## CryptoDB

### André Schrottenloher

#### ORCID: 0000-0002-1329-8630

#### Publications

**Year**

**Venue**

**Title**

2023

EUROCRYPT

Finding many Collisions via Reusable Quantum Walks - Application to Lattice Sieving
Abstract

Given a random function $f$ with domain $[2^n]$ and codomain $[2^m]$, with $m \geq n$, a collision of $f$ is a pair of distinct inputs with the same image. Collision finding is an ubiquitous problem in cryptanalysis, and it has been well studied using both classical and quantum algorithms. Indeed, the quantum query complexity of the problem is well known to be $\Theta(2^{m/3})$, and matching algorithms are known for any value of $m$.
The situation becomes different when one is looking for \emph{multiple} collision pairs. Here, for $2^k$ collisions, a query lower bound of $\Theta(2^{(2k+m)/3})$ was shown by Liu and Zhandry (EUROCRYPT~2019). A matching algorithm is known, but only for relatively small values of $m$, when many collisions exist. In this paper, we improve the algorithms for this problem and, in particular, extend the range of admissible parameters where the lower bound is met.
Our new method relies on a \emph{chained quantum walk} algorithm, which might be of independent interest. It allows to extract multiple solutions of an MNRS-style quantum walk, without having to recompute it entirely: after finding and outputting a solution, the current state is reused as the initial state of another walk.
As an application, we improve the quantum sieving algorithms for the shortest vector problem (SVP), with a complexity of $2^{0.2563d + o(d)}$ instead of the previous $2^{0.2570d + o(d)}$.

2023

CRYPTO

Quantum Linear Key-recovery Attacks Using the QFT
Abstract

The Quantum Fourier Transform is a fundamental tool in
quantum cryptanalysis. In symmetric cryptanalysis, hidden shift algorithms
such as Simon’s, which rely on the QFT, have been
used to obtain structural attacks on some very specific block ciphers.
The Fourier Transform is also used in classical cryptanalysis, for example
in FFT-based linear key-recovery attacks introduced by Collard et al.
(ICISC 2007). Whether such techniques can be adapted to the quantum
setting has remained so far an open question.
In this paper, we introduce a new framework for quantum linear key-recovery
attacks using the QFT. These attacks loosely follow the classical
method of Collard et al., in that they rely on the fast computation of a
correlation state in which experimental correlations, rather than being
directly accessible, are encoded in the amplitudes of a quantum state.
The experimental correlation is a statistic that is expected to be higher
for the good key, and on some conditions, the increased amplitude creates
a speedup with respect to an exhaustive search of the key. The same
method also yields a new family of structural attacks, and new examples
of quantum speedups beyond quadratic using classical known-plaintext
queries.

2023

TOSC

Simplified Modeling of MITM Attacks for Block Ciphers: New (Quantum) Attacks
Abstract

The meet-in-the-middle (MITM) technique has led to many key-recovery attacks on block ciphers and preimage attacks on hash functions. Nowadays, cryptographers use automatic tools that reduce the search of MITM attacks to an optimization problem. Bao et al. (EUROCRYPT 2021) introduced a low-level modeling based on Mixed Integer Linear Programming (MILP) for MITM attacks on hash functions, which was extended to key-recovery attacks by Dong et al. (CRYPTO 2021). However, the modeling only covers AES-like designs. Schrottenloher and Stevens (CRYPTO 2022) proposed a different approach aiming at higher-level simplified models. However, this modeling was limited to cryptographic permutations.In this paper, we extend the latter simplified modeling to also cover block ciphers with simple key schedules. The resulting modeling enables us to target a large array of primitives, typically lightweight SPN ciphers where the key schedule has a slow diffusion, or none at all. We give several applications such as full breaks of the PIPO-256 and FUTURE block ciphers, and reduced-round classical and quantum attacks on SATURNIN-Hash.

