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Inverting HFE Is Quasipolynomial

  • Louis Granboulan
  • Antoine Joux
  • Jacques Stern
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4117)

Abstract

In the last ten years, multivariate cryptography has emerged as a possible alternative to public key cryptosystems based on hard computational problems from number theory. Notably, the HFE scheme [17] appears to combine efficiency and resistance to attacks, as expected from any public key scheme. However, its security is not yet completely understood. On one hand, since the security is related to the hardness of solving quadratic systems of multivariate binary equations, an NP complete problem, there were hopes that the system could be immune to subexponential attacks. On the other hand, several lines of attacks have been explored, based on so-called relinearization techniques [12,5], or on the use of Gröbner basis algorithms [7]. The latter approach was used to break the first HFE Challenge 1 in 96 hours on a 833 MHz Alpha workstation with 4 Gbytes of memory. At a more abstract level, Faugère and Joux discovered an algebraic invariant that explains why the computation finishes earlier than expected. In the present paper, we pursue this line and study the asymptotic behavior of these Gröbner basis based attacks. More precisely, we consider the complexity of the decryption attack which uses Gröbner bases to recover the plaintext and the complexity of a related distinguisher. We show that the decryption attack has a quasipolynomial complexity, where quasipolynomial denotes an subexponential expression much smaller than the classical subexponential expressions encountered in factoring or discrete logarithm computations. The same analysis shows that the related distinguisher has provable quasipolynomial complexity.

Keywords

Discrete Logarithm Security Parameter Random System Discrete Logarithm Problem Quadratic System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Louis Granboulan
    • 1
  • Antoine Joux
    • 2
    • 3
  • Jacques Stern
    • 1
  1. 1.Département d’InformatiqueÉcole normale supérieureParisFrance
  2. 2.DGA 
  3. 3.PRISMUniversité de Versailles St-Quentin-en-YvelinesVersaillesFrance

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