Boomerang Connectivity Table Research Articles

Reconstructing S-Boxes from Cryptographic Tables with Milp

Reconstructing an S-box from a cryptographic table such as difference distribution table (DDT), linear approximation table (LAT), differential-linear connectivity table (DLCT) or boomerang connectivity table (BCT) is one of the fundamental problems in symmetric-key cryptography. Till now, there are only very few known methods which can reconstruct an S-box from a given table: guess-and-determine algorithms of Boura et al. (DCC 2019) and Tian et al. (DCC 2020), sign determination algorithm of Dunkelman et al. (ToSC 2019) and STP based approach of Lu et al. (DCC 2022). In this paper we consider the reconstruction problem in an even more challenging setup where one needs to reconstruct S-boxes from a partial cryptographic table. We are able to reconstruct S-boxes when only a few number of rows of a cryptographic table is given. This problem has never been studied in the literature. We apply mixed integer linear programming (MILP) as the key tool for solving this problem. Needless to say that we can solve the reconstruction problem when the full table is given and this is the first ever application of MILP tool in solving such fundamental problems. As a further application of our method, we provide the generic MILP models which can search for S-boxes with a given cryptographic property such as differential uniformity, linearity, differential-linear uniformity or boomerang uniformity. Additionally, our method can recover a Boolean function from a given Walsh spectrum or a Boolean function with a given nonlinearity. We also introduce a new heuristic called Optimistic MILP objective that guides the model towards obtaining multiple S-boxes or Boolean functions with the same cryptographic property. We give detailed experimental results for up to 6-bit S-boxes showing the effectiveness of our technique.

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.

The c-Differential-Linear Connectivity Table of Vectorial Boolean Functions.

Vectorial Boolean functions and codes are closely related and interconnected. On the one hand, various requirements of binary linear codes are needed for their theoretical interests but, more importantly, for their practical applications (such as few-weight codes or minimal codes for secret sharing, locally recoverable codes for storage, etc.). On the other hand, various criteria and tables have been introduced to analyse the security of S-boxes that are related to vectorial Boolean functions, such as the Differential Distribution Table (DDT), the Boomerang Connectivity Table (BCT), and the Differential-Linear Connectivity Table (DLCT). In previous years, two new tables have been proposed for which the literature was pretty abundant: the c-DDT to extend the DDT and the c-BCT to extend the BCT. In the same vein, we propose extended concepts to study further the security of vectorial Boolean functions, especially the c-Walsh transform, the c-autocorrelation, and the c-differential-linear uniformity and its accompanying table, the c-Differential-Linear Connectivity Table (c-DLCT). We study the properties of these novel functions at their optimal level concerning these concepts and describe the c-DLCT of the crucial inverse vectorial (Boolean) function case. Finally, we draw new ideas for future research toward linear code designs.

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.

SAT-aided Automatic Search of Boomerang Distinguishers for ARX Ciphers

In Addition-Rotation-Xor (ARX) ciphers, the large domain size obstructs the application of the boomerang connectivity table. In this paper, we explore the problem of computing this table for a modular addition and the automatic search of boomerang characteristics for ARX ciphers. We provide dynamic programming algorithms to efficiently compute this table and its variants. These algorithms are the most efficient up to now. For the boomerang connectivity table, the execution time is 42(n − 1) simple operations while the previous algorithm costs 82(n − 1) simple operations, which generates a smaller model in the searching phase. After rewriting these algorithms with boolean expressions, we construct the corresponding Boolean Satisfiability Problem models. Two automatic search frameworks are also proposed based on these models. This is the first time bringing the SAT-aided automatic search techniques into finding boomerang attacks on ARX ciphers. Finally, under these frameworks, we find out the first verifiable 10-round boomerang trail for SPECK32/64 with probability 2−29.15 and a 12-round trail for SPECK48/72 with probability 2−44.15. These are the best distinguishers for them so far. We also perceive that the previous boomerang attacks on LEA are constructed with an incorrect computation of the boomerang connection probability. The result is then fixed by our frameworks.

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.

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.

