## CryptoDB

### Qingju Wang

#### Publications

**Year**

**Venue**

**Title**

2021

ASIACRYPT

Massive Superpoly Recovery with Nested Monomial Predictions
Abstract

Determining the exact algebraic structure or some partial information of the superpoly
for a given cube is a necessary step in the cube attack -- a generic cryptanalytic technique
for symmetric-key primitives with some secret and public tweakable inputs.
Currently, the division property based approach is the most powerful tool
for exact superpoly recovery.
However, as the algebraic normal form (ANF) of the targeted output bit gets
increasingly complicated as the number of rounds grows, existing
methods for superpoly recovery quickly hit their bottlenecks. For example,
previous method stuck at round 842, 190, and 892 for \trivium, \grain, and \kreyvium, respectively.
In this paper, we propose a new framework
for recovering the exact ANFs of massive superpolies
based on the monomial prediction technique (ASIACRYPT 2020, an
alternative language for the division property).
In this framework, the targeted output bit is
first expressed as a polynomial of the bits of some
intermediate states. For each term appearing in
the polynomial, the monomial prediction technique is
applied to determine its superpoly if the corresponding
MILP model can be solved within a preset time limit.
Terms unresolved within the time limit are further
expanded as polynomials of the bits of some deeper intermediate
states with symbolic computation, whose terms are again
processed with monomial predictions. The above procedure
is iterated until all terms are resolved.
Finally, all the sub-superpolies are collected and assembled
into the superpoly of the targeted bit.
We apply the new
framework to \trivium, \grain, and \kreyvium.
As a result, the exact ANFs of the superpolies for
843-, 844- and 845-round \trivium,
191-round \grain and 894-round \kreyvium are recovered.
Moreover, with help of the M\"{o}bius transform, we present a novel key-recovery technique based on
superpolies involving \textit{all} key bits by exploiting the sparse structures, which leads to the best key-recovery attacks on the targets considered.

2020

EUROCRYPT

Modeling for Three-Subset Division Property without Unknown Subset -- Improved Cube Attacks against Trivium and Grain-128AEAD
📺
Abstract

A division property is a generic tool to search for integral distinguishers, and automatic tools such as MILP or SAT/SMT allow us to evaluate the propagation efficiently.
In the application to stream ciphers, it enables us to estimate the security of cube attacks theoretically, and it leads to the best key-recovery attacks against well-known stream ciphers.
However, it was reported that some of the key-recovery attacks based on the division property degenerate to distinguishing attacks due to the inaccuracy of the division property.
Three-subset division property (without unknown subset) is a promising method to solve this inaccuracy problem, and a new algorithm using automatic tools for the three-subset division property was recently proposed at Asiacrypt2019.
In this paper, we first show that this state-of-the-art algorithm is not always efficient and we cannot improve the existing key-recovery attacks.
Then, we focus on the feature of the three-subset division property without unknown subset and propose another new efficient algorithm using automatic tools.
Our algorithm is more efficient than existing algorithms, and it can improve existing key-recovery attacks.
In the application to Trivium, we show a 841-round key-recovery attack.
We also show that a 855-round key-recovery attack, which was proposed at CRYPTO2018, has a critical flaw and does not work.
As a result, our 841-round attack becomes the best key-recovery attack.
In the application to Grain-128AEAD, we show that the known 184-round key-recovery attack degenerates to distinguishing attacks.
Then, the distinguishing attacks are improved up to 189 rounds, and we also show the best key-recovery attack against 190 rounds.

2020

TOSC

Links between Division Property and Other Cube Attack Variants
📺
Abstract

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.

2020

TOSC

Finding Bit-Based Division Property for Ciphers with Complex Linear Layers
📺
Abstract

The bit-based division property (BDP) is the most effective technique for finding integral characteristics of symmetric ciphers. Recently, automatic search tools have become one of the most popular approaches to evaluating the security of designs against many attacks. Constraint-aided automatic tools for the BDP have been applied to many ciphers with simple linear layers like bit-permutation. Constructing models of complex linear layers accurately and efficiently remains hard. A straightforward method proposed by Sun et al. (called the S method), decomposes a complex linear layer into basic operations like COPY and XOR, then models them one by one. However, this method can easily insert invalid division trails into the solution pool, which results in a quicker loss of the balanced property than the cipher itself would. In order to solve this problem, Zhang and Rijmen propose the ZR method to link every valid trail with an invertible sub-matrix of the matrix corresponding to the linear layer, and then generate linear inequalities to represent all the invertible sub-matrices. Unfortunately, the ZR method is only applicable to invertible binary matrices (defined in Definition 3).To avoid generating a huge number of inequalities for all the sub-matrices, we build a new model that only includes that the sub-matrix corresponding to a valid trail should be invertible. The computing scale of our model can be tackled by most of SMT/SAT solvers, which makes our method practical. For applications, we improve the previous BDP for LED and MISTY1. We also give the 7-round BDP results for Camellia with FL/FL−1, which is the longest to date.Furthermore, we remove the restriction of the ZR method that the matrix has to be invertible, which provides more choices for future designs. Thanks to this, we also reproduce 5-round key-dependent integral distinguishers proposed at Crypto 2016 which cannot be obtained by either the S or ZR methods.

