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

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.