Simon's algorithm can detect hidden XOR periods in functions derived from symmetric ciphers. Finding such functions becomes difficult when nonlinear layers and diffusion spread the relevant expressions across many branches, so recent work has used symbolic search to automate the construction. We refine the algebraic SMT model of Liu et al. in two ways. Prefix realization checks whether a symbolic starting state can be reached through preceding rounds and records the round-key nibbles needed...

Integral attacks pose a significant threat to block cipher security, yet providing guarantees against such attacks for a target block cipher is difficult. At ASIACRYPT 2021, Hebborn, Lambin, Leander, and Todo proposed the integral resistance property, which offers strong security guarantees for certain SPN and AND-RX block ciphers, assuming independent round keys. However, limitations remain: they proved a security bound for 13-round Present, while the longest known integral...

In ToSC 2025(1), Jia et al. proposed an SAT-aided automatic search tool for the S-box design. A part of the functionality of this tool is to search for implementations of an S-box with good area and gate-depth complexity. However, it is well-known that the gate depth complexity cannot precisely reflect the latency of an implementation. To overcome this problem, Rasoolzadeh introduced the concept of latency complexity, a more precise metric for the latency cost of implementing an S-box than...

As one of the famous quantum algorithms, Simon's algorithm enables the efficient derivation of the period of periodic functions in polynomial time. However, the complexity of constructing periodic functions has hindered the widespread application of Simon's algorithm in symmetric-key cryptanalysis. Currently, aside from the exhaustive search-based testing method introduced by Canale et al. at CRYPTO 2022, there is no unified model for effectively searching for periodic distinguishers....

Differential cryptanalysis is a powerful technique for attacking block ciphers, wherein the Markov cipher assumption and stochastic hypothesis are commonly employed to simplify the search and probability estimation of differential trails. However, these assumptions often neglect inherent algebraic constraints, potentially resulting in invalid trails and inaccurate probability estimates. Some studies identified violations of these assumptions and explored how they impose constraints on key...

In this paper, we aim to enhance and automate advanced techniques for impossible differential attacks. To demonstrate these advancements, we present improved attacks on the LBlock and HIGHT block ciphers. More precisely, we (a) introduce a methodology to automatically invert symmetric ciphers when represented as directed acyclic graphs, a fundamental step in the search for impossible differential trails and in key recovery techniques; (b) automate the search for impossible differential...

In 1994, Langford and Hellman introduced differential-linear (DL) cryptanalysis, with the idea of decomposing the block cipher E into two parts, EU and EL, such that EU exhibits a high-probability differential trail, while EL has a high-correlation linear trail.Combining these trails forms a distinguisher for E, assuming independence between EU and EL. The dependency between the two parts of DL distinguishers remained unaddressed until EUROCRYPT 2019, where Bar-On et al. introduced the DLCT...

This paper focuses on equivalences between Generalised Feistel Networks (GFN) of type-II. We introduce a new definition of equivalence which captures the concept that two GFNs are identical up to re-labelling of the inputs/outputs, and give a procedure to test this equivalence relation. Such two GFNs are therefore cryptographically equivalent for several classes of attacks. It induces a reduction of the space of possible GFNs: the set of the $(k!)^2$ possible even-odd GFNs with $2k$ branches...

Division property is an effective method for finding integral distinguishers for block ciphers, performing cube attacks on stream ciphers, and studying the algebraic degree of boolean functions. One of the main problems in this field is how to provably find the smallest input multiset leading to a balanced output. In this paper, we propose a new method based on division property for finding integral distinguishers with a provably minimum data complexity on permutation functions and block...

Automatic tools to search for boomerang distinguishers have seen significant advances over the past few years. However, most previous work has focused on ciphers based on a Substitution Permutation Network (SPN), while analyzing the Feistel structure is of great significance. Boukerrou et al. recently provided a theoretical framework to formulate the boomerang switch over multiple Feistel rounds, but they did not provide an automatic tool to find distinguishers. In this paper, by enhancing...

The area is one of the most important criteria for an S-box in hardware implementation when designing lightweight cryptography primitives. The area can be well estimated by the number of gate equivalent (GE). However, to our best knowledge, there is no efficient method to search for an S-box implementation with the least GE. Previous approaches can be classified into two categories, one is a heuristic that aims at finding an implementation with a satisfying but not necessarily the smallest...

