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arXiv:1512.04965

Applying Grover's algorithm to AES: quantum resource estimates

Markus Grassl, Brandon Langenberg, Martin Roetteler, Rainer Steinwandt

Published Dec 15, 2015• 12 citations
arXivHTMLPDF

Abstract

We present quantum circuits to implement an exhaustive key search for the Advanced Encryption Standard (AES) and analyze the quantum resources required to carry out such an attack. We consider the overall circuit size, the number of qubits, and the circuit depth as measures for the cost of the presented quantum algorithms. Throughout, we focus on Clifford$+T$ gates as the underlying fault-tolerant logical quantum gate set. In particular, for all three variants of AES (key size 128, 192, and 256 bit) that are standardized in FIPS-PUB 197, we establish precise bounds for the number of qubits and the number of elementary logical quantum gates that are needed to implement Grover's quantum algorithm to extract the key from a small number of AES plaintext-ciphertext pairs.

DOI
10.48550/arxiv.1512.04965

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We present quantum circuits to implement an exhaustive key search for the Advanced Encryption Standard (AES) and analyze the quantum resources required to carry out such an attack. We consider the overall circuit size, the number of qubits, and the circuit depth as measures for the cost of the presented quantum algorithms. Throughout, we focus on Clifford+T+T+T gates as the underlying fault-tolerant logical quantum gate set. In particular, for all three variants of AES (key size 128, 192, and 256 bit) that are standardized in FIPS-PUB 197, we establish precise bounds for the number of qubits and the number of elementary logical quantum gates that are needed to implement Grover's quantum algorithm to extract the key from a small number of AES plaintext-ciphertext pairs.
To quantify security levels in a postquantum scenario, it is common to use the quantum resources needed to attack the Advanced Encryption Standard (AES) as a reference value. Specifically, in the National Institute of Standards and Technology's ongoing postquantum standardization effort, different security categories are defined that reflect the quantum resources needed to attack AES-128, AES-192, and AES-256. This article presents a quantum circuit to implement the S-box of AES. Also, leveraging an improved implementation of the key expansion, we identify new quantum circuits for all three AES key lengths. For AES-128, the number of Toffoli gates can be reduced by more than 88% compared to Almazrooie et al. 's and Grassl et al. 's estimates while simultaneously reducing the number of qubits. Our circuits can be used to simplify a Grover-based key search for AES.
To quantify security levels in a postquantum scenario, it is common to use the quantum resources needed to attack the Advanced Encryption Standard (AES) as a reference value. Specifically, in the National Institute of Standards and Technology's ongoing postquantum standardization effort, different security categories are defined that reflect the quantum resources needed to attack AES-128, AES-192, and AES-256. This article presents a quantum circuit to implement the S-box of AES. Also, leveraging an improved implementation of the key expansion, we identify new quantum circuits for all three AES key lengths. For AES-128, the number of Toffoli gates can be reduced by more than 88% compared to Almazrooie et al.'s and Grassl et al.'s estimates while simultaneously reducing the number of qubits. Our circuits can be used to simplify a Grover-based key search for AES.
NIST(National Institute of Standards and Technology) estimates the Grover key search cost required for symmetric key cryptography as the postquantum security strength. The Grover search algorithm is applicable to key recovery in symmetric key cryptography, and for this, the target encryption process must be implemented as a quantum circuit. In this paper, we present an optimized implementation of Simplified AES as a quantum circuit. Substitution, Mix Columns, and Key Expansion are implemented efficiently, and as a result of resource analysis, qubits, quantum gates, and circuit depth are significantly reduced compared to the previous result.
An efficient quantum implementation of the advanced encryption standard (AES) is crucial for reducing the complexity of implementing an exhaustive key search through Grover’s algorithm. In this paper, we study how to construct resource-efficient quantum circuits for AES. We consider the product of T-gates depth and width (TDW) and the product of full depth and width (FDW) as optimization targets. We propose a generic method, called the controlled control qubit cascade (CCQC) technique, to construct quantum circuits for nonlinear components with reduced TDW and FDW. Using this, we construct a quantum circuit for the AES S-box. Compared with recent work presented at ASIACRYPT 2023, our S-box quantum circuit achieves reductions of 2.3% in TDW and 45.2% in FDW. Additionally, we propose a new key schedule strategy to reduce the full depth of the AES quantum circuit. Finally, the trade-offs between T-gates depth and width and the parallel numbers of S-box and TDW are analyzed.
Quantum computing poses a significant threat to classical cryptographic algorithms, particularly the Advanced Encryption Standard (AES). This study investigates the impact of Grover's algorithm on the AES S-box, a fundamental component of the AES encryption process. By leveraging quantum mechanics, Grover's algorithm can potentially reduce the time required to break AES encryption from 2128operations to 264operations, thus presenting a substantial security risk. This research aims to evaluate the feasibility of such quantum attacks on AES S-box by conducting experiments using virtual quantum simulators. The study examines the resource requirements, including the number of gates and qubits, and assesses the scalability of Grover's algorithm across different data sizes and types. The findings indicate that, while theoretically effective, the practical implementation of Grover's algorithm is currently hindered by significant resource constraints on existing quantum hardware. These results underscore the need for continued advancements in quantum technology and the development of quantum-resistant cryptographic methods to ensure data security in the quantum era. This research contributes to the understanding of quantum vulnerabilities in classical cryptography and supports the development of robust encryption strategies that can withstand future quantum attacks.
Extensive research is currently underway to determine the security of existing ciphers in light of the advancements in quantum computing. Against symmetric key cryptography, Grover’s search algorithm is a prominent attack, capable of reducing search costs to the square root. For using Grover’s algorithm, it is imperative to embed the target cipher into a quantum circuit. Even so, this area of research is relatively new; it has garnered significant attention from the research community. In this study, we provide the first estimation of the cost of Grover’s key search attack against the AES-based AEAD schemes Rocca-S, AEGIS-128, and Tiaoxin-346. Our analysis considers circuit depth restrictions specified in NIST’s PQC standardization process. Considering NIST’s maximum depth constraints, We present the overall cost of these attacks using gate count and depth-times-width metrics. We observed that for MAXDEPTH=240\textsf{MAXDEPTH} = 2^{40}MAXDEPTH=240 , Rocca-S, AEGIS-128, and Tiaoxin-346 can be retrieved using Grover’s search algorithm with gate count of 1.09 × 2253, 1.14 × 2124, and 1.22 × 2124 respectively. Concerning the current updated values by NIST, these ciphers are secure in terms of the cost of implementing Grover’s attack for key recovery. The quantum circuits of these ciphers are implemented using QISKIT, an open-source software development kit (SDK) designed for working with quantum computers running on the IBM Quantum Experience platform.