A Practical Method to Recover Exact Superpoly in Cube Attack

Cube attack is an important cryptanalytic technique against symmetric cryptosystems, especially for stream ciphers. The key step in cube attack is recovering superpoly. However, when cube size is large, the large time complexity of recovering the exact algebraic normal form (ANF) of superpoly confines cube attack. At CRYPTO 2017, Todo et al. applied conventional bit-based division property (CBDP) into cube attack which could exploit large cube sizes. However, CBDP based cube attacks cannot ensure that the superpoly of a cube is non-constant. Hence the key recovery attack may be just a distinguisher. Moreover, CBDP based cube attacks can only recover partial ANF coefficients of superpoly. The time complexity of recovering the reminding ANF coefficients is very large, because it has to query the encryption oracle and sum over the cube set. To overcome these limits, in this paper, we propose a practical method to recover the ANF coefficients of superpoly. This new method is developed based on bit-based division property using three subsets (BDPT) proposed by Todo at FSE 2016. We apply this new method to reduced-round Trivium. To be specific, the time complexity of recovering the superpoly of 832-round Trivium at CRYPTO 2017 is reduced from $2^{77}$ to practical, and the time complexity of recovering the superpoly of 839-round Trivium at CRYPTO 2018 is reduced from $2^{79}$ to practical. Then, we propose a theoretical attack which can recover the superpoly of Trivium up to 842 round. As far as we know, this is the first time that the superpoly can be recovered for Trivium up to 842 rounds

Fault Analysis of the KATAN Family of Block Ciphers

In this paper, we investigate the security of the KATAN family of block ciphers against differential fault attacks. KATAN consists of three variants with 32, 48 and 64-bit block sizes, called KATAN32,KATAN48 and KATAN64, respectively. All three variants have the same key length of 80 bits. We assume a single-bit fault injection model where the adversary is supposed to be able to corrupt a single random bit of the internal state of the cipher and this fault injection process can be repeated (by resetting the cipher); i.e., the faults are transient rather than permanent. First, we determine suitable rounds for effective fault injections by analyzing distributions of low-degree (mainly, linear and quadratic) polynomial equations obtainable using the cube and extended cube attack techniques. Then, we show how to identify the exact position of faulty bits within the internal state by precomputing difference characteristics for each bit position at a given round and comparing these characteristics with ciphertext differences (XOR of faulty and non-faulty ciphertexts) during the online phase of the attack. The complexity of our attack on KATAN32 is 2^59 computations and about 115 fault injections. For KATAN48 and KATAN64, the attack requires 2^55 computations (for both variants), while the required number of fault injections is 211 and 278, respectively

A Security Analysis of IoT Encryption: Side-channel Cube Attack on Simeck32/64

Simeck, a lightweight block cipher has been proposed to be one of the encryption that can be employed in the Internet of Things (IoT) applications. Therefore, this paper presents the security of the Simeck32/64 block cipher against side-channel cube attack. We exhibit our attack against Simeck32/64 using the Hamming weight leakage assumption to extract linearly independent equations in key bits. We have been able to find 32 linearly independent equations in 32 key variables by only considering the second bit from the LSB of the Hamming weight leakage of the internal state on the fourth round of the cipher. This enables our attack to improve previous attacks on Simeck32/64 within side-channel attack model with better time and data complexity of 2^35 and 2^11.29 respectively.Comment: 12 pages, 6 figures, 4 tables, International Journal of Computer Networks & Communication

An Experimentally Verified Attack on 820-Round Trivium (Full Version)

The cube attack is one of the most important cryptanalytic techniques against Trivium. As the method of recovering superpolies becomes more and more effective, another problem of cube attacks, i.e., how to select cubes corresponding to balanced superpolies, is attracting more and more attention. It is well-known that a balanced superpoly could be used in both theoretical and practical analyses. In this paper, we present a novel framework to search for valuable cubes whose superpolies have an independent secret variable each, i.e., a linear variable not appearing in any nonlinear term. To control online complexity, valuable cubes are selected from very few large cubes. New ideas are given on the large cube construction and the subcube sieve. For the verification of this new algorithm, we apply it to Trivium. For 815-round Trivium, using one cube of size 47, we obtain more than 200 balanced superpolies containing 68 different independent secret variables. To make a trade-off between the number of cubes and computation complexity, we choose 35 balanced superpolies and mount a key-recovery attack on 815-round Trivium with a complexity of $2^{47.32}$. For 820-round Trivium, using two cubes of size 52, we obtain more than 100 balanced superpolies, which contain 54 different independent secret variables. With 30 balanced superpolies, we mount a key-recovery attack on 820-round Trivium with a complexity of $2^{53.17}$. Strong experimental evidence shows that the full key-recovery attacks on 815- and 820-round Trivium could be completed within six hours and two weeks on a PC with two RTX3090 GPUs, respectively