WARP is an energy-efficient lightweight block cipher that is currently the smallest 128-bit block cipher in terms of hardware. It was proposed by Banik et al. in SAC 2020 as a lightweight replacement for AES-128 without changing the mode of operation. This paper proposes key-recovery attacks on WARP based on differential cryptanalysis in single and related-key settings. We searched for differential trails for up to 20 rounds of WARP, with the first 19 having optimal differential...

The introduction of the automatic search boosts the cryptanalysis of symmetric-key primitives to some degree. However, the performance of the automatic search is not always satisfactory for the search of long trails or ciphers with large state sizes. Compared with the extensive attention on the enhancement for the search with the mixed integer linear programming (MILP) method, few works care for the acceleration of the automatic search with the Boolean satisfiability problem (SAT) or...

Light key schedule has found many applications in lightweight blockciphers, e.g. LED, PRINTcipher and LBlock. In this paper, we study an interesting question of how to design a as light as possible key schedule from the view of provable security and revisit the four-round key-alternating Feistel cipher by Guo and Wang in Asiacrypt 18. We optimize the construction by Guo and Wang and propose a four-round key-alternating Feistel cipher with an ultra-light (in fact non-existent) key schedule....

In this paper, we propose a new method to launch a more efficient algebraic cryptanalysis. Algebraic cryptanalysis aims at finding the secret key of a cipher by solving a collection of polynomial equations that describe the internal structure of the cipher, while chosen correlated plaintexts, as what appear in higher order differential cryptanalysis and its derivatives such as cube attack or integral cryptanalysis, forces many linear relation between intermediate state bits in the cipher. In...

In recent years, Mixed Integer Linear Programming (MILP) has been widely used in cryptanalysis of symmetric-key primitives. For differential and linear cryptanalysis, MILP can be used to solve two kinds of problems: calculation of the minimum number of differentially/linearly active S-boxes, and search for the best differential/linear characteristics. There are already numerous papers published in this area. However, the efficiency is not satisfactory enough for many symmetric-key...

Division property is a generalized integral property proposed by Todo at EUROCRYPT 2015, and then conventional bit-based division property (CBDP) and bit-based division property using three subsets (BDPT) were proposed by Todo and Morii at FSE 2016. The huge time and memory complexity that once restricted the applications of CBDP have been solved by Xiang et al. at ASIACRYPT 2016. They extended Mixed Integer Linear Programming (MILP) method to search integral distinguishers based on CBDP....

Cryptanalysis with SAT/SMT, MILP and CP has increased in popularity among symmetric-key cryptanalysts and designers due to its high degree of automation. So far, this approach covers differential, linear, impossible differential, zero-correlation, and integral cryptanalysis. However, the Demirci-Selcuk meet-in-the-middle (DS-MITM) attack is one of the most sophisticated techniques that has not been automated with this approach. By an in-depth study of Derbez and Fouque's work on DS-MITM...

Resistance against differential cryptanalysis is an important design criteria for any modern block cipher and most designs rely on finding some upper bound on probability of single differential characteristics. However, already at EUROCRYPT'91, Lai et al. comprehended that differential cryptanalysis rather uses differentials instead of single characteristics. In this paper, we consider exactly the gap between these two approaches and investigate this gap in the context of recent lightweight...

The division property method is a technique to determine integral distinguishers on block ciphers. While the complexity of finding these distinguishers is higher, it has recently been shown that MILP and SAT solvers can efficiently find such distinguishers. In this paper, we provide a framework to automatically find those distinguishers which solely requires a description of the cryptographic primitive. We demonstrate that by finding integral distinguishers for 30 primitives with different...

In this paper we improve Wu and Wang's method for finding impossible differentials of block cipher structures. This improvement is more general than Wu and Wang's method that it can find more impossible differentials with less time. We apply it on Gen-CAST256, Misty, Gen-Skipjack, Four-Cell, Gen-MARS, SMS4, MIBS, Camellia*, LBlock, E2 and SNAKE block ciphers. All impossible differentials discovered by the algorithm are the same as Wu's method. Besides, for the 8-round MIBS block cipher, we...

Algebraic analysis of block ciphers aims at finding the secret key by solving a collection of polynomial equations that describe the internal structure of a cipher for chosen observations of plaintext/ciphertext pairs. Although algebraic attacks are addressed for cryptanalysis of block and stream ciphers, there is a lack of understanding of the impact of algebraic representation of the cipher on efficiency of solving the resulting collection of equations. The work investigates different...

