Boomerang Connectivity Table Revisited

The boomerang attack is a variant of differential cryptanalysis which regards a block cipher $E$ as the composition of two sub-ciphers, i.e., $E=E_1\circ E_0$, and which constructs distinguishers for $E$ with probability $p^2q^2$ by combining differential trails for $E_0$ and $E_1$ with probability $p$ and $q$ respectively. However, the validity of this attack relies on the dependency between the two differential trails. Murphy has shown cases where probabilities calculated by $p^2q^2$ turn out to be zero, while techniques such as boomerang switches proposed by Biryukov and Khovratovich give rise to probabilities greater than $p^2q^2$. To formalize such dependency to obtain a more accurate estimation of the probability of the distinguisher, Dunkelman et al. proposed the sandwich framework that regards $E$ as $\tilde{E_1}\circ E_m \circ \tilde{E_0}$, where the dependency between the two differential trails is handled by a careful analysis of the probability of the middle part $E_m$. Recently, Cid et al. proposed the Boomerang Connectivity Table (BCT) which unifies the previous switch techniques and incompatibility together and evaluates the probability of $E_m$ theoretically when $E_m$ is composed of a single S-box layer. In this paper, we revisit the BCT and propose a generalized framework which is able to identify the actual boundaries of $E_m$ which contains dependency of the two differential trails and systematically evaluate the probability of $E_m$ with any number of rounds. To demonstrate the power of this new framework, we apply it to two block ciphers SKNNY and AES. In the application to SKNNY, the probabilities of four boomerang distinguishers are re-evaluated. It turns out that $E_m$ involves 5 or 6 rounds and the probabilities of the full distinguishers are much higher than previously evaluated. In the application to AES, the new framework is used to exclude incompatibility and find high probability distinguishers of AES-128 under the related-subkey setting. As a result, a 6-round distinguisher with probability $2^{-109.42}$ is constructed. Lastly, we discuss the relation between the dependency of two differential trails in boomerang distinguishers and the properties of components of the cipher

Boomerang Connectivity Table Revisited. Application to SKINNY and AES

The boomerang attack is a variant of differential cryptanalysis which regards a block cipher E as the composition of two sub-ciphers, i.e., E = E1 o E0, and which constructs distinguishers for E with probability p2q2 by combining differential trails for E0 and E1 with probability p and q respectively. However, the validity of this attack relies on the dependency between the two differential trails. Murphy has shown cases where probabilities calculated by p2q2 turn out to be zero, while techniques such as boomerang switches proposed by Biryukov and Khovratovich give rise to probabilities greater than p2q2. To formalize such dependency to obtain a more accurate estimation of the probability of the distinguisher, Dunkelman et al. proposed the sandwich framework that regards E as Ẽ1 o Em o Ẽ0, where the dependency between the two differential trails is handled by a careful analysis of the probability of the middle part Em. Recently, Cid et al. proposed the Boomerang Connectivity Table (BCT) which unifies the previous switch techniques and incompatibility together and evaluates the probability of Em theoretically when Em is composed of a single S-box layer. In this paper, we revisit the BCT and propose a generalized framework which is able to identify the actual boundaries of Em which contains dependency of the two differential trails and systematically evaluate the probability of Em with any number of rounds. To demonstrate the power of this new framework, we apply it to two block ciphers SKINNY and AES. In the application to SKINNY, the probabilities of four boomerang distinguishers are re-evaluated. It turns out that Em involves5 or 6 rounds and the probabilities of the full distinguishers are much higher than previously evaluated. In the application to AES, the new framework is used to exclude incompatibility and find high probability distinguishers of AES-128 under the related-subkey setting. As a result, a 6-round distinguisher with probability 2−109.42 is constructed. Lastly, we discuss the relation between the dependency of two differential trails in boomerang distinguishers and the properties of components of the cipher

Key Structures: Improved Related-Key Boomerang Attack against the Full AES-256

This paper introduces structure to key, in the related-key attack settings. While the idea of structure has been long used in keyrecovery attacks against block ciphers to enjoy the birthday effect, the same had not been applied to key materials due to the fact that key structure results in uncontrolled differences in key and hence affects the validity or probabilities of the differential trails. We apply this simple idea to improve the related-key boomerang attack against AES-256 by Biryukov and Khovratovich in 2009. Surprisingly, it turns out to be effective, i.e., both data and time complexities are reduced by a factor of about 2^8, to 2^92 and 2^91 respectively, at the cost of the amount of required keys increased from 4 to 2^19. There exist some tradeoffs between the data/time complexity and the number of keys. To the best of our knowledge, this is the first essential improvement of the attack against the full AES-256 since 2009. It will be interesting to see if the structure technique can be applied to other AES-like block ciphers, and to tweaks rather than keys of tweakable block ciphers so the amount of required keys of the attack will not be affected

