Related-Key Boomerang Attacks on GIFT with Automated Trail Search Including BCT Effect

In Eurocrypt 2018, Cid et al. proposed a novel notion called the boomerang connectivity table, which formalised the switch property in the middle round of boomerang distinguishers in a unified approach. In this paper, we present a generic model of the boomerang connectivity table with automatic search technique for the first time, and search for (related-key) boomerang distinguishers directly by combining with the search of (related-key) differential characteristics. With the technique, we are able to find 19-round related-key boomerang distinguishers in the lightweight block cipher \textsc{Gift}-64 and \textsc{Gift}-128. Interestingly, a transition that is not predictable by the conventional switches is realised in a boomerang distinguisher predicted by the boomerang connectivity table. In addition, we experimentally extend the 19-round distinguisher by one more round. A 23-round key-recovery attack is presented on \textsc{Gift}-64 based on the distinguisher, which covers more rounds than previous known results in the single-key setting. Although the designers of \textsc{Gift} do not claim related-key security, bit positions of the key addition and 16-bit rotations were chosen to optimize the related-key differential bound. Indeed, the designers evaluated related-key differential attacks. This is the first work to present better related-key attacks than the simple related-key differential attack

Anomalies and Vector Space Search: Tools for S-Box Analysis

International audienceS-boxes are functions with an input so small that the simplest way to specify them is their lookup table (LUT). How can we quantify the distance between the behavior of a given S-box and that of an S-box picked uniformly at random? To answer this question, we introduce various "anomalies". These real numbers are such that a property with an anomaly equal to should be found roughly once in a set of $2^a$ random S-boxes. First, we present statistical anomalies based on the distribution of the coefficients in the difference distribution table, linear approximation table, and for the first time, the boomerang connectivity table. We then count the number of S-boxes that have block-cipher like structures to estimate the anomaly associated to those. In order to recover these structures, we show that the most general tool for decomposing S-boxes is an algorithm efficiently listing all the vector spaces of a given dimension contained in a given set, and we present such an algorithm. Combining these approaches, we conclude that all permutations that are actually picked uniformly at random always have essentially the same cryptographic properties and the same lack of structure

Boomerang Connectivity Table:A New Cryptanalysis Tool

A boomerang attack is a cryptanalysis framework that regards a block cipher $E$ as the composition of two sub-ciphers $E_1\circ E_0$ and builds a particular characteristic for $E$ with probability $p^2q^2$ by combining differential characteristics for $E_0$ and $E_1$ with probability $p$ and $q$, respectively. Crucially the validity of this figure is under the assumption that the characteristics for $E_0$ and $E_1$ can be chosen independently. Indeed, Murphy has shown that independently chosen characteristics may turn out to be incompatible. On the other hand, several researchers observed that the probability can be improved to $p$ or $q$ around the boundary between $E_0$ and $E_1$ by considering a positive dependency of the two characteristics, e.g.~the ladder switch and S-box switch by Biryukov and Khovratovich. This phenomenon was later formalised by Dunkelman et al.~as a sandwich attack that regards $E$ as $E_1\circ E_m \circ E_0$, where $E_m$ satisfies some differential propagation among four texts with probability $r$, and the entire probability is $p^2q^2r$. In this paper, we revisit the issue of dependency of two characteristics in $E_m$, and propose a new tool called Boomerang Connectivity Table (BCT), which evaluates $r$ in a systematic and easy-to-understand way when $E_m$ is composed of a single S-box layer. With the BCT, previous observations on the S-box including the incompatibility, the ladder switch and the S-box switch are represented in a unified manner. Moreover, the BCT can detect a new switching effect, which shows that the probability around the boundary may be even higher than $p$ or $q$. To illustrate the power of the BCT-based analysis, we improve boomerang attacks against Deoxys-BC, and disclose the mechanism behind an unsolved probability amplification for generating a quartet in SKINNY. Lastly, we discuss the issue of searching for S-boxes having good BCT and extending the analysis to modular addition

