International Association
for Cryptologic Research

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 $2^{67}$.

<jats:title>Abstract</jats:title><jats:p>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 <jats:italic>p</jats:italic> and <jats:italic>q</jats:italic> into a new differential-like property of the whole cryptosystem with probability <jats:inline-formula><jats:alternatives><jats:tex-math>$$p^2q^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> (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 in order to force equalities between certain values on the ciphertext side. In certain cases, this creates a correlation between the four probabilistic events, which increases the probability of the combined property to <jats:inline-formula><jats:alternatives><jats:tex-math>$$p^2q$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>q</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and increases the signal-to-noise ratio of the resultant distinguisher. We call this variant a <jats:italic>retracing boomerang attack</jats:italic> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^{32}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mn>32</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>. At Crypto’18, it was finally reduced to <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^{24}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mn>24</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> (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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^{16.5}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mn>16.5</mml:mn> </mml:mrow> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> (i.e., only 90, 000 encryption/decryption operations are required for a full key recovery). 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.</jats:p>

A central problem in cryptanalysis is to find all the significant deviations from randomness in a given $n$-bit cryptographic primitive. When $n$ is small (e.g., an $8$-bit S-box), this is easy to do, but for large $n$, the only practical way to find such statistical properties was to exploit the internal structure of the primitive and to speed up the search with a variety of heuristic rules of thumb. However, such bottom-up techniques can miss many properties, especially in cryptosystems which are designed to have hidden trapdoors. In this paper we consider the top-down version of the problem in which the cryptographic primitive is given as a structureless black box, and reduce the complexity of the best known techniques for finding all its significant differential and linear properties by a large factor of $2^{n/2}$. Our main new tool is the idea of using {\it surrogate differentiation}. In the context of finding differential properties, it enables us to simultaneously find information about all the differentials of the form $f(x) \oplus f(x \oplus \alpha)$ in all possible directions $\alpha$ by differentiating $f$ in a single arbitrarily chosen direction $\gamma$ (which is unrelated to the $\alpha$'s). In the context of finding linear properties, surrogate differentiation can be combined in a highly effective way with the Fast Fourier Transform. For $64$-bit cryptographic primitives, this technique makes it possible to automatically find in about $2^{64}$ time all their differentials with probability $p \geq 2^{-32}$ and all their linear approximations with bias $|p| \geq 2^{-16}$; previous algorithms for these problems required at least $2^{96}$ time. Similar techniques can be used to significantly improve the best known time complexities of finding related key differentials, second-order differentials, and boomerangs. In addition, we show how to run variants of these algorithms which require no memory, and how to detect such statistical properties even in trapdoored cryptosystems whose designers specifically try to evade our techniques.

Encrypting too much data using the same key is a bad practice from a security perspective. Hence, it is customary to perform re-keying after a given amount of data is transmitted. While in many cases, the re-keying is done using a fresh execution of some key exchange protocol (e.g., in IKE or TLS), there are scenarios where internal re-keying, i.e., without exchange of information, is performed, mostly due to performance reasons.Originally suggested by Abdalla and Bellare, there are several proposals on how to perform this internal re-keying mechanism. For example, Liliya et al. offered the CryptoPro Key Meshing (CPKM) to be used together with GOST 28147-89 (known as the GOST block cipher). Later, ISO and the IETF adopted the Advanced CryptoPro Key Meshing (ACKPM) in ISO 10116 and RFC 8645, respectively.In this paper, we study the security of ACPKM and CPKM. We show that the internal re-keying suffers from an entropy loss in successive repetitions of the rekeying mechanism. We show some attacks based on this issue. The most prominent one has time and data complexities of O(2κ/2) and success rate of O(2−κ/4) for a κ-bit key.Furthermore, we show that a malicious block cipher designer or a faulty implementation can exploit the ACPKM (or the original CPKM) mechanism to significantly hinder the security of a protocol employing ACPKM (or CPKM). Namely, we show that in such cases, the entropy of the re-keyed key can be greatly reduced.

The block cipher GOST 28147-89 was the Russian Federation encryption standard for over 20 years, and is still one of its two standard block ciphers. GOST is a 32-round Feistel construction, whose security benefits from the fact that the S-boxes used in the design are kept secret. In the last 10 years, several attacks on the full 32-round GOST were presented. However, they all assume that the S-boxes are known. When the S-boxes are secret, all published attacks either target a small number of rounds, or apply for small sets of weak keys. In this paper we present the first practical-time attack on GOST with secret S-boxes. The attack works in the related-key model and is faster than all previous attacks in this model which assume that the S-boxes are known. The complexity of the attack is less than $2^{27}$ encryptions. It was fully verified, and runs in a few seconds on a PC. The attack is based on a novel type of related-key differentials of GOST, inspired by local collisions. Our new technique may be applicable to certain GOST-based hash functions as well. To demonstrate this, we show how to find a collision on a Davies-Meyer construction based on GOST with an arbitrary initial value, in less than $2^{10}$ hash function evaluations.