A novel systematic byte substitution method to design strong bijective substitution box (S-box) using piece-wise-linear chaotic map

Cryptography deals with designing practical mathematical algorithms having the two primitive elements of confusion and diffusion. The security of encrypted data is highly dependent on these two primitive elements and a key. S-box is the nonlinear component present in a symmetric encryption algorithm that provides confusion. A cryptographically strong bijective S-box structure in cryptosystem ensures near-optimal resistance against cryptanalytic attacks. It provides uncertainty and nonlinearity that ensures high confidentiality and security against cryptanalysis attacks. The nonlinearity of an S-box is highly dependent on the dispersal of input data using an S-box. Cryptographic performance criteria of chaos-based S-boxes are worse than algebraic S-box design methods, especially differential probability. This article reports a novel approach to design an 8 × 8 S-box using chaos and randomization using dispersion property to S-box cryptographic properties, especially differential probability. The randomization using dispersion property is introduced within the design loop to achieve low differential uniformity possibly. Two steps are involved in generating the proposed S-box. In the first step, a piecewise linear chaotic map (PWLCM) is utilized to generate initial S-box positions. Generally, the dispersion property is a post-processing technique that measures maximum nonlinearity in a given random sequence. However, in the second step, the concept is carefully reverse engineered, and the dispersion property is used within the design loop for systematic dispersal of input substituting sequence. The proposed controlled randomization changes the probability distribution statistics of S-box’s differentials. The proposed methodology systematically substitutes the S-box positions that cause output differences to recur for a given input difference. The proposed S-box is analyzed using well-established and well-known statistical cryptographic criteria of nonlinearity, strict avalanche criteria (SAC), bit independence criteria (BIC), differential probability, and linear probability. Further, the S-box’s boomerang connectivity table (BCT) is generated to analyze its strength against boomerang attack. Boomerang is a relatively new attacking framework for cryptosystem. The proposed S-box is compared with the state-of-the-art latest related publications. Results show that the proposed S-box achieves an upper bound of cryptographic properties, especially differential probability. This work hypothesizes that highly dispersive hamming distances at output difference, generated a systematic S-box. The mixing property of chaos generated trajectories utilized for decimal mapping. To test the randomness of generated chaotic trajectories, a cryptographically secure pseudo-random sequence was generated using a chaotic map that was tested using the National Institute of Standards and Technology (NIST) NIST-800-22 test suit.

The c-Differential Uniformity and Boomerang Uniformity of Two Classes of Permutation Polynomials

The Difference Distribution Table (DDT) and the differential uniformity play a major role for the design of substitution boxes in block ciphers, since they indicate the function’s resistance against differential cryptanalysis. This concept was extended recently to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> -DDT and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> -differential uniformity, which have the potential of extending differential cryptanalysis. Recently, a new theoretical tool, the Boomerang Connectivity Table (BCT) and the corresponding boomerang uniformity were introduced to quantify the resistance of a block cipher against boomerang-style attacks. Here we concentrate on two classes (introduced recently) of permutation polynomials over finite fields of even characteristic. For one of these, which is an involution used to construct a 4-uniform permutation, we explicitly determine the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> -DDT entries and BCT entries. For the second type of function, which is a differentially 4-uniform function, we give bounds for its <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> -differential and boomerang uniformities.

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.

Cryptographically strong permutations from the butterfly structure

Boomerang connectivity table is a new tool to characterize the vulnerability of cryptographic functions against boomerang attacks. Consequently, a cryptographic function is desired to have boomerang uniformity as low as its differential uniformity. Based on generalized butterfly structures recently introduced by Canteaut, Duval and Perrin, this paper presents infinite families of permutations of $${\mathbb {F}}_{2^{2n}}$$ for a positive odd integer n, which have the best known nonlinearity and boomerang uniformity 4. Both open and closed butterfly structures are considered. The open butterflies, according to experimental results, appear not to produce permutations with boomerang uniformity 4. On the other hand, from the closed butterflies we derive a condition on coefficients $$\alpha , \beta \in {\mathbb {F}}_{2^n}$$ such that the functions $$\begin{aligned} V_i(x,y) := (R_i(x,y), R_i(y,x)), \end{aligned}$$ where $$R_i(x,y)=(x+\alpha y)^{2^i+1}+\beta y^{2^i+1}$$ and $$\gcd (i,n)=1$$ , permute $${{\mathbb {F}}}_{2^n}^2$$ and have boomerang uniformity 4. In addition, experimental results for $$n=3, 5$$ indicate that the proposed condition seems to cover all such permutations $$V_i(x,y)$$ with boomerang uniformity 4.

