 1
  • Jean-Gabriel Luque
    • 1
  • Ayoub Otmani
    • 1
  1. 1.Normandie Univ, France; UR, LITISMont-saint-aignanFrance

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