In this paper, we propose a novel approach for optimizing the linear layer used in symmetric cryptography. It is observed that these matrices often have circulant structure. The basic idea of this work is to utilize the property to construct a sequence of transformation matrices, which allows subsequent heuristic algorithms to find more efficient implementations. Our results outperform previous works for various linear layers of block ciphers. For Whirlwind M0 , we obtain two implementations with 159 XOR counts (8% better than Yuan et al. at FSE 2025) and depth 17 (39% better than Shi et al. at AsiaCrypt 2024) respectively. For AES MixColumn, our automated method produces a quantum circuit with depth 10, which nearly matches the manually optimized state-of-the-art result by Zhang et al. at IEEE TC 2024, only with 2 extra CNOTs.
In this paper, we propose a novel approach for optimizing the linear layer used in symmetric cryptography. It is observed that these matrices often have circulant structure. The basic idea of this work is to utilize the property to construct a sequence of transformation matrices, which allows subsequent heuristic algorithms to find more efficient implementations. Our results outperform previous works for various linear layers of block ciphers. For Whirlwind M0 , we obtain two implementations with 159 XOR counts (8% better than Yuan et al. at FSE 2025) and depth 17 (39% better than Shi et al. at AsiaCrypt 2024) respectively. For AES MixColumn, our automated method produces a quantum circuit with depth 10, which nearly matches the manually optimized state-of-the-art result by Zhang et al. at IEEE TC 2024, only with 2 extra CNOTs.
This paper presents a novel approach to optimizing the linear layer of block ciphers using the matrix decomposition framework. It is observed that the reduction properties proposed by Xiang et al. (in FSE 2020) need to be improved. To address these limitations, we propose a new reduction framework with a complete reduction algorithm and swapping algorithm. Our approach formulates matrix decomposition as a new framework with an adaptive objective function and converts the problem to a Graph Isomorphism problem (GI problem). Using the new reduction algorithm, we were able to achieve lower XOR counts and depths of quantum implementations under the s-XOR metric. Our results outperform previous works for many linear layers of block ciphers and hash functions; some of them are better than the current g-XOR implementation. For the AES MixColumn operation, we get two implementations with 91 XOR counts and depth 13 of in-place quantum implementation, respectively.
In this paper, we consider the problem of quantum implementation of the symmetric-key ciphers. The typical ciphers in this category have two main components, the linear and the non-linear layers, we consider both. The linear layer can be described as a non-singular matrix over binary operations. The in-place implementation is one of the main considerations for implementing such layer to minimize the number of qubits. We discuss the research works that have been done to this point on this subject matter and report the improved implementation of some of the matrices. Lists of our modifications include; making it more efficient (specially for the larger matrices), consideration for quantum depth (along with a randomized algorithm for optimization), etc. We report benchmarks for the ASCON and SHA-2 linear matrices. As for the non-linear layer, the constituent block is called an substitution box (an S-box, for short). Our “DORCIS” tool presented in this paper, finds a quantum circuit with an optimized depth for given S-boxes of size 3- and 4-bit. It is a follow-up work on LIGHTER-R (which is applicable for 4-bit S-boxes only) with multiple extensions. Unlike LIGHTER-R, our DORCIS takes a quantum decomposition based on Clifford and T gates. Also, both the full quantum depth and the T depth can be optimized by DORCIS. We compare our implementation with other optimized quantum circuits shown in the other research works and show that we find an implementation with the same cost metric, or find an implementation with lower cost metric, compared other tools proposed in the literature, apart from being simpler and more efficient.
In the post-quantum era, the security level of encryption algorithms is often evaluated based on the quantum resources required to attack AES. In this work, we make thoroughly estimations on various performance metrics of the quantum circuit of AES-128/192/256. Firstly, we introduce a generic round structure for in-place implementation of the AES algorithm, maximizing the parallelism between nonlinear components. Specifically, when employed as an encryption oracle, our structure reduces theT-depth from2rdto(r+1)d. Furthermore, by leveraging the properties of block-cyclic matrices, we present an in-place implementation circuit for MixColumn with depth 10, utilizing 105 CNOT gates. In relation to the S-box, we have assessed its minimum circuit width at differentT-depths and provide multiple versions of circuit implementations for a depth-width trade-off. Finally, based on our optimized S-box circuit, we conduct a comprehensive analysis of the implementation complexity of different round structures, where our structure exhibits significant advantages in terms of lowT-depth.
