Cube attack

About: Cube attack is a research topic. Over the lifetime, 128 publications have been published within this topic receiving 2925 citations.

Higher Order Derivatives and Differential Cryptanalysis

Abstract: High-order derivatives of multi-variable functions are studied in this paper as a natural generalization of the basic concept used in differential cryptanalysis. Possible applications of such derivatives in cryptology are discussed.

Integral Cryptanalysis

Abstract: This paper considers a cryptanalytic approach called integral cryptanalysis. It can be seen as a dual to differential cryptanalysis and applies to ciphers not vulnerable to differential attacks. The method is particularlyapplicable to block ciphers which use bijective components only.

Cube Attacks on Tweakable Black Box Polynomials

Abstract: 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 rounds (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 + log d n . 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.

Cube Testers and Key Recovery Attacks on Reduced-Round MD6 and Trivium

Abstract: CRYPTO 2008 saw the introduction of the hash function MD6 and of cube attacks, a type of algebraic attack applicable to cryptographic functions having a low-degree algebraic normal form over GF(2). This paper applies cube attacks to reduced round MD6, finding the full 128-bit key of a 14-round MD6 with complexity 222 (which takes less than a minute on a single PC). This is the best key recovery attack announced so far for MD6. We then introduce a new class of attacks called cube testers, based on efficient property-testing algorithms, and apply them to MD6 and to the stream cipher Trivium. Unlike the standard cube attacks, cube testers detect nonrandom behavior rather than performing key extraction, but they can also attack cryptographic schemes described by nonrandom polynomials of relatively high degree. Applied to MD6, cube testers detect nonrandomness over 18 rounds in 217 complexity; applied to a slightly modified version of the MD6 compression function, they can distinguish 66 rounds from random in 224 complexity. Cube testers give distinguishers on Trivium reduced to 790 rounds from random with 230 complexity and detect nonrandomness over 885 rounds in 227, improving on the original 767-round cube attack.

Breaking Grain-128 with dynamic cube attacks

Abstract: We present a new variant of cube attacks called a dynamic cube attack. Whereas standard cube attacks [4] find the key by solving a system of linear equations in the key bits, the new attack recovers the secret key by exploiting distinguishers obtained from cube testers. Dynamic cube attacks can create lower degree representations of the given cipher, which makes it possible to attack schemes that resist all previously known attacks. In this paper we concentrate on the well-known stream cipher Grain-128 [6], on which the best known key recovery attack [15] can recover only 2 key bits when the number of initialization rounds is decreased from 256 to 213. Our first attack runs in practical time complexity and recovers the full 128-bit key when the number of initialization rounds in Grain-128 is reduced to 207. Our second attack breaks a Grain-128 variant with 250 initialization rounds and is faster than exhaustive search by a factor of about 228. Finally, we present an attack on the full version of Grain-128 which can recover the full key but only when it belongs to a large subset of 2-10 of the possible keys. This attack is faster than exhaustive search over the 2118 possible keys by a factor of about 215. All of our key recovery attacks are the best known so far, and their correctness was experimentally verified rather than extrapolated from smaller variants of the cipher. This is the first time that a cube attack was shown to be effective against the full version of a well known cipher which resisted all previous attacks.