2020

TOSC

Lightweight AEAD and Hashing using the Sparkle Permutation Family
📺
Abstract

We introduce the Sparkle family of permutations operating on 256, 384 and 512 bits. These are combined with the Beetle mode to construct a family of authenticated ciphers, Schwaemm, with security levels ranging from 120 to 250 bits. We also use them to build new sponge-based hash functions, Esch256 and Esch384. Our permutations are among those with the lowest footprint in software, without sacrificing throughput. These properties are allowed by our use of an ARX component (the Alzette S-box) as well as a carefully chosen number of rounds. The corresponding analysis is enabled by the long trail strategy which gives us the tools we need to efficiently bound the probability of all the differential and linear trails for an arbitrary number of rounds. We also present a new application of this approach where the only trails considered are those mapping the rate to the outer part of the internal state, such trails being the only relevant trails for instance in a differential collision attack. To further decrease the number of rounds without compromising security, we modify the message injection in the classical sponge construction to break the alignment between the rate and our S-box layer.

2020

CRYPTO

Alzette: a 64-bit ARX-box (feat. CRAX and TRAX)
📺
Abstract

S-boxes are the only source of non-linearity in many symmetric cryptographic primitives. While they are often defined as being functions operating on a small space, some recent designs propose the use of much larger ones (e.g., 32 bits). In this context, an S-box is then defined as a subfunction whose cryptographic properties can be estimated precisely.
In this paper, we present a 64-bit ARX-based S-box called Alzette which can be evaluated in constant time using only 12 instructions on modern CPUs. Its parallel application can also leverage vector (SIMD) instructions. One iteration of Alzette has differential and linear properties comparable to those of the AES S-box, while two iterations are at least as secure as the AES super S-box. Since the state size is much larger than the typical 4 or 8 bits, the study of the relevant cryptographic properties of Alzette is not trivial.
We further discuss how such wide S-boxes could be used to construct round functions of 64-, 128- and 256-bit (tweakable) block ciphers with good cryptographic properties that are guaranteed even in the related-tweak setting. We use these structures to design a very lightweight 64-bit block cipher (CRAX) which outerperforms SPECK-64/128 for short messages on micro-controllers, and a 256-bit tweakable block cipher (TRAX) which can be used to obtain strong security guarantees against powerful adversaries (nonce misuse, quantum attacks).

2020

ASIACRYPT

An Algebraic Formulation of the Division Property: Revisiting Degree Evaluations, Cube Attacks, and Key-Independent Sums
📺
Abstract

Since it was proposed in 2015 as a generalization
of integral properties, the division property has
evolved into a powerful tool for probing the
structures of Boolean functions whose
algebraic normal forms are not available.
We capture the most essential elements for the detection of division properties
from a pure algebraic perspective, proposing a technique named as {\it monomial prediction}, which
can be employed to determine the presence or absence of a
monomial in the product of the coordinate functions of a vectorial
Boolean function $\bs f$ by counting the number of the so-called {\it monomial trails}
across a sequence of simpler functions whose composition is $\bs f$.
Under the framework of the monomial prediction, we formally prove that
most algorithms for detecting division properties in previous literature
raise no false alarms but may miss.
We also establish the equivalence between the monomial prediction and
the three-subset bit-based division property without unknown subset presented at EUROCRYPT 2020,
and show that these two techniques are perfectly accurate.
This algebraic formulation gives more insights into division properties
and inspires new search strategies. With the monomial prediction,
we obtain the {\it exact} algebraic degrees of \TRIVIUM up
to 834 rounds for the first time. In the context of cube attacks,
we are able to explore a larger search space in limited time and
recover the exact algebraic normal forms of complex superpolies
with the help of a divide-and-conquer strategy. As a result,
we identify more cubes with smaller dimensions, leading
to improvements of some near-optimal attacks against 840-, 841-
and 842-round \TRIVIUM.

