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Journal of Experimental and Theoretical Physics

Quantum Circuits for S-Box Implementation without Ancilla Qubits

Д. Денисенко

Published Jun 1, 2019• 8 citations
Publisher

Abstract

Abstract A method is presented for implementing substitutions (S-boxes) in the form of quantum circuits without ancilla qubits. Quantum circuits are presented that implement substitutions in the GOST R 34.12-2015 block cipher “Magma” on four logical qubits, and the descriptions of quantum circuits implementing AES and GOST R 34.12-2015 “Kuznechik” S-boxes on 8 logical qubits—quantum circuits with minimum number of logical qubits—are given.

DOI
10.1134/s1063776119050108

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Abstract A method is presented for implementing substitutions (S-boxes) in the form of quantum circuits without ancilla qubits. Quantum circuits are presented that implement substitutions in the GOST R 34.12-2015 block cipher “Magma” on four logical qubits, and the descriptions of quantum circuits implementing AES and GOST R 34.12-2015 “Kuznechik” S-boxes on 8 logical qubits—quantum circuits with minimum number of logical qubits—are given.
Abstract— This article presents a comprehensive analysis of Denisenko's method for constructing quantum circuits for S-boxes without using ancilla qubits. We elaborate on this methodology in terms of practical aspects that can be applied to S-boxes or permutations of arbitrary sizes. In addition, we created an automatic tool in C++ language that allows to construction of quantum circuits for any S-box without ancilla qubits. Furthermore, a quantum circuit for an 8-bit S-box of the MKV block cipher standard in the civilian sector of Vietnam is also provided using this tool.
ABSTRACT Quantum computing environments make block ciphers susceptible to exhaustive key search attacks utilizing Grover's algorithm. However, such quantum‐based attacks remain impractical unless the targeted cipher is directly implemented on a quantum platform. Moreover, their efficiency significantly depends on the quantum circuit design and optimization of the block cipher in question. The cost of a quantum circuit implementation is typically measured by two main metrics: the number of qubits and the circuit depth ( T ‐depth). For most block ciphers, the S‐box is the principal factor contributing to increased T ‐depth and additional qubit requirements. This paper presents a method for generating quantum circuits directly from lookup tables of S‐boxes applicable to arbitrary block ciphers. We illustrate our approach using quantum circuit implementations of the PRESENT and DES ciphers as practical examples. Our proposed method is expected to efficiently implement arbitrary S‐boxes by employing polynomial evaluation, thus balancing time–space complexity.
In recent years, the quantum circuit implementation in symmetric cryptography has attracted widespread attention. The SM4 block cipher algorithm is China's first commercial cipher standard and was accepted as an ISO/IEC international standard in 2021. S-box is the most complex component in the SM4 block cipher algorithm. This study investigates the quantum circuit implementation of the SM4 S-box. Based on the algebraic structure, using tower filed decomposition technology, the quantum circuit of the SM4 S-box is built. Based on the number of NCT gate sets, our quantum circuit requires 20 qubits, 52 Toffoli gates, 196 CNOT gates, and 13 X gates. The Toffoli depth of our circuit is 36. Compared with previous literature, the quantum circuit proposed herein requires fewer quantum resources.
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.
Grover’s search algorithm allows a quantum adversary to find akkk-bit secret key of a block cipher by making O(2k/22^{k/2}2k/2) block cipher queries. Resistance of a block cipher to such an attack is evaluated by quantum resources required to implement Grover’s oracle for the target cipher. The quantum resources are typically estimated by theT\textit {T}T-depth of its circuit implementation and the number of qubits used by the circuit (width). Since the AES S-box is the only component which requiresT\textit {T}T-gates in a quantum implementation of AES, recent research has put its focus on efficient implementation of the AES S-box. However, any efficient implementation with lowT\textit {T}T-depth will not be practical in the real world without considering qubit consumption of the implementation. In this work, we propose three methods of trade-off between time and space for the quantum implementation of the AES S-box. In particular, one of our methods turns out to use the smallest number of qubits among the existing methods, significantly reducing itsT\textit {T}T-depth.