Boomerang Distinguishers Research Articles

On Impossible Boomerang Attacks

The impossible boomerang attack, introduced in 2008 by Jiqiang Lu, is an extension of the impossible differential attack that relies on a boomerang distinguisher of probability 0 for discarding incorrect key guesses. In Lu’s work, the considered impossible boomerang distinguishers were built from 4 (different) probability-1 differentials that lead to 4 differences that do not sum to 0 in the middle, in a miss-in-the-middle way.In this article, we study the possibility of extending this notion by looking at finerlevel contradictions that derive from boomerang switch constraints. We start by discussing the case of quadratic Feistel ciphers and in particular of the Simon ciphers. We exploit their very specific boomerang constraints to enforce a contradiction that creates a new type of impossible boomerang distinguisher that we search with an SMT solver. We next switch to word-oriented ciphers and study how to leverage the Boomerang Connectivity Table contradictions. We apply this idea to SKINNYee, a recent tweakable block cipher proposed at Crypto 2022 and obtain a 21-round distinguisher.After detailing the process and the complexities of an impossible boomerang attack in the single (twea)key and related (twea)key model, we extend our distinguishers into attacks and present a 23-round impossible boomerang attack on Simon-32/64 (out of 32 rounds) and a 29-round impossible boomerang attack on SKINNYee (out of 56 rounds). To the best of our knowledge our analysis covers two more rounds than the (so far, only) other third-party analysis of SKINNYee that has been published to date.

On Boomerang Attacks on Quadratic Feistel Ciphers

The recent introduction of the Boomerang Connectivity Table (BCT) at Eurocrypt 2018 revived interest in boomerang cryptanalysis and in the need to correctly build boomerang distinguishers. Several important advances have been made on this matter, with in particular the study of the extension of the BCT theory to multiple rounds and to different types of ciphers.In this paper, we pursue these investigations by studying the specific case of quadratic Feistel ciphers, motivated by the need to look at two particularly lightweight ciphers, KATAN and Simon. Our analysis shows that their light round function leads to an extreme case, as a one-round boomerang can only have a probability of 0 or 1. We identify six papers presenting boomerang analyses of KATAN or Simon and all use the naive approach to compute the distinguisher’s probability. We are able to prove that several results are theoretically incorrect and we run experiments to check the probability of the others. Many do not have the claimed probability: it fails distinguishing in some cases, but we also identify instances where the experimental probability turns out to be better than the claimed one.To address this shortfall, we propose an SMT model taking into account the boomerang constraints. We present several experimentally-verified related-key distinguishers obtained with our new technique: on KATAN32 a 151-round boomerang and on Simon-32/64 a 17-round boomerang, a 19-round rotational-xor boomerang and a 15-round rotational-xor-differential boomerang.Furthermore, we extend our 19-round distinguisher into a 25-round rotational-xor rectangle attack on Simon-32/64. To the best of our knowledge this attack reaches one more round than previously published results.

New Properties of the 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.

Throwing Boomerangs into Feistel Structures

Automatic tools to search for boomerang distinguishers have seen significant advances over the past few years. However, most previous work has focused on ciphers based on a Substitution Permutation Network (SPN), while analyzing the Feistel structure is of great significance. Boukerrou et al. recently provided a theoretical framework to formulate the boomerang switch over multiple Feistel rounds, but they did not provide an automatic tool to find distinguishers. In this paper, by enhancing the recently proposed method by Hadipour et al., we provide an automatic tool to search for boomerang distinguishers and apply it to block ciphers following the Generalized Feistel Structure (GFS). Applying our tool to a wide range of GFS ciphers, we show that it significantly improves the best previous results on boomerang analysis. In particular, we improve the best previous boomerang distinguishers for 20 and 21 rounds of WARP by a factor of 238.28 and 236.56, respectively. Thanks to he effectiveness of our method, we can extend the boomerang distinguishers of WARP by two rounds and distinguish 23 rounds of this cipher from a random permutation. Applying our method to the internationally-standardized cipher CLEFIA, we achieve a 9-round boomerang distinguisher which improves the best previous boomerang distinguisher by one round. Based on this distinguisher, we build a key-recovery attack on 11 rounds of CLEFIA, which improves the best previous sandwich attack on this cipher by one round. We also apply our method to LBlock, LBlock-s, and TWINE and improve the best previous boomerang distinguisher of these ciphers.

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.