2022

EUROCRYPT

Beyond quadratic speedups in quantum attacks on symmetric schemes
📺
Abstract

In this paper, we report the first quantum key-recovery attack on a symmetric block cipher design, using classical queries only, with a more than quadratic time speedup compared to the best classical attack.
We study the 2XOR-Cascade construction of Ga{\v{z}}i and Tessaro (EUROCRYPT~2012). It is a key length extension technique which provides an n-bit block cipher with 5n/2 bits of security out of an n-bit block cipher with 2n bits of key, with a security proof in the ideal model. We show that the offline-Simon algorithm of Bonnetain et al. (ASIACRYPT~2019) can be extended to, in particular, attack this construction in quantum time $\widetilde{\mathcal{O}}{2^n}$, providing a 2.5 quantum speedup over the best classical attack.
Regarding post-quantum security of symmetric ciphers, it is commonly assumed that doubling the key sizes is a sufficient precaution. This is because Grover's quantum search algorithm, and its derivatives, can only reach a quadratic speedup at most. Our attack shows that the structure of some symmetric constructions can be exploited to overcome this limit. In particular, the 2XOR-Cascade cannot be used to generically strengthen block ciphers against quantum adversaries, as it would offer only the same security as the block cipher itself.

2022

CRYPTO

Simplified MITM Modeling for Permutations: New (Quantum) Attacks
📺
Abstract

Meet-in-the-middle (MITM) is a general paradigm where internal states are computed along two independent paths ('forwards' and 'backwards') that are then matched. Over time, MITM attacks improved using more refined techniques and exploiting additional freedoms and structure, which makes it more involved to find and optimize such attacks. This has led to the use of detailed attack models for generic solvers to automatically search for improved attacks, notably a MILP model developed by Bao et al. at EUROCRYPT 2021.
In this paper, we study a simpler MILP modeling combining a greatly reduced attack representation as input to the generic solver, together with a theoretical analysis that, for any solution, proves the existence and complexity of a detailed attack. This modeling allows to find both classical and quantum attacks on a broad class of cryptographic permutations.
First, Present-like constructions, with the permutations of the Spongent hash functions: we improve the MITM step in distinguishers by up to 3 rounds. Second, AES-like designs: despite being much simpler than Bao et al.'s, our model allows to recover the best previous results. The only limitation is that we do not use degrees of freedom from the key schedule. Third, we show that the model can be extended to target more permutations, like Feistel networks. In this context we give new Guess-and-determine attacks on reduced Simpira v2 and Sparkle.
Finally, using our model, we find several new quantum preimage and pseudo-preimage attacks (e.g. Haraka v2, Simpira v2 ... ) targeting the same number of rounds as the classical attacks.

2021

ASIACRYPT

Quantum Linearization Attacks
📺
Abstract

Recent works have shown that quantum period-finding can be used to break many popular constructions (some block ciphers such as Even-Mansour, multiple MACs and AEs...) in the superposition query model. So far, all the constructions broken exhibited a strong algebraic structure, which enables to craft a periodic function of a single input block. The recovery of the secret period allows to recover a key, distinguish, break the confidentiality or authenticity of these modes.
In this paper, we introduce the \emph{quantum linearization attack}, a new way of using Simon's algorithm to target MACs in the superposition query model. Specifically, we use inputs of multiple blocks as an interface to a function hiding a linear structure. The recovery of this structure allows to perform forgeries.
We also present some variants of this attack that use other quantum algorithms, which are much less common in quantum symmetric cryptanalysis: Deutsch's, Bernstein-Vazirani's, and Shor's. To the best of our knowledge, this is the first time these algorithms have been used in quantum forgery or key-recovery attacks.
Our attack breaks many parallelizable MACs such as {\sf LightMac}, {\sf PMAC}, and numerous variants with (classical) beyond-birthday-bound security ({\sf LightMAC+}, {\sf PMAC+}) or using tweakable block ciphers ({\sf ZMAC}). More generally, it shows that constructing parallelizable quantum-secure PRFs might be a challenging task.

