The Retracing Boomerang Attack

Boomerang attacks are extensions of differential attacks, that make it possible to combine two unrelated differential properties of the first and second part of a cryptosystem with probabilities $p$ and $q$ into a new differential-like property of the whole cryptosystem with probability $p^2q^2$ (since each one of the properties has to be satisfied twice). In this paper we describe a new version of boomerang attacks which uses the counterintuitive idea of throwing out most of the data (including potentially good cases) in order to force equalities between certain values on the ciphertext side. This creates a correlation between the four probabilistic events, which increases the probability of the combined property to $p^2q$ and increases the signal to noise ratio of the resultant distinguisher. We call this variant a retracing boomerang attack since we make sure that the boomerang we throw follows the same path on its forward and backward directions. To demonstrate the power of the new technique, we apply it to the case of 5-round AES. This version of AES was repeatedly attacked by a large variety of techniques, but for twenty years its complexity had remained stuck at $2^{32}$. At Crypto\u2718 it was finally reduced to $2^{24}$ (for full key recovery), and with our new technique we can further reduce the complexity of full key recovery to the surprisingly low value of $2^{16.5}$ (i.e., only 90,000 encryption/decryption operations are required for a full key recovery on half the rounds of AES). In addition to improving previous attacks, our new technique unveils a hidden relationship between boomerang attacks and two other cryptanalytic techniques, the yoyo game and the recently introduced mixture differentials

Truncated Boomerang Attacks and Application to AES-based Ciphers

The boomerang attack is a cryptanalysis technique that combines two short differentials instead of using a single long differential. It has been applied to many primitives, and results in the best known attacks against several AES-based ciphers (Kiasu-BC, Deoxys-BC). In this paper, we introduce a general framework for boomerang attacks with truncated differentials. While the underlying ideas are already known, we show that a careful analysis provides a significant improvement over the best boomerang attacks in the literature. In particular, we take into account structures on the plaintext and ciphertext sides, and include an analysis of the key recovery step. On 6-round AES, we obtain a structural distinguisher with complexity $2^{87}$ and a key recovery attack with complexity $2^{61}$. The truncated boomerang attacks is particularly effective against tweakable AES variants. We apply it to 8-round Kiasu-BC, resulting in the best known attack with complexity $2^{83}$ (rather than $2^{103}$). We also show an interesting use of the 6-round distinguisher on TNT-AES, a tweakable block-cipher using 6-round AES as a building block. Finally, we apply this framework to Deoxys-BC, using a MILP model to find optimal trails automatically. We obtain the best attacks against round-reduced versions of all variants of Deoxys-BC

Revisiting Related-Key Boomerang attacks on AES using computer-aided tool

In recent years, several MILP models were introduced to search automatically for boomerang distinguishers and boomerang attacks on block ciphers. However, they can only be used when the key schedule is linear. Here, a new model is introduced to deal with nonlinear key schedules as it is the case for AES. This model is more complex and actually it is too slow for exhaustive search. However, when some hints are added to the solver, it found the current best related-key boomerang attack on AES-192 with $2^{124}$ time, $2^{124}$ data, and $2^{79.8}$ memory complexities, which is better than the one presented by Biryukov and Khovratovich at ASIACRYPT 2009 with complexities $2^{176}/2^{123}/2^{152}$ respectively. This represents a huge improvement for the time and memory complexity, illustrating the power of MILP in cryptanalysis

Structure Evaluation of AES-like Ciphers against Mixture Differential Cryptanalysis

In ASIACRYPT 2017, Rønjom et al. analyzed AES with yoyo attack. Inspired by their 4-round AES distinguisher, Grassi proposed the mixture differential cryptanalysis as well as a key recovery attack on 5-round AES, which was shown to be better than the classical square attack in computation complexity. After that, Bardeh et al. combined the exchange attack with the 4-round mixture differential distinguisher of AES, leading to the first secret-key chosen plaintext distinguisher for 6-round AES. Unlike the attack on 5-round AES, the result of 6-round key-recovery attack on AES has extremely large complexity, which implies the weakness of mixture difference to a certain extent. Our work aims at evaluating the security of AES-like ciphers against mixture differential cryptanalysis. We propose a new structure called a boomerang struncture and illustrate that a differential distinguisher of a boomerang struncture just corresponds to a mixture differential distinguisher for AES-like ciphers. Based on the boomerang structure, it is shown that the mixture differential cryptanalysis is not suitable to be applied to AES-like ciphers with high round number. In specific, we associate the primitive index with our framework built on the boomerang structure and give the upper bound for the length of mixture differentials with probability 1 on AES-like ciphers. It can be directly deduced from our framework that there is no mixture differential distinguisher for 6-round AES

Quantum Boomerang Attacks and Some Applications

International audienceIn this paper, we study quantum key-recovery attacks on block ciphers. While it is well known that a quantum adversary can generically speed up an exhaustive search of the key, much less is known on how to use specific vulnerabilities of the cipher to accelerate this procedure. In this context, we show how to convert classical boomerang and mixing boomerang attacks into efficient quantum key-recovery attacks. In some cases, we can even obtain a quadratic speedup, the same as simple differential attacks. We apply this technique to a 5-round attack on SAFER++