Improved Rectangle Attacks on SKINNY and CRAFT

The boomerang and rectangle attacks are adaptions of differential cryptanalysis that regard the target cipher E as a composition of two sub-ciphers, i.e., E = E1 ∘ E0, to construct a distinguisher for E with probability p2q2 by concatenating two short differential trails for E0 and E1 with probability p and q respectively. According to the previous research, the dependency between these two differential characteristics has a great impact on the probability of boomerang and rectangle distinguishers. Dunkelman et al. proposed the sandwich attack to formalise such dependency that regards E as three parts, i.e., E = E1 ∘ Em ∘ E0, where Em contains the dependency between two differential trails, satisfying some differential propagation with probability r. Accordingly, the entire probability is p2q2r. Recently, Song et al. have proposed a general framework to identify the actual boundaries of Em and systematically evaluate the probability of Em with any number of rounds, and applied their method to accurately evaluate the probabilities of the best SKINNY’s boomerang distinguishers. In this paper, using a more advanced method to search for boomerang distinguishers, we show that the best previous boomerang distinguishers for SKINNY can be significantly improved in terms of probability and number of rounds. More precisely, we propose related-tweakey boomerang distinguishers for up to 19, 21, 23, and 25 rounds of SKINNY-64-128, SKINNY-128-256, SKINNY-64-192 and SKINNY-128-384 respectively, which improve the previous boomerang distinguishers of these variants of SKINNY by 1, 2, 1, and 1 round respectively. Based on the improved boomerang distinguishers for SKINNY, we provide related-tweakey rectangle attacks on 23 rounds of SKINNY-64-128, 24 rounds of SKINNY-128-256, 29 rounds of SKINNY-64-192, and 30 rounds of SKINNY-128-384. It is worth noting that our improved related-tweakey rectangle attacks on SKINNY-64-192, SKINNY-128-256 and SKINNY-128-384 can be directly applied for the same number of rounds of ForkSkinny-64-192, ForkSkinny-128-256 and ForkSkinny-128-384 respectively. CRAFT is another SKINNY-like tweakable block cipher for which we provide the security analysis against rectangle attack for the first time. As a result, we provide a 14-round boomerang distinguisher for CRAFT in the single-tweak model based on which we propose a single-tweak rectangle attack on 18 rounds of this cipher. Moreover, following the previous research regarding the evaluation of switching in multiple rounds of boomerang distinguishers, we also introduce new tools called Double Boomerang Connectivity Table (DBCT), LBCT⫤, and UBCT⊨ to evaluate the boomerang switch through the multiple rounds more accurately.

Catching the Fastest Boomerangs

In this paper we describe a new tool to search for boomerang distinguishers. One limitation of the MILP model of Liu et al. is that it handles only one round for the middle part while Song et al. have shown that dependencies could affect much more rounds, for instance up to 6 rounds for SKINNY. Thus we describe a new approach to turn an MILP model to search for truncated characteristics into an MILP model to search for truncated boomerang characteristics automatically handling the middle rounds. We then show a new CP model to search for the best possible instantiations to identify good boomerang distinguishers. Finally we systematized the method initiated by Song et al. to precisely compute the probability of a boomerang. As a result, we found many new boomerang distinguishers up to 24 rounds in the TK3 model. In particular, we improved by a factor 230 the probability of the best known distinguisher against 18-round SKINNY-128/256.

On the Feistel Counterpart of the Boomerang Connectivity Table

At Eurocrypt 2018, Cid et al. introduced the Boomerang Connectivity Table (BCT), a tool to compute the probability of the middle round of a boomerang distinguisher from the description of the cipher’s Sbox(es). Their new table and the following works led to a refined understanding of boomerangs, and resulted in a series of improved attacks. Still, these works only addressed the case of Substitution Permutation Networks, and completely left out the case of ciphers following a Feistel construction. In this article, we address this lack by introducing the FBCT, the Feistel counterpart of the BCT. We show that the coefficient at row Δi, ∇o corresponds to the number of times the second order derivative at points Δi, ∇o) cancels out. We explore the properties of the FBCT and compare it to what is known on the BCT. Taking matters further, we show how to compute the probability of a boomerang switch over multiple rounds with a generic formula.