TinyJAMBU is one of the finalists in the NIST lightweight cryptography competition. It is considered to be one of the more efficient ciphers in the competition and has undergone extensive analysis in recent years as both the keyed permutation as well as the mode are new designs. In this paper we present a related-key forgery attack on the updated TinyJAMBU-v2 scheme with 256- and 192-bit keys. We introduce a high probability related-key differential attack where the differences are only introduced into the key state. Therefore, the characteristic is applicable to the TinyJAMBU mode and can be used to mount a forgery attack. The time and data complexity of the forgery are 233 using 214 related-keys for the 256-bit key version, and 243 using 216 related-keys for the 192-bit key version.For the 128-bit key we construct a related-key differential characteristic on the full keyed permutation of TinyJAMBU with a probability of 2−16. We extend the relatedkey differential characteristics on TinyJAMBU to practical-time key-recovery attacks that extract the full key from the keyed permutation with a time and data complexity of 224, 221, and 219 for respectively the 128-, 192-, and 256-bit key variants.All characteristics are experimentally verified and we provide key nonce pairs that produce the same tag to show the feasibility of the forgery attack. We note that the designers do not claim related-key security, however, the attacks proposed in this paper suggest that the scheme is not key-commiting, which has been recently identified as a favorable property for AEAD schemes.

We introduce the QARMAv2 family of tweakable block ciphers. It is a redesign of QARMA (from FSE 2017) to improve its security bounds and allow for longer tweaks, while keeping similar latency and area. The wider tweak input caters to both specific use cases and the design of modes of operation with higher security bounds. This is achieved through new key and tweak schedules, revised S-Box and linear layer choices, and a more comprehensive security analysis. QARMAv2 offers competitive latency and area in fully unrolled hardware implementations.Some of our results may be of independent interest. These include: new MILP models of certain classes of diffusion matrices; the comparative analysis of a full reflection cipher against an iterative half-cipher; our boomerang attack framework; and an improved approach to doubling the width of a block cipher.

The Keccak sponge function family, designed by Bertoni et al. in 2007, was selected by the U.S. National Institute of Standards and Technology (NIST) in 2012 as the next generation of Secure Hash Algorithm (SHA-3). Due to its theoretical and practical importance, cryptanalysis of SHA-3 has attracted a lot of attention. Currently, the most powerful collision attack on SHA-3 is Jian Guo et al.’s linearisation technique. However, this technique is infeasible for variants with asmaller input space, such as SHA-3-384.In this work we improve upon previous results by utilising three ideas which were not used in previous works on collision attacks against SHA-3. First, we use 2-block messages instead of 1-block messages, to reduce constraints and increase flexibility in our solutions. Second, we reduce the connectivity problem into a satisfiability (SAT) problem, instead of applying the linearisation technique. Finally, we propose an efficient deduce-and-sieve algorithm on the basis of two new non-random propertiesof the Keccak non-linear layer.The resulting collision-finding algorithm on 4-round SHA-3-384 has a practical time complexity of 259.64 (and a memory complexity of 245.94). This greatly improves upon the best known collision attack so far: Dinur et al. achieved an impractical 2147 time complexity. Our attack does not threaten the security margin of the SHA-3 hash function. However, the tools developed in this paper could be used to analyse other cryptographic primitives as well as to develop new and faster SAT solvers.

Format-Preserving Encryption (FPE) schemes accept plaintexts from any finite set of values (such as social security numbers or birth dates) and produce ciphertexts that belong to the same set. They are extremely useful in practice since they make it possible to encrypt existing databases or communication packets without changing their format. Due to industry demand, NIST had standardized in 2016 two such encryption schemes called FF1 and FF3. They immediately attracted considerable cryptanalytic attention with decreasing attack complexities. The best currently known attack on the Feistel construction FF3 has data and memory complexity of ${O}(N^{11/6})$ and time complexity of ${O}(N^{17/6})$, where the input belongs to a domain of size $N \times N$. In this paper, we present and experimentally verify three improved attacks on FF3. Our best attack achieves the tradeoff curve $D=M=\tilde{O}(N^{2-t})$, $T=\tilde{O}(N^{2+t})$ for all $t \leq 0.5$. In particular, we can reduce the data and memory complexities to the more practical $\tilde{O}(N^{1.5})$, and at the same time, reduce the time complexity to $\tilde{O}(N^{2.5})$. We also identify another attack vector against FPE schemes, the {\em related-domain} attack. We show how one can mount powerful attacks when the adversary is given access to the encryption under the same key in different domains, and show how to apply it to efficiently distinguish FF3 and FF3-1 instances.