2020

ASIACRYPT

An Algebraic Attack on Ciphers with Low-Degree Round Functions: Application to Full MiMC
📺
Abstract

Algebraically simple PRFs, ciphers, or cryptographic hash functions are becoming increasingly popular, for example due to their attractive properties for MPC and new proof systems (SNARKs, STARKs, among many others).
In this paper, we focus on the algebraically simple construction MiMC, which became an attractive cryptanalytic target due to its simplicity, but also due to its use as a baseline in a competition for more recent algorithms exploring this design space.
For the first time, we are able to describe key-recovery attacks on all full-round versions of MiMC over GF(2^n), requiring half the code book. In the chosen-ciphertext scenario, recovering the key from this data for the n-bit full version of MiMC takes the equivalent of less than 2^(n - log_2(n) + 1) calls to MiMC and negligible amounts of memory.
The attack procedure is a generalization of higher-order differential cryptanalysis, and it is based on two main ingredients. First, we present a higher-order distinguisher which exploits the fact that the algebraic degree of MiMC grows significantly slower than originally believed. Secondly, we describe an approach to turn this distinguisher into a key-recovery attack without guessing the full subkey. Finally, we show that approximately ceil(log_3(2 * R)) more rounds (where R = ceil(n * log_3(2)) is the current number of rounds of MiMC-n/n) can be necessary and sufficient to restore the security against the key-recovery attack presented here.
The attack has been practically verified on toy versions of MiMC. Note that our attack does not affect the security of MiMC over prime fields.

2018

CRYPTO

Improved Division Property Based Cube Attacks Exploiting Algebraic Properties of Superpoly
📺
Abstract

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, $$2^{|I|+|J|}$$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 I’s even if $$|I|+|J|\ge n$$|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 Acornrespectively.

2017

TOSC

Design of Lightweight Linear Diffusion Layers from Near-MDS Matrices
Abstract

Near-MDS matrices provide better trade-offs between security and efficiency compared to constructions based on MDS matrices, which are favored for hardwareoriented designs. We present new designs of lightweight linear diffusion layers by constructing lightweight near-MDS matrices. Firstly generic n×n near-MDS circulant matrices are found for 5 ≤ n ≤9. Secondly, the implementation cost of instantiations of the generic near-MDS matrices is examined. Surprisingly, for n = 7, 8, it turns out that some proposed near-MDS circulant matrices of order n have the lowest XOR count among all near-MDS matrices of the same order. Further, for n = 5, 6, we present near-MDS matrices of order n having the lowest XOR count as well. The proposed matrices, together with previous construction of order less than five, lead to solutions of n×n near-MDS matrices with the lowest XOR count over finite fields F2m for 2 ≤ n ≤ 8 and 4 ≤ m ≤ 2048. Moreover, we present some involutory near-MDS matrices of order 8 constructed from Hadamard matrices. Lastly, the security of the proposed linear layers is studied by calculating lower bounds on the number of active S-boxes. It is shown that our linear layers with a well-chosen nonlinear layer can provide sufficient security against differential and linear cryptanalysis.

2015

EPRINT

2013

CHES

#### Coauthors

- Hoda AlKhzaimi (2)
- Christof Beierle (2)
- Begül Bilgin (2)
- Alex Biryukov (2)
- Andrey Bogdanov (2)
- Luan Cardoso dos Santos (2)
- Lei Cheng (2)
- Itai Dinur (2)
- Maria Eichlseder (1)
- Lorenzo Grassi (1)
- Johann Großschädl (2)
- Yonglin Hao (3)
- Kai Hu (3)
- Takanori Isobe (1)
- Lin Jiao (1)
- Miroslav Knezevic (2)
- Gregor Leander (1)
- Chaoyun Li (3)
- Ruilin Li (2)
- Chao Li (2)
- Yunwen Liu (2)
- Zhiqiang Liu (2)
- Reinhard Lüftenegger (1)
- Willi Meier (5)
- Florian Mendel (2)
- Léo Perrin (2)
- Christian Rechberger (1)
- Vincent Rijmen (2)
- Markus Schofnegger (1)
- Bing Sun (2)
- Siwei Sun (2)
- Yosuke Todo (4)
- Aleksei Udovenko (2)
- Vesselin Velichkov (2)
- Meiqin Wang (3)
- Morten Øygarden (1)