Division property is a generalized integral property proposed by Todo at EUROCRYPT 2015, and very recently, Todo et al. proposed bit-based division property and applied to SIMON32 at FSE 2016. However, this technique can only be applied to block ciphers with block size no larger than 32 due to its high time and memory complexity. In this paper, we extend Mixed Integer Linear Programming (MILP) method, which is used to search differential characteristics and linear trails of block ciphers, to...

Impossible differential and zero-correlation linear cryptanalysis are two of the most powerful cryptanalysis methods in the field of symmetric key cryptography. There are several automatic tools to search such trails for ciphers with S-boxes. These tools focus on the properties of linear layers, and idealize the underlying S-boxes, i.e., assume any input and output difference pairs are possible. In reality, such S-box never exists, and the possible output differences with any fixed input...

Key schedules in block ciphers are often highly simplified, which causes weakness that can be exploited in many attacks. At ASIACRYPT 2011, Dunkelman et al. proposed a technique using the weakness in the key schedule of AES, called key-bridging technique, to improve the overall complexity. The advantage of key-bridging technique is that it allows the adversary to deduce some sub-key bits from some other sub-key bits, even though they are separated by many key mixing steps. Although the...

At EUROCRYPT 2015, Todo proposed the division property. Since then, many researches about the division property had occurred in succession. Inspired by the bit-based division property on SIMON introduced by Todo and Morri at FSE 2016, we give a further understanding of bit-based division property and come up with a new method to reconsider the \textbf{Substitution} rule given by Todo. By integrating the method of division property with the concrete boolean function expressions of S-box, this...

TWINE is a recent lightweight block cipher based on a Feistel structure. We first present two new attacks on TWINE-128 reduced to 25 rounds that have a slightly higher overall complexity than the 25-round attack presented by Wang and Wu at ACISP 2014, but a lower data complexity. Then, we introduce alternative representations of both the round function of this block cipher and of a sequence of 4 rounds. LBlock, another lightweight block cipher, turns out to exhibit the same behaviour. Then,...

In this paper we introduce a framework for the benchmarking of lightweight block ciphers on a multitude of embedded platforms. Our framework is able to evaluate the execution time, RAM footprint, as well as binary code size, and allows one to define a custom "figure of merit" according to which all evaluated candidates can be ranked. We used the framework to benchmark implementations of 19 lightweight ciphers, namely AES, Chaskey, Fantomas, HIGHT, LBlock, LEA, LED, Piccolo, PRESENT, PRIDE,...

In this paper, we investigate the Mixed-integer Linear Programming (MILP) modelling of the differential and linear behavior of a wide range of block ciphers. We point out that the differential behavior of an arbitrary S-box can be exactly described by a small system of linear inequalities. ~~~~~Based on this observation and MILP technique, we propose an automatic method for finding high probability (related-key) differential or linear characteristics of block ciphers. Compared with Sun {\it...

Impossible differential cryptanalysis has shown to be a very powerful form of cryptanalysis against block ciphers. These attacks, even if extensively used, remain not fully understood because of their high technicality. Indeed, numerous are the applications where mistakes have been discovered or where the attacks lack optimality. This paper aims in a first step at formalizing and improving this type of attacks and in a second step at applying our work to block ciphers based on the Feistel...

Impossible differential attacks are among the most powerful forms of cryptanalysis against block ciphers. We present in this paper an in-depth complexity analysis of these attacks. We show an unified way to mount such attacks and provide generic formulas for estimating their time, data and memory complexities. LBlock is a well studied lightweight block cipher with respect to impossible differential attacks. While previous single-key cryptanalysis reached up to 22 rounds, by applying our...

Impossible differential cryptanalysis has been proved to be one of the most powerful techniques to attack block ciphers. Based on the impossible differential paths, we can usually add several rounds before or after to launch the key recovery attack. Impossible differential cryptanalysis is powerful not only because the number of rounds it can break is very competitive compared to other attacks, but also unlike differential attacks which are statistical attacks in the essential, impossible...

We propose two systematic methods to describe the differential property of an S-box with linear inequalities based on logical condition modelling and computational geometry respectively. In one method, inequalities are generated according to some conditional differential properties of the S-box; in the other method, inequalities are extracted from the H-representation of the convex hull of all possible differential patterns of the S-box. For the second method, we develop a greedy algorithm...