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.
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.
Grover’s search algorithm allows a quantum adversary to find ak-bit secret key of a block cipher by making O(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-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-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-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-depth.
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.
Optimizing the quantum circuit for implementing Advanced Encryption Standard (AES) is crucial for estimating the necessary resources in attacking AES by the Grover algorithm. Previous studies have reduced the number of qubits required for the quantum circuits of AES-128/-192/-256 from 984/1112/1336 to 270/334/398, which is close to the optimal value of 256/320/384. It becomes a challenging task to further optimize them. AimTaking aim at this task, we find a method for how the quantum circuit of AES S-box can be designed with the help of the automation tool LIGHTER-R. Particularly, the multiplicative inversion in F28 , which is the main part of the S-box, is converted into the multiplicative inversion (and multiplication) in F24 , then the latter can be implemented by LIGHTER-R because its search space is small enough. By this method, we construct the quantum circuits of S-box for mapping |a⟩|0⟩ to |a⟩|S(a)⟩ and |a⟩|b⟩ to |a⟩|b ⊕ S(a)⟩ with 20 qubits instead of 22 in the previous studies. In addition, we introduce new techniques to reduce the number of qubits required by the S-box circuit for mapping |a⟩ to |S(a)⟩ from 22 in the previous studies to 16. Accordingly, we synthesize the quantum circuits of AES-128/-192/-256 with 264/328/392 qubits, which implies a new record.
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.
With the advancement of quantum computing, quantum circuit implementations of symmetric cryptographic algorithms are essential for evaluating their resistance to quantum attacks. As an ISO-standardized cipher, the efficient quantum circuit construction of Camellia is of great significance. Focusing on minimizing the DW-cost (qubits × Toffoli-depth) while balancing ancilla qubits and Toffoli-depth, this paper proposes an optimized quantum circuit design for Camellia. Specifically: (1) using a serial-multiplication and out-of-place inversion architecture under composite-field decomposition, combined with hierarchical screening for parallelizing bilinear multiplications, a compact in-place S-box quantum circuit is constructed with a DW-cost of 276, 28% lower than the state of the art; (2) a low-gate-count linear-layer circuit is obtained via heuristic search, and gate-sequence reordering then reduces the full depth from 9 to 7 without increasing the gate count; (3) Camellia-128/192/256 quantum circuits are constructed with DW-costs of 119,168, 257,664, and 257,664, achieving reductions of 48%, 52%, and 52% respectively over prior work. A pipelined Grover oracle is further designed and the total resources required for a full Grover search attacking Camellia are evaluated. For Camellia-128, the complete Grover search requires a T-depth of 2.10 × 2⁷⁶ and a total TDW-cost (qubits × T-depth) of 1.00 × 2⁸⁶, reducing T-depth by 75% and TDW-cost by 84% over prior work. The first full Grover search resource estimates for Camellia-192/256 are also presented, with T-depths of 7.20 × 2¹⁰⁸ and 8.11 × 2¹⁴⁰ and TDW-costs of 2.03 × 2¹²³ and 2.79 × 2¹⁵³, respectively. These results show that Grover's attack on Camellia requires fewer quantum resources than on AES at the 128-bit level, implying comparatively weaker post-quantum security at that key size, while the picture reverses at larger key sizes (Section 4).
Quantum search algorithms, such as Grover's algorithm, will soon pose a threat to current encryption standards as quantum computing advances toward practical implementation. These algorithms rely on quantum circuits that map cryptographic algorithms into appropriate Hilbert spaces, thus linking algorithmic inputs and outputs through quantum superposition. However, implementing and testing such circuits on existing quantum simulators, such as Qiskit, encounters significant memory and computational bottlenecks that limit research progress. To address these simulation constraints, we present a resource-optimized quantum circuit implementation of a complete two-round Mini-AES encryption algorithm. Our approach systematically converts the cipher's algebraic structure using algebraic normal form (ANF) equations, mapping XOR operations to CNOT gates and AND operations to Toffoli gates. To reduce memory overhead, we implement an ancilla-reuse strategy utilizing mid-circuit measurements and resets. Additionally, we introduce an automated procedure that generates Qiskitcompatible quantum circuits for arbitrary S-boxes and cryptographic primitives, streamlining the development of quantum cryptanalysis implementations. Our implementation achieves full encryption and decryption in a quantum circuit with modest gate requirements that can be achieved by the current or near-future state of the art quantum computers. In addition, we provide system utilization metrics to validate our resource-constrained implementation. This compact and pedagogically accessible implementation establishes a practical foundation for quantum cryptanalysis research and post-quantum cryptographic analysis.