Generalized Related-Key Rectangle Attacks on Block Ciphers with Linear Key Schedule: Applications to SKINNY and GIFT

This paper gives a new generalized key-recovery model of related-key rectangle attacks on block ciphers with linear key schedules. The model is quite optimized and applicable to various block ciphers with linear key schedule. As a proof of work, we apply the new model to two very important block ciphers, i.e. SKINNY and GIFT, which are basic modules of many candidates of the Lightweight Cryptography (LWC) standardization project by NIST. For SKINNY, we reduce the complexity of the best previous 27-round related-tweakey rectangle attack on SKINNY-128-384 from $2^{331}$ to $2^{294}$. In addition, the first 28-round related-tweakey rectangle attack on SKINNY-128-384 is given, which gains one more round than before. For the case of GIFT-64, we give the first 24-round related-key rectangle attack with a time complexity $2^{91.58}$, while the best previous attack on GIFT-64 only reaches 23 rounds at most

Differential Cryptanalysis of WARP

WARP is an energy-efficient lightweight block cipher that is currently the smallest 128-bit block cipher in terms of hardware. It was proposed by Banik et al. in SAC 2020 as a lightweight replacement for AES-128 without changing the mode of operation. This paper proposes key-recovery attacks on WARP based on differential cryptanalysis in single and related-key settings. We searched for differential trails for up to 20 rounds of WARP, with the first 19 having optimal differential probabilities. We also found that the cipher has a strong differential effect, whereby 16 to 20-round differentials have substantially higher probabilities than their corresponding individual trails. A 23-round key-recovery attack was then realized using an 18-round differential distinguisher. Next, we formulated an automatic boomerang search using SMT that relies on the Feistel Boomerang Connectivity Table to identify valid switches. We designed the search as an add-on to the CryptoSMT tool, making it applicable to other Feistel-like ciphers such as TWINE and LBlock-s. For WARP, we found a 21-round boomerang distinguisher which was used in a 24-round rectangle attack. In the related-key setting, we describe a family of 2-round iterative differential trails, which we used in a practical related-key attack on the full 41-round WARP

Autocorrelations of Vectorial Boolean Functions

International audienceRecently, BarOn et al. introduced at Eurocrypt'19 a new tool, called the differential-linear connectivity table (DLCT), which allows for taking into account the dependency between the two subciphers E0 and E1 involved in differential-linear attacks. This paper presents a theoretical characterization of the DLCT, which corresponds to an autocorrelation table (ACT) of a vectorial Boolean function. We further provide some new theoretical results on ACTs of vectorial Boolean functions

Switching the Top Slice of the Sandwich with Extra Filling Yields a Stronger Boomerang for NLFSR-based Block Ciphers

The Boomerang attack was one of the first attempts to visualize a cipher ($E$) as a composition of two sub-ciphers ($E_0\circ E_1$) to devise and exploit two high-probability (say $p,q$) shorter trails instead of relying on a single low probability (say $s$) longer trail for differential cryptanalysis. The attack generally works whenever $p^2 \cdot q^2 > s$. However, it was later succeeded by the so-called ``sandwich attack\u27\u27 which essentially splits the cipher in three parts $E\u27_0\circ E_m \circ E\u27_1$ adding an additional middle layer ($E_m$) with distinguishing probability of $p^2\cdot r\cdot q^2$. It is primarily the generalization of a body of research in this direction that investigate what is referred to as the switching activity and capture the dependencies and potential incompatibilities of the layers that the middle layer separates. This work revisits the philosophy of the sandwich attack over multiple rounds for NLFSR-based block ciphers and introduces a new method to find high probability boomerang distinguishers. The approach formalizes boomerang attacks using only ladder, And switches. The cipher is treated as $E = E_m \circ E_1$, a specialized form of a sandwich attack which we called as the ``open-sandwich attack\u27\u27. The distinguishing probability for this attack configuration is $r \cdot q^2$. Using this innovative approach, the study successfully identifies a deterministic boomerang distinguisher for the keyed permutation of the TinyJambu cipher over 320 rounds. Additionally, a 640-round boomerang with a probability of $2^{-22}$ is presented with 95% success rate. In the related-key setting, we unveil full-round boomerangs with probabilities of $2^{-19}$, $2^{-18}$, and $2^{-12}$ for all three variants, demonstrating a 99% success rate. Similarly, for Katan-32, a more effective related-key boomerang spanning 140 rounds with a probability of $2^{-15}$ is uncovered with 70% success rate. Further, in the single-key setting, a 84-round boomerang with probability $2^{-30}$ found with success rate of 60%. This research deepens the understanding of boomerang attacks, enhancing the toolkit for cryptanalysts to develop efficient and impactful attacks on NLFSR-based block ciphers