Quantum boomerang attacks and some applications

In this paper, we study quantum key-recovery attacks on block ciphers. While it is well known that a quantum adversary can generically speed up an exhaustive search of the key, much less is known on how to use specific vulnerabilities of the cipher to accelerate this procedure. In this context, we show how to convert classical boomerang and mixing boomerang attacks into efficient quantum key-recovery attacks. In some cases, we can even obtain a quadratic speedup, the same as simple differential attacks. We apply this technique to a 5-round attack on SAFER++

Revisiting the Boomerang Attack from a Perspective of 3-differential

In this paper, inspired by the work of Beyne and Rijmen at CRYPTO 2022, we explore the accurate probability of $d$-differential in the fixed-key model. The theoretical foundations of our method are based on a special matrix $-$ quasi-$d$-differential transition matrix, which is a natural extension of the quasidifferential transition matrix. The role of quasi-$d$-differential transition matrices in polytopic cryptananlysis is analogous to that of correlation matrices in linear cryptanalysis. Therefore, the fixed-key probability of a $d$-differential can be exactly expressed as the sum of the correlations of its quasi-$d$-differential trails. Then we revisit the boomerang attack from a perspective of 3-differential. Different from previous works, the probability of a boomerang distinguisher can be exactly expressed as the sum of the correlations of its quasi-$3$-differential trails without any assumptions in our work. In order to illustrate our theory, we apply it to the lightweight block cipher GIFT. It is interesting to find the probability of every optimal 3-differential characteristic of an existing 2-round boomerang is zero, which can be seen as an evidence that the security of block ciphers adopting half-round key XOR might be overestimated previously to some extent in differential-like attacks

New Properties of Double Boomerang Connectivity Table

The double boomerang connectivity table (DBCT) is a new table proposed recently to capture the behavior of two consecutive S-boxes in boomerang attacks. In this paper, we observe an interesting property of DBCT of S-box that the ladder switch and the S-box switch happen in most cases for two continuous S-boxes, and for some S-boxes only S-box switch and ladder switch are possible. This property implies an additional criterion for S-boxes to resist the boomerang attacks and provides as well a new evaluation direction for an S-box. Using an extension of the DBCT, we verify that some boomerang distinguishers of TweAES and Deoxys are flawed. On the other hand, inspired by the property, we put forward a formula for estimating boomerang cluster probabilities. Furthermore, we introduce the first model to search for boomerang distinguishers with good cluster probabilities. Applying the model to CRAFT, we obtain 9-round and 10-round boomerang distinguishers with a higher probability than that of previous works

Approximate Modeling of Signed Difference and Digraph based Bit Condition Deduction: New Boomerang Attacks on BLAKE

The signed difference is a powerful tool for analyzing the Addition, XOR, Rotation (ARX) cryptographic primitives. Currently, solving the accurate model for the signed difference propagation is infeasible. We propose an approximate MILP modeling method capturing the propagation rules of signed differences. Unlike the accurate signed difference model, the approximate model only focuses on active bits and ignores the possible bit conditions on inactive bits. To overcome the negative effect of a lower accuracy arising from ignoring bit conditions on inactive bits, we propose an additional tool for deducing all bit conditions automatically. Such a tool is based on a directed-graph capturing the whole computation process of ARX primitives by drawing links among intermediate words and operations. The digraph is also applicable in the MILP model construction process: it enables us to identify the parameters upper bounding the number of bit conditions so as to define the objective function; it is further used to connect the boomerang top and bottom signed differential paths by introducing proper constraints to avoid incompatible intersections. Benefiting from the approximate model and the directed-graph based tool, the solving time of the new MILP model is significantly reduced, enabling us to deduce signed differential paths efficiently and accurately. To show the utility of our method, we propose boomerang attacks on the keyed permutations of three ARX hash functions of BLAKE. For the first time we mount an attack on the full 7 rounds of BLAKE3, with the complexity as low as $2^{180}$. Our best attack on BLAKE2s can improve the previously best result by 0.5 rounds but with lower complexity. The attacks on BLAKE-256 cover the same 8 rounds with the previous best result but with complexity $2^{16}$ times lower. All our results are verified practically with round-reduced boomerang quartets

Partial Sums Meet FFT: Improved Attack on 6-Round AES

The partial sums cryptanalytic technique was introduced in 2000 by Ferguson et al., who used it to break 6-round AES with time complexity of $2^{52}$ S-box computations -- a record that has not been beaten ever since. In 2014, Todo and Aoki showed that for 6-round AES, partial sums can be replaced by a technique based on the Fast Fourier Transform (FFT), leading to an attack with a comparable complexity. In this paper we show that the partial sums technique can be combined with an FFT-based technique, to get the best of the two worlds. Using our combined technique, we obtain an attack on 6-round AES with complexity of about $2^{46.4}$ additions. We fully implemented the attack experimentally, along with the partial sums attack and the Todo-Aoki attack, and confirmed that our attack improves the best known attack on 6-round AES by a factor of more than 32. We expect that our technique can be used to significantly enhance numerous attacks that exploit the partial sums technique. To demonstrate this, we use our technique to improve the best known attack on 7-round Kuznyechik by a factor of more than 80, and to reduce the complexity of the best known attack on the full MISTY1 from $2^{69.5}$ to $2^{67}$