With the continuous development of quantum technology and the gradual advancement of quantum computers, the challenges faced by modern cryptography are becoming increasingly diverse. When analyzing the security strength of cryptographic algorithms against quantum attacks, the construction of quantum implementation circuits is a crucial aspect. Symmetric-key cryptography is an important component of modern cryptography. This study reviews the research results of optimizing the quantum implementation of symmetric-key cryptography under quantum computing models. Specifically, this work summarizes the construction of reversible circuits for linear layers, Substitution-boxes (S-boxes), and various addition operations, as well as the quantum implementation of symmetric-key ciphers such as AES, SM4, and Camellia.
In recent years, quantum technology has been rapidly developed. As security analyses for symmetric ciphers continue to emerge, many require an evaluation of the resources needed for the quantum circuit implementation of the encryption algorithm. In this regard, we propose the quantum circuit decision problem, which requires us to determine whether there exists a quantum circuit for a given permutation f using M ancilla qubits and no more than K quantum gates within the circuit depth D. Firstly, we investigate heuristic algorithms and classical SAT-based models in previous works, revealing their limitations in solving the problem. Hence, we innovatively propose an improved SAT-based model incorporating three metrics of quantum circuits. The model enables us to find the optimal quantum circuit of an arbitrary 3 or 4-bit S-box under a given optimization goal based on SAT solvers, which has proved the optimality of circuits constructed by the tool, LIGHTER-R. Then, by combining different criteria in the model, we find more compact quantum circuit implementations of S-boxes such as RECTANGLE and GIFT. For GIFT S-box, our model provides the optimal quantum circuit that only requires 8 gates with a depth of 31. Furthermore, our model can be generalized to linear layers and improve the previous SAT-based model proposed by Huang et al. in ASIACRYPT 2022 by adding the criteria on the number of qubits and the circuit depth.
Our work explores the key recovery attack using the Grover's search on the three variants of AES (-128, -192, -256). In total, we develop a pool of 26 implementations per AES variant (totaling 78), by taking the state-of-the-art advancements in the relevant fields into account. We present the least Toffoli depth and full depth implementations of AES, thereby improving from Zou et al.'s Asiacrypt'20 paper by more than 97 percent for each variant of AES. We show that the qubit count - Toffoli depth product is reduced from theirs by more than 87 percent. Furthermore, we analyze the Jaques et al.'s Eurocrypt'20 implementations in detail, fix the bugs (arising from some problem of the quantum computing tool used), and report corrected benchmarks (which seem to improve from the authors' own bug-fixing, thanks to our architecture consideration). To the best of our finding, our work improves from all the previous works (including the Asiacrypt'22 paper by Huang and Sun, the Asiacrypt'23 paper by Liu et al. and the Asiacrypt'24 paper by Shi and Feng) in terms of various quantum circuit complexity metrics. To be more precise, we estimate the currently best-known quantum attack complexities for AES-128 ( 2 156.2630 ), AES-192 ( 2 221.5801 ) and AES-256 ( 2 286.0731 ). Additionally, we achieve the least Toffoli depth - qubit count product for AES-128 ( 121920 , improving from 130720 by Shi and Feng in Asiacrypt'24), AES-192 ( 161664 , improving from 188880 by Liu et al. in Asiacrypt'23) and AES-256 ( 206528 , improving from 248024 by Liu et al. in Asiacrypt'23) so far. We further investigate the prospect of the Grover's search. We propose four new implementations of the S-box, one new implementation of the MixColumn; as well as five new architecture (one is motivated by the architecture by Jaques et al. in Eurocrypt'20, and the rest four are entirely our innovation). Under the MAXDEPTH constraint (specified by NIST), the circuit depth metrics (Toffoli depth, T -depth and full depth) become crucial factors and parallelization for often becomes necessary. We provide the least depth implementation in this respect that offers the best performance in terms of metrics like depth-squared - qubit count product, depth - gate count product.