Links between Division Property and Other Cube Attack Variants

A theoretically reliable key-recovery attack should evaluate not only the non-randomness for the correct key guess but also the randomness for the wrong ones as well. The former has always been the main focus but the absence of the latter can also cause self-contradicted results. In fact, the theoretic discussion of wrong key guesses is overlooked in quite some existing key-recovery attacks, especially the previous cube attack variants based on pure experiments. In this paper, we draw links between the division property and several variants of the cube attack. In addition to the zero-sum property, we further prove that the bias phenomenon, the non-randomness widely utilized in dynamic cube attacks and cube testers, can also be reflected by the division property. Based on such links, we are able to provide several results: Firstly, we give a dynamic cube key-recovery attack on full Grain-128. Compared with Dinur et al.’s original one, this attack is supported by a theoretical analysis of the bias based on a more elaborate assumption. Our attack can recover 3 key bits with a complexity 297.86 and evaluated success probability 99.83%. Thus, the overall complexity for recovering full 128 key bits is 2125. Secondly, now that the bias phenomenon can be efficiently and elaborately evaluated, we further derive new secure bounds for Grain-like primitives (namely Grain-128, Grain-128a, Grain-V1, Plantlet) against both the zero-sum and bias cube testers. Our secure bounds indicate that 256 initialization rounds are not able to guarantee Grain-128 to resist bias-based cube testers. This is an efficient tool for newly designed stream ciphers for determining the number of initialization rounds. Thirdly, we improve Wang et al.’s relaxed term enumeration technique proposed in CRYPTO 2018 and extend their results on Kreyvium and ACORN by 1 and 13 rounds (reaching 892 and 763 rounds) with complexities 2121.19 and 2125.54 respectively. To our knowledge, our results are the current best key-recovery attacks on these two primitives

An Algebraic Method to Recover Superpolies in Cube Attacks

Cube attacks are an important type of key recovery attacks against NFSR-based cryptosystems. The key step in cube attacks closely related to key recovery is recovering superpolies. However, in the previous cube attacks including original, division property based, and correlation cube attacks, the algebraic normal form of superpolies could hardly be shown to be exact due to an unavoidable failure probability or a requirement of large time complexity. In this paper, we propose an algebraic method aiming at recovering the exact algebraic normal forms of superpolies practically. Our method is developed based on degree evaluation method proposed by Liu in Crypto-2017. As an illustration, we apply our method to Trivium. As a result, we recover the algebraic normal forms of some superpolies for the 818-, 835-, 837-, and 838-round Trivium. Based on these superpolies, on a large set of weak keys, we can recover at least five key bits equivalently for up to the 838-round Trivium with a complexity of about $2^{37}$. Besides, for the cube proposed by Liu in Crypto-2017 as a zero-sum distinguisher for the 838-round Trivium, it is proved that its superpoly is not zero-constant. Hopefully, our method would provide some new insights on cube attacks against NFSR-based ciphers

Improved Division Property Based Cube Attacks Exploiting Algebraic Properties of Superpoly (Full Version)

The cube attack is an important technique for the cryptanalysis of symmetric key primitives, especially for stream ciphers. Aiming at recovering some secret key bits, the adversary reconstructs a superpoly with the secret key bits involved, by summing over a set of the plaintexts/IV which is called a cube. Traditional cube attack only exploits linear/quadratic superpolies. Moreover, for a long time after its proposal, the size of the cubes has been largely confined to an experimental range, e.g., typically 40. These limits were first overcome by the division property based cube attacks proposed by Todo et al. at CRYPTO 2017. Based on MILP modelled division property, for a cube (index set) $I$, they identify the small (index) subset $J$ of the secret key bits involved in the resultant superpoly. During the precomputation phase which dominates the complexity of the cube attacks, $2^{|I|+|J|}$ encryptions are required to recover the superpoly. Therefore, their attacks can only be available when the restriction $|I|+|J|<n$ is met. In this paper, we introduced several techniques to improve the division property based cube attacks by exploiting various algebraic properties of the superpoly. 1. We propose the ``flag\u27\u27 technique to enhance the preciseness of MILP models so that the proper non-cube IV assignments can be identified to obtain a non-constant superpoly. 2. A degree evaluation algorithm is presented to upper bound the degree of the superpoly. With the knowledge of its degree, the superpoly can be recovered without constructing its whole truth table. This enables us to explore larger cubes $I$\u27s even if $|I|+|J|\geq n$. 3. We provide a term enumeration algorithm for finding the monomials of the superpoly, so that the complexity of many attacks can be further reduced. As an illustration, we apply our techniques to attack the initialization of several ciphers. To be specific, our key recovery attacks have mounted to 839-round TRIVIUM, 891-round Kreyvium, 184-round Grain-128a and 750-round ACORN respectively

