Cube attacks on tweakable black boxp Polynomials

Itai Dinur, Adi Shamir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

309 Scopus citations


Almost any cryptographic scheme can be described by tweakable polynomials over GF(2), which contain both secret variables (e.g., key bits) and public variables (e.g., plaintext bits or IV bits). The cryptanalyst is allowed to tweak the polynomials by choosing arbitrary values for the public variables, and his goal is to solve the resultant system of polynomial equations in terms of their common secret variables. In this paper we develop a new technique (called a cube attack) for solving such tweakable polynomials, which is a major improvement over several previously published attacks of the same type. For example, on the stream cipher Trivium with a reduced number of initialization rounds, the best previous attack (due to Fischer, Khazaei, and Meier) requires a barely practical complexity of 255 to attack 672 initialization rounds, whereas a cube attack can find the complete key of the same variant in 219 bit operations (which take less than a second on a single PC). Trivium with 735 initialization round (which could not be attacked by any previous technique) can now be broken with 230 bit operations. Trivium with 767 initialization rounds can now be broken with 245 bit operations, and the complexity of the attack can almost certainly be further reduced to about 236 bit operations. Whereas previous attacks were heuristic, had to be adapted to each cryptosystem, had no general complexity bounds, and were not expected to succeed on random looking polynomials, cube attacks are provably successful when applied to random polynomials of degree d over n secret variables whenever the number m of public variables exceeds d + logdn. Their complexity is 2d-1n + n2 bit operations, which is polynomial in n and amazingly low when d is small. Cube attacks can be applied to any block cipher, stream cipher, or MAC which is provided as a black box (even when nothing is known about its internal structure) as long as at least one output bit can be represented by (an unknown) polynomial of relatively low degree in the secret and ublic variables.

Original languageEnglish
Title of host publicationAdvances in Cryptology - EUROCRYPT 2009 - 28th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings
Number of pages22
StatePublished - 23 Jul 2009
Externally publishedYes
Event28th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2009 - Cologne, Germany
Duration: 26 Apr 200930 Apr 2009

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5479 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference28th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2009


  • Algebraic attacks
  • Cryptanalysis
  • Cube attacks
  • Stream ciphers
  • Trivium
  • Tweakable black box polynomials

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)


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