Optimizing the Depth of Quantum Implementations of Linear Layers

András Czégel

Buji Xu, Xiaoming Sun

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.

Anubhab Baksi, Sumanta Chakraborty, Anupam Chattopadhyay +8 more

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.

Hermann Fürntratt, Paul Schnabl, Florian Krebs +2 more

Yufei Yuan, Wenling Wu, Tairong Shi +2 more

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.

Zhenyu Huang, Fuxin Zhang, Dongdai Lin