Correlation Cube Attack Revisited

Abstract

In this paper, we improve the cube attack by exploiting low-degree factors of the superpoly w.r.t. certain “special” index set of cube (ISoC). This can be viewed as a special case of the correlation cube attack proposed at Eurocrypt 2018, but under our framework more beneficial equations on the key variables can be obtained in the key-recovery phase. To mount our attack, one has two challenging problems: (1) effectively recover algebraic normal form of the superpoly and extract out its low-degree factors; and (2) efficiently search a large quantity of good ISoCs. We bring in new techniques to solve both of them.

First, we propose the variable substitution technique for middle rounds of a cipher, in which polynomials on the key variables in the algebraic expressions of internal states are substituted by new variables. This will improve computational complexity of the superpoly recovery and promise more compact superpolys that can be easily decomposed with respect to the new variables. Second, we propose the vector numeric mapping technique, which seeks out a tradeoff between efficiency of the numeric mapping technique (Crypto 2019) and accuracy of the monomial prediction technique (Asiacrypt 2020) in degree evaluation of superpolys. Combining with this technique, a fast pruning method is given and modeled by MILP to filter good ISoCs of which the algebraic degree satisfies some fixed threshold. Thanks to automated MILP solvers, it becomes practical to comprehensively search for good cubes across the entire search space.

To illustrate the power of our techniques, we apply all of them to Trivium stream cipher. As a result, we have recovered the superpolys of three cubes given by Kesarwani et al. in 2020, only to find they do not have zero-sum property up to 842 rounds as claimed in their paper. To our knowledge, the previous best practical key recovery attack was on 820-round Trivium with complexity $2^{53.17}$. We put forward 820-, 825- and 830-round practical key-recovery attacks, in which there are $\mathbf {2^{80}\times 87.8\%}$, $\mathbf {2^{80}\times 83\%}$ and $\mathbf {2^{80}\times 65.7\%}$ keys that could be practically recovered, respectively, if we consider $\mathbf {2^{60}}$ as the upper bound for practical computational complexity. Besides, even for computers with computational power not exceeding $\mathbf {2^{52}}$ (resp. $\mathbf {2^{55}}$), we can still recover $\mathbf {58\%}$ (resp. $\mathbf {46.6\%}$) of the keys in the key space for 820 rounds (resp. 830 rounds). Our attacks have led 10 rounds more than the previous best practical attack.

Supported by National Natural Science Foundation of China (Grant No. 61972393).