Paper 2018/972

Observations on the Dynamic Cube Attack of 855-Round TRIVIUM from Crypto'18

Yonglin Hao, Lin Jiao, Chaoyun Li, Willi Meier, Yosuke Todo, and Qingju Wang

Abstract

Recently, another kind of dynamic cube attack is proposed by Fu et al. With some key guesses and a transformation in the output bit, they claim that, when the key guesses are correct, the degree of the transformed output bit can drop so significantly that the cubes of lower dimension can not exist, making the output bit vulnerable to the zero-sum cube tester using slightly higher dimensional cubes. They applied their method to 855-round TRIVIUM. In order to verify the correctness of their result, they even proposed a practical attack on 721-round TRIVIUM claiming that the transformed output bit after 721-rounds of initialization does not contain cubes of dimensions 31 and below. However, the degree evaluation algorithm used by Fu et al. is innovative and complicated, and its complexity is not given. Their algorithm can only be implemented on huge clusters and cannot be verified by existing theoretic tools. In this paper, we theoretically analyze the dynamic cube attack method given by Fu et al. using the division property and MILP modeling technique. Firstly, we draw links between the division property and Fu et al.'s dynamic cube attack so that their method can be described as a theoretically well founded and computationally economic MILP-aided division-property-based cube attack. With the MILP model drawn according to the division property, we analyzed the 721-round TRIVIUM in detail and find some interesting results: \begin​{enumerate} \item The degree evaluation using our MILP method is more accurate than that of Fu et al.'s. Fu et al. prove that the degree of pure z721z721 is 40 while our method gives 29. We practically proved the correctness of our method by trying thousands of random keys, random 30-dimensional cubes and random assignments to non-cube IVs finding that the summations are constantly 0. \item For the transformed output bit (1+s2901)⋅z721(1+s1290)⋅z721, we proved the same degree 31 as Fu et al. and we also find 32-dimensional cubes have zero-sum property for correct key guesses. But since the degree of pure z721z721 is only 29, the 721-round practical attack on TRIVIUM is violating the principle of Fu et al.'s work: after the transformation in the output bit, when the key guesses are correct, the degree of the transformed output bit has not dropped but risen. \item Now that the degree theoretic foundation of the 721-round attack has been violated, we also find out that the key-recovery attack cannot be carried out either. We theoretically proved and practically verified that no matter the key guesses are correct or incorrect, the summation over 32-dimensional cube are always 0. So, no key bit can be recovered at all. \end{enumerate} All these analysis on 721-round TRIVIUM can be verified practically and we open our C++ source code for implementation as well. Secondly, we revisit their 855-round result. Our MILP model reveal that the 855-round result suffers from the same problems with its 721-round counterpart. We provide theoretic evidence that, after their transformation, the degree of the output bit is more likely to rise rather than drop. Furthermore, since Fu \etal's degree evaluation is written in an unclear manner and no complexity analysis is given, we rewrite the algorithm according to their main ideas and supplement a detailed complexity analysis. Our analysis indicates that a precise evaluation to the degree requires complexities far beyond practical reach. We also demonstrate that further abbreviation to our rewritten algorithm can result in wrong evaluation. This might be the reason why Fu \etal give such a degree evaluation. This is also an additional argument against Fu \etal's dynamic cube attack method. Thirdly, the selection of Fu \etal's cube dimension is also questionable. According to our experiments and existing theoretic results, there is high risk that the correct key guesses and wrong ones share the same zero-sum property using Fu \etal's cube testers. As a remedy, we suggest that concrete cubes satisfying particular conditions should be identified rather than relying on the IV-degree drop hypothesis. To conclude, Fu \etal's dynamic cube attack on 855-round TRIVIUM is questionable. 855-round as well as 840-and-up-round TRIVIUM should still be open for further convincible cryptanalysis.