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Reducing Key Length of the McEliece Cryptosystem

  • Thierry P. Berger
  • Pierre-Louis Cayrel
  • Philippe Gaborit
  • Ayoub Otmani
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5580)

Abstract

The McEliece cryptosystem is one of the oldest public-key cryptosystems ever designed. It is also the first public-key cryptosystem based on linear error-correcting codes. Its main advantage is to have very fast encryption and decryption functions. However it suffers from a major drawback. It requires a very large public key which makes it very difficult to use in many practical situations. A possible solution is to advantageously use quasi-cyclic codes because of their compact representation. On the other hand, for a fixed level of security, the use of optimal codes like Maximum Distance Separable ones allows to use smaller codes. The almost only known family of MDS codes with an efficient decoding algorithm is the class of Generalized Reed-Solomon (GRS) codes. However, it is well-known that GRS codes and quasi-cyclic codes do not represent secure solutions. In this paper we propose a new general method to reduce the public key size by constructing quasi-cyclic Alternant codes over a relatively small field like \({\mathbb{F}}_{2^8}\). We introduce a new method of hiding the structure of a quasi-cyclic GRS code. The idea is to start from a Reed-Solomon code in quasi-cyclic form defined over a large field. We then apply three transformations that preserve the quasi-cyclic feature. First, we randomly block shorten the RS code. Next, we transform it to get a Generalised Reed Solomon, and lastly we take the subfield subcode over a smaller field. We show that all existing structural attacks are infeasible. We also introduce a new NP-complete decision problem called quasi-cyclic syndrome decoding. This result suggests that decoding attack against our variant has little chance to be better than the general one against the classical McEliece cryptosystem. We propose a system with several sizes of parameters from 6,800 to 20,000 bits with a security ranging from 280 to 2120.

Keywords

public-key cryptography McEliece cryptosystem Alternant code quasi-cyclic 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Thierry P. Berger
    • 1
  • Pierre-Louis Cayrel
    • 2
  • Philippe Gaborit
    • 1
  • Ayoub Otmani
    • 3
  1. 1.Université de Limoges, XLIM-DMILimoges CedexFrance
  2. 2.Département de mathématiquesUniversité Paris 8SAINT-DENIS cedex 02France
  3. 3.GREYC – EnsicaenUniversité de CaenCaen CedexFrance

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