New Related-Tweakey Boomerang and Rectangle Attacks on Deoxys-BC Including BDT Effect

In the CAESAR competition, Deoxys-I and Deoxys-II are two important authenticated encryption schemes submitted by Jean et al. Recently, Deoxys-II together with Ascon, ACORN, AEGIS-128, OCB and COLM have been selected as the final CAESAR portfolio. Notably, Deoxys-II is also the primary choice for the use case “Defense in depth”. However, Deoxys-I remains to be one of the third-round candidates of the CAESAR competition. Both Deoxys-I and Deoxys-II adopt Deoxys-BC-256 and Deoxys-BC-384 as their internal tweakable block ciphers.In this paper, we investigate the security of round-reduced Deoxys-BC-256/-384 and Deoxys-I against the related-tweakey boomerang and rectangle attacks with some new boomerang distinguishers. For Deoxys-BC-256, we present 10-round related-tweakey boomerang and rectangle attacks for the popular setting (|tweak|, |key|) = (128, 128), which reach one more round than the previous attacks in this setting. Moreover, an 11-round related-tweakey rectangle attack on Deoxys-BC-256 is given for the first time. We also put forward a 13-round related-tweakey boomerang attack in the popular setting (|tweak|, |key|) = (128, 256) for Deoxys-BC-384, while the previous attacks in this setting only work for 12 rounds at most. In addition, the first 14-round relatedtweakey rectangle attack on Deoxys-BC-384 is given when (|tweak| 286), that attacks one more round than before. Besides, we give the first 10-round rectangle attack on the authenticated encryption mode Deoxys-I-128-128 with one more round than before, and we also reduce the complexity of the related-tweakey rectangle attack on 12-round Deoxys-I-256-128 by a factor of 228. Our attacks can not be applied to (round-reduced) Deoxys-II.

New Related-Tweakey Boomerang and Rectangle Attacks on Deoxys-BC Including BDT Effect

In the CAESAR competition, Deoxys-I and Deoxys-II are two important authenticated encryption schemes submitted by Jean et al. Recently, Deoxys-II together with Ascon, ACORN, AEGIS-128, OCB and COLM have been selected as the final CAESAR portfolio. Notably, Deoxys-II is also the primary choice for the use case “Defense in depth”. However, Deoxys-I remains to be one of the third-round candidates of the CAESAR competition. Both Deoxys-I and Deoxys-II adopt Deoxys-BC-256 and Deoxys-BC-384 as their internal tweakable block ciphers.In this paper, we investigate the security of round-reduced Deoxys-BC-256/-384 and Deoxys-I against the related-tweakey boomerang and rectangle attacks with some new boomerang distinguishers. For Deoxys-BC-256, we present 10-round related-tweakey boomerang and rectangle attacks for the popular setting (|tweak|, |key|) = (128, 128), which reach one more round than the previous attacks in this setting. Moreover, an 11-round related-tweakey rectangle attack on Deoxys-BC-256 is given for the first time. We also put forward a 13-round related-tweakey boomerang attack in the popular setting (|tweak|, |key|) = (128, 256) for Deoxys-BC-384, while the previous attacks in this setting only work for 12 rounds at most. In addition, the first 14-round relatedtweakey rectangle attack on Deoxys-BC-384 is given when (|tweak| < 98, |key| > 286), that attacks one more round than before. Besides, we give the first 10-round rectangle attack on the authenticated encryption mode Deoxys-I-128-128 with one more round than before, and we also reduce the complexity of the related-tweakey rectangle attack on 12-round Deoxys-I-256-128 by a factor of 228. Our attacks can not be applied to (round-reduced) Deoxys-II.

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.

Evaluate the security margins of SHA-512, SHA-256 and DHA-256 against the boomerang attack

For an $n$-bit random permutation, there are three types of boomerang distinguishers, denoted as Type I, II and III, with generic complexities $2^{n}$, $2^{n/3}$ and $2^{n/2}$ respectively. in this paper, we try to evaluate the security margins of three hash functions namely SHA-512, SHA-256 and DHA-256 against the boomerang attack. firstly, we give a boomerang attack on 48-step SHA-512 with a practical complexity of $2^{51}$. the correctness of this attack is verified by providing a Type III boomerang quartet. then, we extend the existing differential characteristics of the three hash functions to more rounds. we deduce the sufficient conditions and give thorough evaluations to the security margins as follows: type I boomerang method can attack 54-step SHA-512, 51-step SHA-256 and 46-step DHA-256 with complexities $2^{480}$, $2^{218}$ and $2^{236}$ respectively. type II boomerang method can attack 51-step SHA-512, 49-step SHA-256 and 43-step DHA-256 with complexities $2^{158.50}$, $2^{72.91}$ and $2^{74.50}$ respectively. type III boomerang method can attack 52-step SHA-512, 50-step SHA-256 and 44-step DHA-256 with complexities $2^{223.80}$, $2^{123.63}$ and $2^{99.85}$ respectively.