Boomerang uniformity of popular S-box constructions

In order to study the resistance of a block cipher against boomerang attacks, a tool called the Boomerang Connectivity Table (BCT) for S-boxes was recently introduced. Very little is known today about the properties of this table especially for bijective S-boxes defined for n variables with $$n\equiv 0 \mod 4$$ . In this work we study the boomerang uniformity of some popular constructions used for building large S-boxes, e.g. for eight variables, from smaller ones. We show that the BCTs of all the studied constructions have abnormally high values in some positions. This remark permits us in some cases to link the boomerang properties of an S-box with other well-known cryptanalytic techniques on such constructions while in other cases it leads to the discovery of new ones. A surprising outcome concerns notably the Feistel and MISTY networks. While these two structures are very similar, their boomerang uniformity can be very different. Indeed, we show that the boomerang uniformity of a Feistel network is always the worst one possible, while for a MISTY construction this quantity depends on its inner building blocks. In the second part, we investigate the boomerang uniformity under EA-equivalence for Gold and the inverse function (as used respectively in MPC-friendly ciphers and the AES) and we prove that the boomerang uniformity is EA-invariant in these cases. Finally, we present an algorithm for inverting a given BCT and provide experimental results on the size of the BCT-equivalence classes for some 4 and 8-bit S-boxes.

On the boomerang uniformity of quadratic permutations

At Eurocrypt’18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool called Boomerang Connectivity Table (BCT) for measuring the resistance of a block cipher against the boomerang attack which is an important cryptanalysis technique introduced by Wagner in 1999 against block ciphers. Next, Boura and Canteaut introduced an important parameter related to the BCT for cryptographic S-boxes called boomerang uniformity. The purpose of this paper is to present a brief state-of-the-art on the notion of boomerang uniformity of vectorial Boolean functions (or S-boxes) and provide new results. More specifically, we present a slightly different but more convenient formulation of the boomerang uniformity and prove some new identities. Moreover, we focus on quadratic permutations in even dimension and obtain general criteria by which they have optimal BCT. As a consequence of the new criteria, two previously known results can be derived, and many new quadratic permutations with optimal BCT (optimal means that the maximal value in the Boomerang Connectivity Table equals the lowest known differential uniformity) can be found. In particular, we show that the boomerang uniformity of the binomial differentially 4-uniform permutations presented by Bracken, Tan, and Tan equals 4. Furthermore, we show a link between the boomerang uniformity and the nonlinearity for some special quadratic permutations. Finally, we present a characterization of quadratic permutations with boomerang uniformity 4. With this characterization, we show that the boomerang uniformity of a quadratic permutation with boomerang uniformity 4 is preserved by the extended affine (EA) equivalence.

On the boomerang uniformity of some permutation polynomials

The boomerang attack, introduced by Wagner in 1999, is a cryptanalysis technique against block ciphers based on differential cryptanalysis. In particular it takes into consideration two differentials, one for the upper part of the cipher and one for the lower part, and it exploits the dependency of these two differentials. At Eurocrypt’18, Cid et al. introduced a new tool, called the Boomerang Connectivity Table (BCT), that permits to simplify this analysis. Next, Boura and Canteaut introduced an important parameter for cryptographic S-boxes called boomerang uniformity, that is the maximum value in the BCT. Very recently, the boomerang uniformity of some classes of permutations (in particular quadratic functions) have been studied by Li, Qu, Sun and Li, and by Mesnager, Tang and Xiong. In this paper we further study the boomerang uniformity of some non-quadratic differentially 4-uniform functions. In particular, we consider the case of the Bracken-Leander cubic function and three classes of 4-uniform functions constructed by Li, Wang and Yu, obtained from modifying the inverse functions.