2021

ASIACRYPT

Clustering Effect in Simon and Simeck
📺
Abstract

Simon and Simeck are two lightweight block ciphers with a
simple round function using only word rotations and a bit-wise AND
operation. Previous work has shown a strong clustering effect for
differential and linear cryptanalysis, due to the existence of many
trails with the same inputs and outputs.
In this paper, we explore this clustering effect by exhibiting a class
of high probability differential and linear trails where the active
bits stay in a fixed window of w bits. Instead of enumerating a set
of good trails contributing to a differential or a linear
approximation, we compute the probability distribution over this
space, including all trails in the class.
This results in stronger distinguishers than previously proposed, and
we describe key recovery attacks against Simon and Simeck improving
the previous results by up to 7 rounds. In particular, we obtain an
attack against 42-round Simeck-64, leaving only two rounds of security
margin, and an attack against 45-round Simon-96/144, reducing the
security margin from 16 rounds to 9 rounds.

2021

ASIACRYPT

QCB: Efficient Quantum-secure Authenticated Encryption
📺
Abstract

It was long thought that symmetric cryptography was only mildly affected by quantum attacks, and that doubling the key length was sufficient to restore security. However, recent works have shown that Simon's quantum period finding algorithm breaks a large number of MAC and authenticated encryption algorithms when the adversary can query the MAC/encryption oracle with a quantum superposition of messages. In particular, the OCB authenticated encryption mode is broken in this setting, and no quantum-secure mode is known with the same efficiency (rate-one and parallelizable).
In this paper we generalize the previous attacks, show that a large class of OCB-like schemes is unsafe against superposition queries, and discuss the quantum security notions for authenticated encryption modes. We propose a new rate-one parallelizable mode named QCB inspired by TAE and OCB and prove its security against quantum superposition queries.

2021

JOFC

Internal Symmetries and Linear Properties: Full-permutation Distinguishers and Improved Collisions on Gimli
Abstract

$$\mathsf {Gimli}$$ Gimli is a family of cryptographic primitives (both a hash function and an AEAD scheme) that has been selected for the second round of the NIST competition for standardizing new lightweight designs. The candidate $$\mathsf {Gimli}$$ Gimli is based on the permutation $$\mathsf {Gimli}$$ Gimli , which was presented at CHES 2017. In this paper, we study the security of both the permutation and the constructions that are based on it. We exploit the slow diffusion in $$\mathsf {Gimli}$$ Gimli and its internal symmetries to build, for the first time, a distinguisher on the full permutation of complexity $$2^{64}$$ 2 64 . We also provide a practical distinguisher on 23 out of the full 24 rounds of $$\mathsf {Gimli}$$ Gimli that has been implemented. Next, we give (full state) collision and semi-free start collision attacks on $$\mathsf {Gimli}$$ Gimli -Hash, reaching, respectively, up to 12 and 18 rounds. On the practical side, we compute a collision on 8-round $$\mathsf {Gimli}$$ Gimli -Hash. In the quantum setting, these attacks reach 2 more rounds. Finally, we perform the first study of linear trails in $$\mathsf {Gimli}$$ Gimli , and we find a linear distinguisher on the full permutation.

2020

CRYPTO

Cryptanalysis Results on Spook: Bringing Full-round Shadow-512 to the Light
📺
Abstract

Spook is one of the 32 candidates that has made it to the second round of the NIST Lightweight Cryptography Standardization process, and is particularly interesting since it proposes differential side channel resistance. In this paper, we present practical distinguishers of the full 6-step version of the underlying permutations of Spook, namely Shadow-512 and Shadow-384, solving challenges proposed by the designers on the permutation. We also propose practical forgeries with 4-step Shadow for the S1P mode of operation in the nonce misuse scenario, which is allowed by the CIML2 security game considered by the authors. All the results presented in this paper have been implemented.