Improved boomerang attacks on round‐reduced SM3 and keyed permutation of BLAKE‐256

In this study, the authors study the security of hash functions SM3 and BLAKE-256 against boomerang attack. SM3 is designed by Wang et al. and published by Chinese Commercial Cryptography Administration Office for the use of electronic certification service system in China. BLAKE is one of the five finalists of the NIST SHA-3 competition submitted by Aumasson et al. For SM3, they present boomerang distinguishers for the compression function reduced to 34/35/36/37 steps out of 64 steps, with time complexities 231.4, 233.6, 273.4 and 2192, respectively. Then, they show some incompatible problems existed in the previous boomerang attacks on SM3. Meanwhile, they launch boomerang attacks on up to 7- and 8-round keyed permutation of BLAKE-256, which are the first valid 7-round and 8-round boomerangs for BLAKE-256. Especially, since the author's distinguishers on 34/35-steps compression function of SM3 and 7-round keyed permutation of BLAKE-256 are practical, they are able to obtain boomerang quartets of these attacks. As far as they know, these are the best results against round-reduced SM3 and BLAKE-256.

Revisiting iterated attacks in the context of decorrelation theory

Iterated attacks are comprised of iterating adversaries who can make d plaintext queries, in each iteration to compute a bit, and are trying to distinguish between a random cipher C and the perfect cipher C � based on all bits. Vaudenay showed that a 2d-decorrelated cipher resists to iterated attacks of order d when iterations have almost no common queries. Then, he first asked what the necessary conditions are for a cipher to resist a non-adaptive iterated attack of order d. I.e., whether decorrelation of order 2d � 1 could be sufficient. Secondly, he speculated that repeating a plaintext query in different iterations does not provide any advantage to a non-adaptive distinguisher. We close here these two long-standing open problems negatively. For those questions, we provide two counter-intuitive examples.W e also deal with adaptive iterated adversaries who can make both plaintext and ciphertext queries in which the future queries are dependent on the past queries. We show that decorrelation of order 2d protects against these attacks of order d. We also study the generalization of these distinguishers for iterations making non-binary outcomes. Finally, we measure the resistance against two well-known statistical distinguishers, namely, differential-linear and boomerang distinguishers and show that 4-decorrelation degree protects against these attacks.

Boomerang Distinguishers on MD4-Based Hash Functions: First Practical Results on Full 5-Pass HAVAL Compression Function

Boomerang Distinguishers on MD4-Based Hash Functions: First Practical Results on Full 5-Pass HAVAL Compression Function

Cryptanalysis of Reduced-Round DASH

In ACISP 2008, the hash family DASH has been proposed by Billet et al., which considers the design of Rijndael and RC6. DASH family has two variants that support 256-bit and 512-bit output length respectively. This paper presents the first third-party cryptanalysis of DASH-256 with a focus on the underlying block cipher \( {{\mathcal{A}}_{256 }} \). In particular, we study the distinguisher using differential and boomerang attack. As a result, we build a distinguishing attack for the compression function of DASH-256 with 8-round \( {{\mathcal{A}}_{256 }} \) using the differential cryptanalysis. Finally, we obtain a boomerang distinguisher of 9-round \( {{\mathcal{A}}_{256 }} \).

An Asymmetric Impossible Boomerang Attack on 7-Round AES-128

Block cipher is the core of cryptography that provides data encryption,authentication and key management in information security.The security of block cipher is an important issue in the cryptanalysis.Based on the principle of differential cryptanalysis,this paper introduces a new cryptanalytic technique on block cipher: asymmetric impossible boomerang attack.The technique used asymmetric impossible boomerang distinguisher to eliminate wrong key material and leave the right key candidate.With key schedule considerations,techniques of looking up differential tables and re-using the data,the authors apply asymmetric impossible boomerang attack to AES-128.It is shown that attack on 7-round AES-128 requires data complexity of about 2105.18 chosen plaintexts,time complexity of about 2115.2 encryptions and memory complexity of about 2106.78 AES blocks.The presented result is better than any previous published cryptanalytic results on AES-128 in terms of the numbers of attacked rounds,the data complexity and the time complexity.

Related-Key Boomerang Attack on Block Cipher SQUARE

Square is an 8-round SPN structure block cipher and its round function and key schedule have been slightly modified to design building blocks of Rijndael. Key schedule of Square is simple and efficient but fully affine, so we apply a related-key attack on it. We find a 3-round related-key differential trail with probability 2-28, which has zero differences both on its input and output states, which is called local collision in [6]. By extending of this related-key differential, we construct a successful attack on full rounds of Square. In this paper, we present a key recovery attack on full rounds of Square using a related-key boomerang distinguisher. We construct a 7-round related-key boomerang distinguisher with probability 2-119 by finding local collision, and calculate its probability using ladder switch and multiple path estimation techniques. As a result, one round on top of the distinguisher is added to construct an attack on full rounds of Square which recovers 16-bit key information with 2123 encryptions and 2121 data.