A Class of Quadrinomial Permutations With Boomerang Uniformity Four

In Eurocrypt’18, Cid et al. proposed a new cryptanalysis tool called Boomerang Connectivity Table (BCT), to evaluate S-boxes of block ciphers. Later, Boura and Canteaut further investigated the new parameter Boomerang uniformity for cryptographic S-boxes. It is of great interest to find new S-boxes with low Boomerang uniformity for even dimensions. In this paper, we prove that a class of permutation quadrinomials over $\mathbb {F}_{2^{2m}}$ with $m$ odd has Boomerang uniformity four, which gives the fifth class of such kind of permutation polynomials. Further, the occurrences of 0 and 4 in the BCTs of the investigated permutation polynomials are also completely determined.

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 Results About the Boomerang Uniformity of Permutation Polynomials

In EUROCRYPT 2018, Cid et al. introduced a new concept on the cryptographic property of S-boxes: boomerang connectivity table (BCT for short) for evaluating the subtleties of boomerang-style attacks. Very recently, BCT and the boomerang uniformity, the maximum value in BCT, were further studied by Boura and Canteaut. In this paper, aiming at providing new insights, we show some new results about BCT and the boomerang uniformity of permutations in terms of theory and experiment. First, we present an equivalent technique to compute BCT and the boomerang uniformity, which seems to be much simpler than the original definition by Cid et al. Second, thanks to Carlet's idea, we give a characterization of functions f from F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> to itself with boomerang uniformity δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> by means of the Walsh transform. Third, by our method, we consider boomerang uniformities of some specific permutations, mainly the ones with low differential uniformity. Finally, we obtain another class of 4-uniform BCT permutation polynomials over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> n.

Boomerang Switch in Multiple Rounds. Application to AES Variants and Deoxys

The boomerang attack is a cryptanalysis technique that allows an attacker to concatenate two short differential characteristics. Several research results (ladder switch, S-box switch, sandwich attack, Boomerang Connectivity Table (BCT), ...) showed that the dependency between these two characteristics at the switching round can have a significant impact on the complexity of the attack, or even potentially invalidate it. In this paper, we revisit the issue of boomerang switching effect, and exploit it in the case where multiple rounds are involved. To support our analysis, we propose a tool called Boomerang Difference Table (BDT), which can be seen as an improvement of the BCT and allows a systematic evaluation of the boomerang switch through multiple rounds. In order to illustrate the power of this technique, we propose a new related-key attack on 10-round AES-256 which requires only 2 simple related-keys and 275 computations. This is a much more realistic scenario than the state-of-the-art 10-round AES-256 attacks, where subkey oracles, or several related-keys and high computational power is needed. Furthermore, we also provide improved attacks against full AES-192 and reduced-round Deoxys.

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.

On the Boomerang Uniformity of Cryptographic Sboxes

The boomerang attack is a cryptanalysis technique against block ciphers which combines two differentials for the upper part and the lower part of the cipher. The dependency between these two differentials then highly affects the complexity of the attack and all its variants. Recently, Cid et al. introduced at Eurocrypt’18 a new tool, called the Boomerang Connectivity Table (BCT) that permits to simplify this complexity analysis, by storing and unifying the different switching probabilities of the cipher’s Sbox in one table. In this seminal paper a brief analysis of the properties of these tables is provided and some open questions are raised. It is being asked in particular whether Sboxes with optimal BCTs exist for even dimensions, where optimal means that the maximal value in the BCT equals the lowest known differential uniformity. When the dimension is even and differs from 6, such optimal Sboxes correspond to permutations such that the maximal value in their DDT and in their BCT equals 4 (unless APN permutations for such dimensions exist). We provide in this work a more in-depth analysis of boomerang connectivity tables, by studying more closely differentially 4-uniform Sboxes. We first completely characterize the BCT of all differentially 4-uniform permutations of 4 bits and then study these objects for some cryptographically relevant families of Sboxes, as the inverse function and quadratic permutations. These two families provide us with the first examples of differentially 4-uniform Sboxes optimal against boomerang attacks for an even number of variables, answering the above open question.