2020

EUROCRYPT

Optimal Merging in Quantum $k$-xor and $k$-sum Algorithms
📺
Abstract

The $k$-xor or Generalized Birthday Problem aims at finding, given $k$ lists of bit-strings, a $k$-tuple among them XORing to 0. If the lists are unbounded, the best classical (exponential) time complexity has withstood since Wagner's CRYPTO 2002 paper. If the lists are bounded (of the same size) and such that there is a single solution, the \emph{dissection algorithms} of Dinur \emph{et al.} (CRYPTO 2012) improve the memory usage over a simple meet-in-the-middle.
In this paper, we study quantum algorithms for the $k$-xor problem. With unbounded lists and quantum access, we improve previous work by Grassi \emph{et al.} (ASIACRYPT 2018) for almost all $k$. Next, we extend our study to lists of any size and with classical access only.
We define a set of ``merging trees'' which represent the best known strategies for quantum and classical merging in $k$-xor algorithms, and prove that our method is optimal among these. Our complexities are confirmed by a Mixed Integer Linear Program that computes the best strategy for a given $k$-xor problem. All our algorithms apply also when considering modular additions instead of bitwise xors.
This framework enables us to give new improved quantum $k$-xor algorithms for all $k$ and list sizes. Applications include the subset-sum problem, LPN with limited memory and the multiple-encryption problem.

2020

EUROCRYPT

Quantum Security Analysis of CSIDH
📺
Abstract

CSIDH is a recent proposal for post-quantum non-interactive key-exchange, based on supersingular elliptic curve isogenies. It is similar in design to a previous scheme by Couveignes, Rostovtsev and Stolbunov, but aims at an improved balance between efficiency and security.
In the proposal, the authors suggest concrete parameters in order to meet some desired levels of quantum security. These parameters are based on the hardness of recovering a hidden isogeny between two elliptic curves, using a quantum subexponential algorithm of Childs, Jao and Soukharev. This algorithm combines two building blocks: first, a quantum algorithm for recovering a hidden shift in a commutative group. Second, a computation in superposition of all isogenies originating from a given curve, which the algorithm calls as a black box.
In this paper, we give a comprehensive security analysis of CSIDH. Our first step is to revisit three quantum algorithms for the abelian hidden shift problem from the perspective of non-asymptotic cost, with trade-offs between their quantum and classical complexities. Second, we complete the non-asymptotic study of the black box in the hidden shift algorithm. We give a quantum procedure that evaluates CSIDH-512 using less than 40~000 logical qubits.
This allows us to show that the parameters proposed by the authors of CSIDH do not meet their expected quantum security.

2020

TOSC

Saturnin: a suite of lightweight symmetric algorithms for post-quantum security
📺
Abstract

The cryptographic algorithms needed to ensure the security of our communications have a cost. For devices with little computing power, whose number is expected to grow significantly with the spread of the Internet of Things (IoT), this cost can be a problem. A simple answer to this problem is a compromise on the security level: through a weaker round function or a smaller number of rounds, the security level can be decreased in order to cheapen the implementation of the cipher. At the same time, quantum computers are expected to disrupt the state of the art in cryptography in the near future. For public-key cryptography, the NIST has organized a dedicated process to standardize new algorithms. The impact of quantum computing is harder to assess in the symmetric case but its study is an active research area.In this paper, we specify a new block cipher, Saturnin, and its usage in different modes to provide hashing and authenticated encryption in such a way that we can rigorously argue its security in the post-quantum setting. Its security analysis follows naturally from that of the AES, while our use of components that are easily implemented in a bitsliced fashion ensures a low cost for our primitives. Our aim is to provide a new lightweight suite of algorithms that performs well on small devices, in particular micro-controllers, while providing a high security level even in the presence of quantum computers. Saturnin is a 256-bit block cipher with a 256-bit key and an additional 9-bit parameter for domain separation. Using it, we built two authenticated ciphers and a hash function.• Saturnin-CTR-Cascade is an authenticated cipher using the counter mode and a separate MAC. It requires two passes over the data but its implementation does not require the inverse block cipher.• Saturnin-Short is an authenticated cipher intended for messages with a length strictly smaller than 128 bits which uses only one call to Saturnin to providenconfidentiality and integrity.• Saturnin-Hash is a 256-bit hash function. In this paper, we specify this suite of algorithms and argue about their security in both the classical and the post-quantum setting.
https://project.inria.fr/saturnin/

