Quantum reversible circuits for $\mathrm{GF}(2^{8})$ multiplicative inverseQuantum reversible circuits for $\mathrm{GF}(2^{8})$ multiplicative inverse

Qing-bin Luo, Guowu Yang, Xiaoyu Li +1 more

Abstract The synthesis of quantum circuits for multiplicative inverse over $\operatorname{GF}(2^{8})$ GF ( 2 8 ) are discussed in this paper. We first convert the multiplicative inverse operation in $\operatorname{GF}(2^{8})$ GF ( 2 8 ) to arithmetic operations in the composite field $\operatorname{GF}((2^{4})^{2})$ GF ( ( 2 4 ) 2 ) , and then discuss the expressions of the square calculation, the inversion calculation and the multiplication calculation separately in the finite field $\operatorname{GF}(2^{4})$ GF ( 2 4 ) , where the expressions of multiplication calculation in $\operatorname{GF}(2^{4})$ GF ( 2 4 ) are given directly in $\operatorname{GF}(2^{4})$ GF ( 2 4 ) and given through being transformed into the composite field $\operatorname{GF}((2^{2})^{2})$ GF ( ( 2 2 ) 2 ) . Then the quantum circuits of these calculations are realized one by one. Finally, two quantum circuits for multiplicative inverse over $\operatorname{GF}(2^{8})$ GF ( 2 8 ) are synthesized. They both use 21 qubits, the first quantum circuit uses 55 Toffoli gates and 107 CNOT gates and the second one uses 37 Toffoli gates and 209 CNOT gates. As an example of the application of multiplication inverse, we apply these quantum circuits to the implementations of the S-box quantum circuit of the AES cryptographic algorithm. Two quantum circuits for implementing the S-box of the AES cryptographic algorithm are presented. The first quantum circuit uses 21 qubits, 55 Toffoli gates, 131 CNOT gates and 4 NOT gates and the second one uses 21 qubits, 37 Toffoli gates, 233 CNOT gates and 4 NOT gates. Through the evaluation of quantum cost, the two quantum circuits of the S-box of AES cryptographic algorithm use less quantum resources than the existing schemes.Abstract The synthesis of quantum circuits for multiplicative inverse over $\operatorname{GF}(2^{8})$ GF ( 2 8 ) are discussed in this paper. We first convert the multiplicative inverse operation in $\operatorname{GF}(2^{8})$ GF ( 2 8 ) to arithmetic operations in the composite field $\operatorname{GF}((2^{4})^{2})$ GF ( ( 2 4 ) 2 ) , and then discuss the expressions of the square calculation, the inversion calculation and the multiplication calculation separately in the finite field $\operatorname{GF}(2^{4})$ GF ( 2 4 ) , where the expressions of multiplication calculation in $\operatorname{GF}(2^{4})$ GF ( 2 4 ) are given directly in $\operatorname{GF}(2^{4})$ GF ( 2 4 ) and given through being transformed into the composite field $\operatorname{GF}((2^{2})^{2})$ GF ( ( 2 2 ) 2 ) . Then the quantum circuits of these calculations are realized one by one. Finally, two quantum circuits for multiplicative inverse over $\operatorname{GF}(2^{8})$ GF ( 2 8 ) are synthesized. They both use 21 qubits, the first quantum circuit uses 55 Toffoli gates and 107 CNOT gates and the second one uses 37 Toffoli gates and 209 CNOT gates. As an example of the application of multiplication inverse, we apply these quantum circuits to the implementations of the S-box quantum circuit of the AES cryptographic algorithm. Two quantum circuits for implementing the S-box of the AES cryptographic algorithm are presented. The first quantum circuit uses 21 qubits, 55 Toffoli gates, 131 CNOT gates and 4 NOT gates and the second one uses 21 qubits, 37 Toffoli gates, 233 CNOT gates and 4 NOT gates. Through the evaluation of quantum cost, the two quantum circuits of the S-box of AES cryptographic algorithm use less quantum resources than the existing schemes.

EPJ Quantum TechnologyOct 7, 202210 citations

An Automated Synthesis Framework for Benchmarking Quantum Resource Costs of Symmetric-Key Cryptography

Chanho Choi, Jinseob Oh, SangMan Lee +2 more

Modern information security relies heavily on symmetric-key cryptography such as AES. As quantum computing advances, these classical schemes face increasing pressure from quantum key-search attacks, most notably Grover’s algorithm. To evaluate and compare quantum security quantitatively, the core components of symmetric-key algorithms must be implemented and optimized as quantum circuits. Among them, the S-box is a key source of nonlinearity and often dominates the circuit cost. In this paper, we introduce ADOQ (Automatic Depth Optimizer for Quantum circuits), a modular Python (version 3.13.3) framework that automatically synthesizes reversible quantum circuits from S-box specifications and applies a sequence of depth optimization techniques to produce optimized QASM circuits. Our experiments show that ADOQ achieves circuit depths comparable to prior work on 4-qubit S-boxes, and it also supports synthesis for larger S-boxes.Modern information security relies heavily on symmetric-key cryptography such as AES. As quantum computing advances, these classical schemes face increasing pressure from quantum key-search attacks, most notably Grover’s algorithm. To evaluate and compare quantum security quantitatively, the core components of symmetric-key algorithms must be implemented and optimized as quantum circuits. Among them, the S-box is a key source of nonlinearity and often dominates the circuit cost. In this paper, we introduce ADOQ (Automatic Depth Optimizer for Quantum circuits), a modular Python (version 3.13.3) framework that automatically synthesizes reversible quantum circuits from S-box specifications and applies a sequence of depth optimization techniques to produce optimized QASM circuits. Our experiments show that ADOQ achieves circuit depths comparable to prior work on 4-qubit S-boxes, and it also supports synthesis for larger S-boxes.

