Cryptology ePrint Archive: Report 2008/385

Cube Attacks on Tweakable Black Box Polynomials

Itai Dinur and Adi Shamir

Abstract: Almost any cryptographic scheme can be described by \emph{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 \emph{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 $2^{55}$ to attack $672$ initialization rounds, whereas a cube attack can find the complete key of the same variant in $2^{19}$ bit operations (which take less than a second on a single PC). Trivium with $735$ initialization rounds (which could not be attacked by any previous technique) can now be broken with $2^{30}$ bit operations, and by extrapolating our experimentally verified complexities for various sizes, we have reasons to believe that cube attacks will remain faster than exhaustive search even for $1024$ initialization rounds. 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 $d+log_dn$. Their complexity is $2^{d-1}n+n^2$ 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 public variables. In particular, they can be easily and automatically combined with any type of side channel attack that leaks some partial information about the early stages of the encryption process (which can typically be represented by a very low degree polynomial), such as the Hamming weight of a byte written into a register.

Category / Keywords: secret-key cryptography / Cryptanalysis, algebraic attacks, cube attacks,

Date: received 13 Sep 2008, last revised 14 Sep 2008

Contact author: itai dinur at weizmann ac il

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Version: 20080914:160327 (All versions of this report)

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