Yoyo Tricks with AES

In this paper we present new fundamental properties of SPNs. These properties turn out to be particularly useful in the adaptive chosen ciphertext/plaintext setting and we show this by introducing for the first time key-independent yoyo-distinguishers for 3- to 5-rounds of AES. All of our distinguishers beat previous records and require respectively $3, 4$ and $2^{25.8}$ data and essentially zero computation except for observing differences. In addition, we present the first key-independent distinguisher for 6-rounds AES based on yoyos that preserve impossible zero differences in plaintexts and ciphertexts. This distinguisher requires an impractical amount of $2^{122.83}$ plaintext/ciphertext pairs and essentially no computation apart from observing the corresponding differences. We then present a very favorable key-recovery attack on 5-rounds of AES that requires only $2^{11.3}$ data complexity and $2^{31}$ computational complexity, which as far as we know is also a new record. All our attacks are in the adaptively chosen plaintext/ciphertext scenario. Our distinguishers for AES stem from new and fundamental properties of generic SPNs, including generic SAS and SASAS, that can be used to preserve zero differences under the action of exchanging values between existing ciphertext and plaintext pairs. We provide a simple distinguisher for 2 generic SP-rounds that requires only 4 adaptively chosen ciphertexts and no computation on the adversaries side. We then describe a generic and deterministic yoyo-game for 3 generic SP-rounds which preserves zero differences in the middle but which we are not capable of exploiting in the generic setting

Polytopic Cryptanalysis

Standard differential cryptanalysis uses statistical dependencies between the difference of two plaintexts and the difference of the respective two ciphertexts to attack a cipher. Here we introduce polytopic cryptanalysis which considers interdependencies between larger sets of texts as they traverse through the cipher. We prove that the methodology of standard differential cryptanalysis can unambiguously be extended and transferred to the polytopic case including impossible differentials. We show that impossible polytopic transitions have generic advantages over impossible differentials. To demonstrate the practical relevance of the generalization, we present new low-data attacks on round-reduced DES and AES using impossible polytopic transitions that are able to compete with existing attacks, partially outperforming these

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

A Bit-Vector Differential Model for the Modular Addition by a Constant

ARX algorithms are a class of symmetric-key algorithms constructed by Addition, Rotation, and XOR, which achieve the best software performances in low-end microcontrollers. To evaluate the resistance of an ARX cipher against differential cryptanalysis and its variants, the recent automated methods employ constraint satisfaction solvers, such as SMT solvers, to search for optimal characteristics. The main difficulty to formulate this search as a constraint satisfaction problem is obtaining the differential models of the non-linear operations, that is, the constraints describing the differential probability of each non-linear operation of the cipher. While an efficient bit-vector differential model was obtained for the modular addition with two variable inputs, no differential model for the modular addition by a constant has been proposed so far, preventing ARX ciphers including this operation from being evaluated with automated methods. In this paper, we present the first bit-vector differential model for the n-bit modular addition by a constant input. Our model contains O(log2(n)) basic bit-vector constraints and describes the binary logarithm of the differential probability. We also represent an SMT-based automated method to look for differential characteristics of ARX, including constant additions, and we provide an open-source tool ArxPy to find ARX differential characteristics in a fully automated way. To provide some examples, we have searched for related-key differential characteristics of TEA, XTEA, HIGHT, and LEA, obtaining better results than previous works. Our differential model and our automated tool allow cipher designers to select the best constant inputs for modular additions and cryptanalysts to evaluate the resistance of ARX ciphers against differential attacks.acceptedVersio

One-dimensional Model of a Gamma Klystron

A new scheme for amplification of coherent gamma rays is proposed. The key elements are crystalline undulators - single crystals with periodically bent crystallographic planes exposed to a high energy beam of charged particles undergoing channeling inside the crystals. The scheme consists of two such crystals separated by a vacuum gap. The beam passes the crystals successively. The particles perform undulator motion inside the crystals following the periodic shape of the crystallographic planes. Gamma rays passing the crystals parallel to the beam get amplified due to interaction with the particles inside the crystals. The term `gamma klystron' is proposed for the scheme because its operational principles are similar to those of the optical klystron. A more simple one-crystal scheme is considered as well for the sake of comparison. It is shown that the gamma ray amplification in the klystron scheme can be reached at considerably lower particle densities than in the one-crystal scheme, provided that the gap between the crystals is sufficiently large.Comment: RevTeX4, 22 pages, 4 figure