International Association for Cryptologic Research

International Association
for Cryptologic Research


Chaoyun Li


Constructing Low-latency Involutory MDS Matrices with Lightweight Circuits
MDS matrices are important building blocks providing diffusion functionality for the design of many symmetric-key primitives. In recent years, continuous efforts are made on the construction of MDS matrices with small area footprints in the context of lightweight cryptography. Just recently, Duval and Leurent (ToSC 2018/FSE 2019) reported some 32 × 32 binary MDS matrices with branch number 5, which can be implemented with only 67 XOR gates, whereas the previously known lightest ones of the same size cost 72 XOR gates.In this article, we focus on the construction of lightweight involutory MDS matrices, which are even more desirable than ordinary MDS matrices, since the same circuit can be reused when the inverse is required. In particular, we identify some involutory MDS matrices which can be realized with only 78 XOR gates with depth 4, whereas the previously known lightest involutory MDS matrices cost 84 XOR gates with the same depth. Notably, the involutory MDS matrix we find is much smaller than the AES MixColumns operation, which requires 97 XOR gates with depth 8 when implemented as a block of combinatorial logic that can be computed in one clock cycle. However, with respect to latency, the AES MixColumns operation is superior to our 78-XOR involutory matrices, since the AES MixColumns can be implemented with depth 3 by using more XOR gates.We prove that the depth of a 32 × 32 MDS matrix with branch number 5 (e.g., the AES MixColumns operation) is at least 3. Then, we enhance Boyar’s SLP-heuristic algorithm with circuit depth awareness, such that the depth of its output circuit is limited. Along the way, we give a formula for computing the minimum achievable depth of a circuit implementing the summation of a set of signals with given depths, which is of independent interest. We apply the new SLP heuristic to a large set of lightweight involutory MDS matrices, and we identify a depth 3 involutory MDS matrix whose implementation costs 88 XOR gates, which is superior to the AES MixColumns operation with respect to both lightweightness and latency, and enjoys the extra involution property.
Correlation of Quadratic Boolean Functions: Cryptanalysis of All Versions of Full $\mathsf {MORUS}$
We show that the correlation of any quadratic Boolean function can be read out from its so-called disjoint quadratic form. We further propose a polynomial-time algorithm that can transform an arbitrary quadratic Boolean function into its disjoint quadratic form. With this algorithm, the exact correlation of quadratic Boolean functions can be computed efficiently.We apply this method to analyze the linear trails of $$\mathsf {MORUS}$$ (one of the seven finalists of the CAESAR competition), which are found with the help of a generic model for linear trails of $$\mathsf {MORUS}$$-like key-stream generators. In our model, any tool for finding linear trails of block ciphers can be used to search for trails of $$\mathsf {MORUS}$$-like key-stream generators. As a result, a set of trails with correlation $$2^{-38}$$ is identified for all versions of full $$\mathsf {MORUS}$$, while the correlations of previously published best trails for $$\mathsf {MORUS}$$-640 and $$\mathsf {MORUS}$$-1280 are $$2^{-73}$$ and $$2^{-76}$$ respectively (ASIACRYPT 2018). This significantly improves the complexity of the attack on $$\mathsf {MORUS}$$-1280-256 from $$2^{152}$$ to $$2^{76}$$. These new trails also lead to the first distinguishing and message-recovery attacks on $$\mathsf {MORUS}$$-640-128 and $$\mathsf {MORUS}$$-1280-128 with surprisingly low complexities around $$2^{76}$$.Moreover, we observe that the condition for exploiting these trails in an attack can be more relaxed than previously thought, which shows that the new trails are superior to previously published ones in terms of both correlation and the number of ciphertext blocks involved.
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) 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.
Design of Lightweight Linear Diffusion Layers from Near-MDS Matrices
Chaoyun Li Qingju Wang
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.