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 in order to force equalities between certain values on the ciphertext side. In certain cases, 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 {\it 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'18 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.

The slide attack is a powerful cryptanalytic tool which has the unusual property that it can break iterated block ciphers with a complexity that does not depend on their number of rounds. However, it requires complete self similarity in the sense that all the rounds must be identical. While this can be the case in Feistel structures, this rarely happens in SP networks since the last round must end with an additional post-whitening subkey. In addition, in many SP networks the final round has additional asymmetries - for example, in AES the last round omits the MixColumns operation. Such asymmetry in the last round can make it difficult to utilize most of the advanced tools which were developed for slide attacks, such as deriving from one slid pair additional slid pairs by repeatedly re-encrypting their ciphertexts. Consequently, almost all the successful applications of slide attacks against real cryptosystems (e.g., FF3, GOST, SHACAL-1, etc.) had targeted Feistel structures rather than SP networks. In this paper we overcome this last round problem by developing four new types of slide attacks. We demonstrate their power by applying them to many types of AES-like structures (with and without linear mixing in the last round, with known or secret S-boxes, with periodicity of 1,2 and 3 in their subkeys, etc). In most of these cases, the time complexity of our attack is close to $2^{n/2}$, the smallest possible complexity for most slide attacks. Our new slide attacks have several unique properties: The first uses slid sets in which each plaintext from the first set forms a slid pair with some plaintext from the second set, but without knowing the exact correspondence. The second makes it possible to create from several slid pairs an exponential number of new slid pairs which form a hypercube spanned by the given pairs. The third has the unusual property that it is always successful, and the fourth can use known messages instead of chosen messages, with only slightly higher time complexity.

Differential cryptanalysis and linear cryptanalysis are the two best-known techniques for cryptanalysis of block ciphers. In 1994, Langford and Hellman introduced the differential-linear (DL) attack based on dividing the attacked cipher E into two subciphers $$E_0$$E0 and $$E_1$$E1 and combining a differential characteristic for $$E_0$$E0 with a linear approximation for $$E_1$$E1 into an attack on the entire cipher E. The DL technique was used to mount the best known attacks against numerous ciphers, including the AES finalist Serpent, ICEPOLE, COCONUT98, Chaskey, CTC2, and 8-round DES.Several papers aimed at formalizing the DL attack, and formulating assumptions under which its complexity can be estimated accurately. These culminated in a recent work of Blondeau, Leander, and Nyberg (Journal of Cryptology, 2017) which obtained an accurate expression under the sole assumption that the two subciphers $$E_0$$E0 and $$E_1$$E1 are independent.In this paper we show that in many cases, dependency between the two subcipher s significantly affects the complexity of the DL attack, and in particular, can be exploited by the adversary to make the attack more efficient. We present the Differential-Linear Connectivity Table (DLCT) which allows us to take into account the dependency between the two subciphers, and to choose the differential characteristic in $$E_0$$E0 and the linear approximation in $$E_1$$E1 in a way that takes advantage of this dependency. We then show that the DLCT can be constructed efficiently using the Fast Fourier Transform. Finally, we demonstrate the strength of the DLCT by using it to improve differential-linear attacks on ICEPOLE and on 8-round DES, and to explain published experimental results on Serpent and on the CAESAR finalist Ascon which did not comply with the standard differential-linear framework.

In this paper we study the problem of recovering a secret S-box from its difference distribution table (DDT). While being an interesting theoretical problem on its own, the ability to recover the S-box from the DDT of a secret S-box can be used in cryptanalytic attacks where the attacker can obtain the DDT (e.g., in Bar-On et al.’s attack on GOST), in supporting theoretical analysis of the properties of difference distribution tables (e.g., in Boura et al.’s work), or in some analysis of S-boxes with unknown design criteria (e.g., in Biryukov and Perrin’s analysis).We show that using the well established relation between the DDT and the linear approximation table (LAT), one can devise an algorithm different from the straightforward guess-and-determine (GD) algorithm proposed by Boura et al. Moreover, we show how to exploit this relation, and embed the knowledge obtained from it in the GD algorithm. We tested our new algorithm on random S-boxes of different sizes, and for random 14-bit bijective S-boxes, our results outperform the GD attack by several orders of magnitude.