2020

ASIACRYPT

Improved Classical and Quantum Algorithms for Subset-Sum
📺
Abstract

We present new classical and quantum algorithms for solving random subset-sum instances. First, we improve over the Becker-Coron-Joux algorithm (EUROCRYPT 2011) from $\widetilde{O}(2^{0.291 n})$ down to $\widetilde{O}(2^{0.283 n})$, using more general representations with values in $\{0,1,-1,2\}$.
Next, we improve the state of the art of quantum algorithms for this problem in several directions. By combining the Howgrave-Graham-Joux algorithm (EUROCRYPT 2010) and quantum search, we devise an algorithm with asymptotic cost $\widetilde{O}(2^{0.236 n})$, lower than the cost of the quantum walk based on the same classical algorithm proposed by Bernstein, Jeffery, Lange and Meurer (PQCRYPTO 2013). This algorithm has the advantage of using \emph{classical} memory with quantum random access, while the previously known algorithms used the quantum walk framework, and required \emph{quantum} memory with quantum random access.
We also propose new quantum walks for subset-sum, performing better than the previous best time complexity of $\widetilde{O}(2^{0.226 n})$ given by Helm and May (TQC 2018). We combine our new techniques to reach a time $\widetilde{O}(2^{0.216 n})$. This time is dependent on a heuristic on quantum walk updates, formalized by Helm and May, that is also required by the previous algorithms. We show how to partially overcome this heuristic, and we obtain an algorithm with quantum time $\widetilde{O}(2^{0.218 n})$ requiring only the standard classical subset-sum heuristics.

2020

ASIACRYPT

New results on Gimli: full-permutation distinguishers and improved collisions
📺 ★
Abstract

Gimli is a family of cryptographic primitives (both a hash function and an AEAD scheme) that has been selected for the second round of the NIST competition for standardizing new lightweight designs. The candidate Gimli is based on the permutation Gimli, which was presented at CHES 2017. In this paper, we study the security of both the permutation and the constructions that are based on it. We exploit the slow diffusion in Gimli and its internal symmetries to build, for the first time, a distinguisher on the full permutation of complexity $2^{64}$. We also provide a practical distinguisher on 23 out of the full 24 rounds of Gimli that has been implemented.
Next, we give (full state) collision and semi-free-start collision attacks on Gimli-Hash, reaching respectively up to 12 and 18 rounds. On the practical side, we compute a collision on 8-round Gimli-Hash. In the quantum setting, these attacks reach 2 more rounds. Finally, we perform the first study of linear trails in the permutation, and we propose differential-linear cryptanalysis that reach up to 17 rounds of Gimli.

2019

TOSC

Quantum Security Analysis of AES
📺
Abstract

In this paper we analyze for the first time the post-quantum security of AES. AES is the most popular and widely used block cipher, established as the encryption standard by the NIST in 2001. We consider the secret key setting and, in particular, AES-256, the recommended primitive and one of the few existing ones that aims at providing a post-quantum security of 128 bits. In order to determine the new security margin, i.e., the lowest number of non-attacked rounds in time less than 2128 encryptions, we first provide generalized and quantized versions of the best known cryptanalysis on reduced-round AES, as well as a discussion on attacks that don’t seem to benefit from a significant quantum speed-up. We propose a new framework for structured search that encompasses both the classical and quantum attacks we present, and allows to efficiently compute their complexity. We believe this framework will be useful for future analysis.Our best attack is a quantum Demirci-Selçuk meet-in-the-middle attack. Unexpectedly, using the ideas underlying its design principle also enables us to obtain new, counter-intuitive classical TMD trade-offs. In particular, we can reduce the memory in some attacks against AES-256 and AES-128.One of the building blocks of our attacks is solving efficiently the AES S-Box differential equation, with respect to the quantum cost of a reversible S-Box. We believe that this generic quantum tool will be useful for future quantum differential attacks. Judging by the results obtained so far, AES seems a resistant primitive in the post-quantum world as well as in the classical one, with a bigger security margin with respect to quantum generic attacks.

