Cryptanalysis of Low-Data Instances of Full LowMCv2

LowMC is a family of block ciphers designed for a low multiplicative complexity. The specification allows a large variety of instantiations, differing in block size, key size, number of S-boxes applied per round and allowed data complexity. The number of rounds deemed secure is determined by evaluating a number of attack vectors and taking the number of rounds still secure against the best of these. In this paper, we demonstrate that the attacks considered by the designers of LowMC in the version 2 of the round-formular were not sufficient to fend off all possible attacks. In the case of instantiations of LowMC with one of the most useful settings, namely with few applied S-boxes per round and only low allowable data complexities, efficient attacks based on difference enumeration techniques can be constructed. We show that it is most effective to consider tuples of differences instead of simple differences, both to increase the range of the distinguishers and to enable key recovery attacks. All applications for LowMC we are aware of, including signature schemes like Picnic and more recent (ring/group) signature schemes have used version 3 of the roundformular for LowMC, which takes our attack already into account

Cryptanalysis of the MALICIOUS Framework

This note describes several attacks on the MALICIOUS framework for creating backdoored tweakable block ciphers. It is shown that, although the embedded malicious tweak pair itself is hard to recover, it is feasible to find additional weak tweak pairs that can be used to mount key-recovery attacks. Full-round attacks on most instances of LowMC-M are given. Our attacks are far from optimized and significant future improvements are to be expected. We focus on low-data attacks, since these are the most relevant for typical use-cases of LowMC. In addition, this implies that our attacks can not be prevented by limiting the amount of data that can be encrypted using the weak tweak pair. Despite our findings, we believe that the MALICIOUS framework can be used to create backdoored variants of LowMC provided that the parameters are modified

Improved Attacks on LowMC with Algebraic Techniques

The LowMC family of SPN block cipher proposed by Albrecht et al. was designed specifically for MPC-/FHE-/ZKP-friendly use cases. It is especially used as the underlying block cipher of PICNIC, one of the alternate third-round candidate digital signature algorithms for NIST post-quantum cryptography standardization. The security of PICNIC is highly related to the difficulty of recovering the secret key of LowMC from a given plaintext/ciphertext pair, which raises new challenges for security evaluation under extremely low data complexity. In this paper, we improve the attacks on LowMC under low data complexity, i.e. 1 or 2 chosen plaintext/ciphertext pairs. For the difference enumeration attack with 2 chosen plaintexts, we propose new algebraic methods to better exploit the nonlinear relation inside the introduced variables based on the attack framework proposed by Liu et al. at ASIACRYPT 2022. With this technique, we significantly extend the number of attack rounds for LowMC with partial nonlinear layers and improve the success probability from around 0.5 to over 0.9. The security margin of some instances can be reduced to only 3/4 rounds. For the key-recovery attack using a single plaintext, we adopt a different linearization strategy to reduce the huge memory consumption caused by the polynomial methods for solving multivariate equation systems. The memory complexity reduces drastically for all 5-/6-round LowMC instances with full nonlinear layers at the sacrifice of a small factor of time complexity. For 5-round LowMC instances with a block size of 129, the memory complexity decreases from 286.46 bits to 248.18 bits while the time complexity even slightly reduces. Our results indicate that the security for different instances of LowMC under extremely low data complexity still needs further exploration

Rasta: A cipher with low ANDdepth and few ANDs per bit

Recent developments in multi party computation (MPC) and fully homomorphic encryption (FHE) promoted the design and analysis of symmetric cryptographic schemes that minimize multiplications in one way or another. In this paper, we propose with Rasta a design strategy for symmetric encryption that has ANDdepth d and at the same time only needs d ANDs per encrypted bit. Even for very low values of d between 2 and 6 we can give strong evidence that attacks may not exist. This contributes to a better understanding of the limits of what concrete symmetric-key constructions can theoretically achieve with respect to AND-related metrics, and is to the best of our knowledge the first attempt that minimizes both metrics simultaneously. Furthermore, we can give evidence that for choices of d between 4 and 6 the resulting implementation properties may well be competitive by testing our construction in the use-case of removing the large ciphertext-expansion when using the BGV scheme