In this paper, we show that a large class of diverse problems have a bicomposite structure which makes it possible to solve them with a new type of algorithm called dissection , which has much better time/memory tradeoffs than previously known algorithms. A typical example is the problem of finding the key of multiple encryption schemes with r independent n -bit keys. All the previous error-free attacks required time T and memory M satisfying $$\textit{TM} = 2^{rn}$$ TM = 2 rn , and even if “false negatives” are allowed, no attack could achieve $$\textit{TM}<2^{3rn/4}$$ TM < 2 3 r n / 4 . Our new technique yields the first algorithm which never errs and finds all the possible keys with a smaller product of $$\textit{TM}$$ TM , such as $$T=2^{4n}$$ T = 2 4 n time and $$M=2^{n}$$ M = 2 n memory for breaking the sequential execution of $$\hbox {r}=7$$ r = 7 block ciphers. The improvement ratio we obtain increases in an unbounded way as r increases, and if we allow algorithms which can sometimes miss solutions, we can get even better tradeoffs by combining our dissection technique with parallel collision search. To demonstrate the generality of the new dissection technique, we show how to use it in a generic way in order to improve rebound attacks on hash functions and to solve with better time complexities (for small memory complexities) hard combinatorial search problems, such as the well-known knapsack problem.

Determining the security of AES is a central problem in cryptanalysis, but progress in this area had been slow and only a handful of cryptanalytic techniques led to significant advancements. At Eurocrypt 2017 Grassi et al. presented a novel type of distinguisher for AES-like structures, but so far all the published attacks which were based on this distinguisher were inferior to previously known attacks in their complexity. In this paper we combine the technique of Grassi et al. with several other techniques in a novel way to obtain the best known key recovery attack on 5-round AES in the single-key model, reducing its overall complexity from about $$2^{32}$$ 2 32 to less than $$2^{22}$$ 2 22 . Extending our techniques to 7-round AES, we obtain the best known attacks on reduced-round AES-192 which use practical amounts of data and memory, breaking the record for such attacks which was obtained in 2000 by the classical Square attack. In addition, we use our techniques to improve the Gilbert–Minier attack (2000) on 7-round AES, reducing its memory complexity from $$2^{80}$$ 2 80 to $$2^{40}$$ 2 40 .

Lilliput-AE is a tweakable block cipher submitted as a candidate to the NIST lightweight cryptography standardization process. It is based upon the lightweight block cipher Lilliput, whose cryptanalysis so far suggests that it has a large security margin. In this note, we present an extremely efficient forgery attack on Lilliput-AE: Given a single arbitrary message of length about $$2^{36}$$ 2 36 bytes, we can instantly produce another valid message that leads to the same tag, along with the corresponding ciphertext. The attack uses a weakness in the tweakey schedule of Lilliput-AE which leads to the existence of a related-tweak differential characteristic with probability 1 in the underlying block cipher. The weakness we exploit, which does not exist in Lilliput, demonstrates the potential security risk in using a very simple tweakey schedule in which the same part of the key/tweak is reused in every round, even when round constants are employed to prevent slide attacks. Following this attack, the Lilliput-AE submission to NIST was tweaked.

Determining the security of AES is a central problem in cryptanalysis, but progress in this area had been slow and only a handful of cryptanalytic techniques led to significant advancements. At Eurocrypt 2017 Grassi et al. presented a novel type of distinguisher for AES-like structures, but so far all the published attacks which were based on this distinguisher were inferior to previously known attacks in their complexity. In this paper we combine the technique of Grassi et al. with several other techniques to obtain the best known key recovery attack on 5-round AES in the single-key model, reducing its overall complexity from about $$2^{32}$$ to about $$2^{22.5}$$. Extending our techniques to 7-round AES, we obtain the best known attacks on AES-192 which use practical amounts of data and memory, breaking the record for such attacks which was obtained 18 years ago by the classical Square attack.

GOST 28147 is a 256-bit key 64-bit block cipher developed by the USSR, later adopted by the Russian government as a national standard. In 2010, GOST was suggested to be included in ISO/IEC 18033-3, but was rejected due to weaknesses found in its key schedule. In 2015, a new version of GOST was suggested with the purpose of mitigating such attacks. In this paper, we show that similar weaknesses exist in the new version as well. More specifically, we present a fixed-point attack on the full cipher with time complexity of 2237 encryptions. We also present a reflection attack with time complexity of 2192 for a key that is chosen from a class of 2224 weak keys. Finally, we discuss an impossible reflection attack which improves on exhaustive search by a factor of 2e, and several possible related-key attacks.