2019

ASIACRYPT

Quantum Attacks Without Superposition Queries: The Offline Simon’s Algorithm
Abstract

In symmetric cryptanalysis, the model of superposition queries has led to surprising results, with many constructions being broken in polynomial time thanks to Simon’s period-finding algorithm. But the practical implications of these attacks remain blurry. In contrast, the results obtained so far for a quantum adversary making classical queries only are less impressive.In this paper, we introduce a new quantum algorithm which uses Simon’s subroutines in a novel way. We manage to leverage the algebraic structure of cryptosystems in the context of a quantum attacker limited to classical queries and offline quantum computations. We obtain improved quantum-time/classical-data tradeoffs with respect to the current literature, while using only as much hardware requirements (quantum and classical) as a standard exhaustive search with Grover’s algorithm. In particular, we are able to break the Even-Mansour construction in quantum time $$\tilde{O}(2^{n/3})$$, with $$O(2^{n/3})$$ classical queries and $$O(n^2)$$ qubits only. In addition, we improve some previous superposition attacks by reducing the data complexity from exponential to polynomial, with the same time complexity.Our approach can be seen in two complementary ways: reusing superposition queries during the iteration of a search using Grover’s algorithm, or alternatively, removing the memory requirement in some quantum attacks based on a collision search, thanks to their algebraic structure.We provide a list of cryptographic applications, including the Even-Mansour construction, the FX construction, some Sponge authenticated modes of encryption, and many more.

2018

ASIACRYPT

Quantum Algorithms for the $k$-xor Problem
Abstract

The $$k$$-xor (or generalized birthday) problem is a widely studied question with many applications in cryptography. It aims at finding k elements of n bits, drawn at random, such that the xor of all of them is 0. The algorithms proposed by Wagner more than fifteen years ago remain the best known classical algorithms for solving them, when disregarding logarithmic factors.In this paper we study these problems in the quantum setting, when considering that the elements are created by querying a random function (or k random functions) $$H~: \{0,1\}^n \rightarrow \{0,1\}^n$$. We consider two scenarios: in one we are able to use a limited amount of quantum memory (i.e. a number O(n) of qubits, the same as the one needed by Grover’s search algorithm), and in the other we consider that the algorithm can use an exponential amount of qubits. Our newly proposed algorithms are of general interest. In both settings, they provide the best known quantum time complexities.In particular, we are able to considerately improve the $$3$$-xor algorithm: with limited qubits, we reach a complexity considerably better than what is currently possible for quantum collision search. Furthermore, when having access to exponential amounts of quantum memory, we can take this complexity below $$O(2^{n/3})$$, the well-known lower bound of quantum collision search, clearly improving the best known quantum time complexity also in this setting.We illustrate the importance of these results with some cryptographic applications.

2017

ASIACRYPT

#### Program Committees

- Crypto 2023
- FSE 2023

#### Coauthors

- Ritam Bhaumik (1)
- Xavier Bonnetain (8)
- Rémi Bricout (1)
- Anne Canteaut (1)
- André Chailloux (3)
- Patrick Derbez (1)
- Sébastien Duval (1)
- Lorenzo Grassi (1)
- Antonio Flórez Gutiérrez (2)
- Akinori Hosoyamada (1)
- Paul Huynh (1)
- Virginie Lallemand (1)
- Gaëtan Leurent (6)
- María Naya-Plasencia (11)
- Clara Pernot (1)
- Léo Perrin (4)
- Thomas Pornin (1)
- Yu Sasaki (1)
- Yannick Seurin (1)
- Yixin Shen (2)
- Ferdinand Sibleyras (3)
- Marc Stevens (2)