Improved Attacks on LowMC with Algebraic Techniques

The LowMC family of SPN block cipher proposed by Albrecht et al. was designed specifically for MPC-/FHE-/ZKP-friendly use cases. It is especially used as the underlying block cipher of PICNIC, one of the alternate third-round candidate digital signature algorithms for NIST post-quantum cryptography standardization. The security of PICNIC is highly related to the difficulty of recovering the secret key of LowMC from a given plaintext/ciphertext pair, which raises new challenges for security evaluation under extremely low data complexity. In this paper, we improve the attacks on LowMC under low data complexity, i.e. 1 or 2 chosen plaintext/ciphertext pairs. For the difference enumeration attack with 2 chosen plaintexts, we propose new algebraic methods to better exploit the nonlinear relation inside the introduced variables based on the attack framework proposed by Liu et al. at ASIACRYPT 2022. With this technique, we significantly extend the number of attack rounds for LowMC with partial nonlinear layers and improve the success probability from around 0.5 to over 0.9. The security margin of some instances can be reduced to only 3/4 rounds. For the key-recovery attack using a single plaintext, we adopt a different linearization strategy to reduce the huge memory consumption caused by the polynomial methods for solving multivariate equation systems. The memory complexity reduces drastically for all 5-/6-round LowMC instances with full nonlinear layers at the sacrifice of a small factor of time complexity. For 5-round LowMC instances with a block size of 129, the memory complexity decreases from $2^{86.46}$ bits to $2^{48.18}$ bits while the time complexity even slightly reduces. Our results indicate that the security for different instances of LowMC under extremely low data complexity still needs further exploration

Algebraic Meet-in-the-Middle Attack on LowMC

By exploiting the feature of partial nonlinear layers, we propose a new technique called algebraic meet-in-the-middle (MITM) attack to analyze the security of LowMC, which can reduce the memory complexity of the simple difference enumeration attack over the state-of-the-art. Moreover, while an efficient algebraic technique to retrieve the full key from a differential trail of LowMC has been proposed at CRYPTO 2021, its time complexity is still exponential in the key size. In this work, we show how to reduce it to constant time when there are a sufficiently large number of active S-boxes in the trail. With the above new techniques, the attacks on LowMC and \mbox{LowMC-M} published at CRYPTO 2021 are further improved, and some LowMC instances could be broken for the first time. Our results seem to indicate that partial nonlinear layers are still not well-understood

New Attacks on LowMC instances with a Single Plaintext/Ciphertext pair

Cryptanalysis of the LowMC block cipher when the attacker has access to a single known plaintext/ciphertext pair is a mathematically challenging problem. This is because the attacker is unable to employ most of the standard techniques in symmetric cryptography like linear and differential cryptanalysis. This scenario is particularly relevant while arguing the security of the \picnic digital signature scheme in which the plaintext/ciphertext pair generated by the LowMC block cipher serves as the public (verification) key and the corresponding LowMC encryption key also serves as the secret (signing) key of the signature scheme. In the paper by Banik et al. (IACR ToSC 2020:4), the authors used a linearization technique of the LowMC S-box to mount attacks on some instances of the block cipher. In this paper, we first make a more precise complexity analysis of the linearization attack. Then, we show how to perform a 2-stage MITM attack on LowMC. The first stage reduces the key candidates corresponding to a fraction of key bits of the master key. The second MITM stage between this reduced candidate set and the remaining fraction of key bits successfully recovers the master key. We show that the combined computational complexity of both these stages is significantly lower than those reported in the ToSC paper by Banik et al

Algebraic Attack on FHE-Friendly Cipher HERA Using Multiple Collisions

Fully homomorphic encryption (FHE) is an advanced cryptography technique to allow computations (i.e., addition and multiplication) over encrypted data. After years of effort, the performance of FHE has been significantly improved and it has moved from theory to practice. The transciphering framework is another important technique in FHE to address the issue of ciphertext expansion and reduce the client-side computational overhead. Motivated by this framework, several FHE-friendly symmetric-key primitives have been proposed since the publication of LowMC at EUROCRYPT 2015. To apply the transciphering framework to the CKKS scheme, a new transciphering framework called the Real-to-Finite-Field (RtF) framework and a corresponding FHE-friendly symmetric-key primitive called HERA were proposed at ASIACRYPT 2021. Although HERA has a very similar structure to AES, it is considerably different in the following aspects: 1) the power map $x\mapsto x^3$ is used as the S-box; 2) a randomized key schedule is used; 3) it is over a prime field $\mathbb F_p$ with $p>2^{16}$. In this work, we perform the first third-party cryptanalysis of HERA, by showing how to mount new algebraic attacks with multiple collisions in the round keys. Specifically, according to the special way to randomize the round keys in HERA, we find it possible to peel off the last nonlinear layer by using collisions in the last-round key and a simple property of the power map. In this way, we could construct an overdefined system of equations of a much lower degree in the key, and efficiently solve the system via the linearization technique. As a result, for HERA with 192 and 256 bits of security, respectively, we could break some parameters under the same assumption made by designers that the algebra constant $\omega$ for Gaussian elimination is $\omega=2$, i.e., Gaussian elimination on an $n\times n$ matrix takes $\mathcal{O}(n^{\omega})$ field operations. If using more conservative choices like $\omega\in\{2.8,3\}$, our attacks can also successfully reduce the security margins of some variants of \hera to only 1 round. However, the security of HERA with 80 and 128 bits of security is not affected by our attacks due to the high cost to find multiple collisions. In any case, our attacks reveal a weakness of HERA caused by the randomized key schedule and its small state size