MathematicsFeb 19, 20260 citations

Optimal T-depth Quantum Circuits for Implementing Arbitrary Boolean Functions

Suman Dutta, Anik Bhaumik, Anupam Chattopadhyay +1 more

Abstract In this paper, we present a generic construction for synthesizing an optimal T-depth quantum circuit for any arbitrary n -input, m -output Boolean function f : {0, 1 } n → {0, 1} m with algebraic degree k ≤ n , achieving an exact Toffoli (consequently T) depth of ⌈log 2 k ⌉. This broadly generalizes the recent result establishing the optimal Toffoli (and T) depth for multi-controlled Toffoli decompositions (Dutta et al., Phys. Rev. A, 2025). The optimality of T-depth in this initiative is considered in the context of implementing an n -MCT, assuming the decomposition via Clifford plus Toffoli gates. The key technique involves inspecting the Algebraic Normal Form (ANF) of the Boolean function. Obtaining a benchmark for the minimum T-depth of such circuits is crucial for the efficient implementation of quantum algorithms by enabling greater parallelism, reducing time complexity, and minimizing circuit latency, making them suitable for near-term quantum devices with limited coherence times. The broader implications of our results include a provable lower bounds on T-depth for S-box and block cipher implementations, such as AES. Finally, we also explain the impact of our result in identifying the T-depth for the generic cryptanalysis of block ciphers using Grover’s algorithm.Abstract In this paper, we present a generic construction for synthesizing an optimal T-depth quantum circuit for any arbitrary n -input, m -output Boolean function f : {0, 1 } n → {0, 1} m with algebraic degree k ≤ n , achieving an exact Toffoli (consequently T) depth of ⌈log 2 k ⌉. This broadly generalizes the recent result establishing the optimal Toffoli (and T) depth for multi-controlled Toffoli decompositions (Dutta et al., Phys. Rev. A, 2025). The optimality of T-depth in this initiative is considered in the context of implementing an n -MCT, assuming the decomposition via Clifford plus Toffoli gates. The key technique involves inspecting the Algebraic Normal Form (ANF) of the Boolean function. Obtaining a benchmark for the minimum T-depth of such circuits is crucial for the efficient implementation of quantum algorithms by enabling greater parallelism, reducing time complexity, and minimizing circuit latency, making them suitable for near-term quantum devices with limited coherence times. The broader implications of our results include a provable lower bounds on T-depth for S-box and block cipher implementations, such as AES. Finally, we also explain the impact of our result in identifying the T-depth for the generic cryptanalysis of block ciphers using Grover’s algorithm.

Quantum Information and ComputationDec 1, 20252 citations

Quantum reversible circuits for

\mathrm{GF}(2^{8})

multiplicative inverse

Abstract The synthesis of quantum circuits for multiplicative inverse over

\operatorname{GF}(2^{8})

GF ( 2 8 ) are discussed in this paper. We first convert the multiplicative inverse operation in

\operatorname{GF}(2^{8})

GF ( 2 8 ) to arithmetic operations in the composite field

\operatorname{GF}((2^{4})^{2})

GF ( ( 2 4 ) 2 ) , and then discuss the expressions of the square calculation, the inversion calculation and the multiplication calculation separately in the finite field

\operatorname{GF}(2^{4})

GF ( 2 4 ) , where the expressions of multiplication calculation in

\operatorname{GF}(2^{4})

GF ( 2 4 ) are given directly in

\operatorname{GF}(2^{4})

GF ( 2 4 ) and given through being transformed into the composite field

\operatorname{GF}((2^{2})^{2})

GF ( ( 2 2 ) 2 ) . Then the quantum circuits of these calculations are realized one by one. Finally, two quantum circuits for multiplicative inverse over

\operatorname{GF}(2^{8})

GF ( 2 8 ) are synthesized. They both use 21 qubits, the first quantum circuit uses 55 Toffoli gates and 107 CNOT gates and the second one uses 37 Toffoli gates and 209 CNOT gates. As an example of the application of multiplication inverse, we apply these quantum circuits to the implementations of the S-box quantum circuit of the AES cryptographic algorithm. Two quantum circuits for implementing the S-box of the AES cryptographic algorithm are presented. The first quantum circuit uses 21 qubits, 55 Toffoli gates, 131 CNOT gates and 4 NOT gates and the second one uses 21 qubits, 37 Toffoli gates, 233 CNOT gates and 4 NOT gates. Through the evaluation of quantum cost, the two quantum circuits of the S-box of AES cryptographic algorithm use less quantum resources than the existing schemes.