Improved Division Property Based Cube Attacks Exploiting Algebraic Properties of Superpoly

The cube attack is an important technique for the cryptanalysis of symmetric key primitives, especially for stream ciphers. Aiming at recovering some secret key bits, the adversary reconstructs a superpoly with the secret key bits involved, by summing over a set of the plaintexts/IV which is called a cube. Traditional cube attack only exploits linear/quadratic superpolies. Moreover, for a long time after its proposal, the size of the cubes has been largely confined to an experimental range, e.g., typically 40. These limits were first overcome by the division property based cube attacks proposed by Todo et al. at CRYPTO 2017. Based on MILP modelled division property, for a cube (index set) II, they identify the small (index) subset JJ of the secret key bits involved in the resultant superpoly. During the precomputation phase which dominates the complexity of the cube attacks, 2|I|+|J|2|I|+|J| encryptions are required to recover the superpoly. Therefore, their attacks can only be available when the restriction |I|+|J|<n|I|+|J|<n is met. In this paper, we introduced several techniques to improve the division property based cube attacks by exploiting various algebraic properties of the superpoly. 1. We propose the ``flag'' technique to enhance the preciseness of MILP models so that the proper non-cube IV assignments can be identified to obtain a non-constant superpoly. 2. A degree evaluation algorithm is presented to upper bound the degree of the superpoly. With the knowledge of its degree, the superpoly can be recovered without constructing its whole truth table. This enables us to explore larger cubes II's even if |I|+|J|≥n|I|+|J|≥n. 3. We provide a term enumeration algorithm for finding the monomials of the superpoly, so that the complexity of many attacks can be further reduced. As an illustration, we apply our techniques to attack the initialization of several ciphers. To be specific, our key recovery attacks have mounted to 839-round TRIVIUM, 891-round Kreyvium, 184-round Grain-128a and 750-round ACORN respectively

Key Filtering in Cube Attacks from the Implementation Aspect

In cube attacks, key filtering is a basic step of identifying the correct key candidates by referring to the truth tables of superpolies. When terms of superpolies get massive, the truth table lookup complexity of key filtering increases significantly. In this paper, we propose the concept of implementation dependency dividing all cube attacks into two categories: implementation dependent and implementation independent. The implementation dependent cube attacks can only be feasible when the assumption that one encryption oracle query is more complicated than one table lookup holds. On the contrary, implementation independent cube attacks remain feasible in the extreme case where encryption oracles are implemented in the full codebook manner making one encryption query equivalent to one table lookup. From this point of view, we scrutinize existing cube attack results of stream ciphers Trivium, Grain-128AEAD, Acorn and Kreyvium. As a result, many of them turn out to be implementation dependent. Combining with the degree evaluation and divide-and-conquer techniques used for superpoly recovery, we further propose new cube attack results on Kreyvium reduced to 898, 899 and 900 rounds. Such new results not only mount to the maximal number of rounds so far but also are implementation independent

A Practical Key-Recovery Attack on 805-Round Trivium

The cube attack is one of the most important cryptanalytic techniques against Trivium. Many improvements have been proposed and lots of key-recovery attacks based on cube attacks have been established. However, among these key-recovery attacks, few attacks can recover the 80-bit full key practically. In particular, the previous best practical key-recovery attack was on 784-round Trivium proposed by Fouque and Vannet at FSE 2013 with on-line complexity about $2^{39}$. To mount a practical key-recovery attack against Trivium on a PC, a sufficient number of low-degree superpolies should be recovered, which is around 40. This is a difficult task both for experimental cube attacks and division property based cube attacks with randomly selected cubes due to lack of efficiency. In this paper, we give a new algorithm to construct candidate cubes targeting at linear superpolies in cube attacks. It is shown by our experiments that the new algorithm is very effective. In our experiments, the success probability is $100\%$ for finding linear superpolies using the constructed cubes. As a result, we mount a practical key-recovery attack on 805-round Trivium, which increases the number of attacked initialisation rounds by 21. We obtain over 1000 cubes with linear superpolies for 805-round Trivium, where 42 linearly independent ones could be selected. With these superpolies, for 805-round Trivium, the 80-bit key could be recovered within on-line complexity $2^{41.40}$, which could be carried out on a single PC equipped with a GTX-1080 GPU in several hours. Furthermore, the new algorithm is applied to 810-round Trivium, a cube of size 43 is constructed and two subcubes of size 42 with linear superpolies for 810-round Trivium are found