Cryptanalysis of Full LowMC and LowMC-M with Algebraic Techniques

In this paper, we revisit the difference enumeration technique for LowMC and develop new algebraic techniques to achieve efficient key-recovery attacks. In the original difference enumeration attack framework, an inevitable step is to precompute and store a set of intermediate state differences for efficient checking via the binary search. Our first observation is that Bar-On et al.\u27s general algebraic technique developed for SPNs with partial nonlinear layers can be utilized to fulfill the same task, which can make the memory complexity negligible as there is no need to store a huge set of state differences any more. Benefiting from this technique, we could significantly improve the attacks on LowMC when the block size is much larger than the key size and even break LowMC with such a kind of parameter. On the other hand, with our new key-recovery technique, we could significantly improve the time to retrieve the full key if given only a single pair of input and output messages together with the difference trail that they take, which was stated as an interesting question by Rechberger et al. at ToSC 2018. Combining both techniques, with only 2 chosen plaintexts, we could break 4 rounds of LowMC adopting a full S-Box layer with block size of 129, 192 and 255 bits, respectively, which are the 3 recommended parameters for Picnic3, an alternative third-round candidate in NIST\u27s Post-Quantum Cryptography competition. We have to emphasize that our attacks do not indicate that Picnic3 is broken as the Picnic use-case is very different and an attacker cannot even freely choose 2 plaintexts to encrypt for a concrete LowMC instance. However, such parameters are deemed as secure in the latest LowMC. Moreover, much more rounds of seven instances of the backdoor cipher LowMC-M as proposed by Peyrin and Wang in CRYPTO 2020 can be broken without finding the backdoor by making full use of the allowed $2^{64}$ data. The above mentioned attacks are all achieved with negligible memory

New Low-Memory Algebraic Attacks on LowMC in the Picnic Setting

The security of the post-quantum signature scheme Picnic is highly related to the difficulty of recovering the secret key of LowMC from a single plaintext-ciphertext pair. Since Picnic is one of the alternate third-round candidates in NIST post-quantum cryptography standardization process, it has become urgent and important to evaluate the security of LowMC in the Picnic setting. The best attacks on LowMC with full S-box layers used in Picnic3 were achieved with Dinur’s algorithm. For LowMC with partial nonlinear layers, e.g. 10 S-boxes per round adopted in Picnic2, the best attacks on LowMC were published by Banik et al. with the meet-in-the-middle (MITM) method. In this paper, we improve the attacks on LowMC in a model where memory consumption is costly. First, a new attack on 3-round LowMC with full S-box layers with negligible memory complexity is found, which can outperform Bouillaguet et al.’s fast exhaustive search attack and can achieve better time-memory tradeoffs than Dinur’s algorithm. Second, we extend the 3-round attack to 4 rounds to significantly reduce the memory complexity of Dinur’s algorithm at the sacrifice of a small factor of time complexity. For LowMC instances with 1 S-box per round, our attacks are shown to be much faster than the MITM attacks. For LowMC instances with 10 S-boxes per round, we can reduce the memory complexity from 32GB (238 bits) to only 256KB (221 bits) using our new algebraic attacks rather than the MITM attacks, while the time complexity of our attacks is about 23.2 ∼ 25 times higher than that of the MITM attacks. A notable feature of our new attacks (apart from the 4-round attack) is their simplicity. Specifically, only some basic linear algebra is required to understand them and they can be easily implemented