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arXiv:2305.11410 · Cybersecurity

Minimizing CNOT-count in quantum circuit of the extended Shor’s algorithm for ECDLP

Xia Liu, Huan Yang, Li Yang

Published Dec 6, 2023• 6 citations
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Abstract

Abstract The elliptic curve discrete logarithm problem (ECDLP) is a popular choice for cryptosystems due to its high level of security. However, with the advent of the extended Shor’s algorithm, there is concern that ECDLP may soon be vulnerable. While the algorithm does offer hope in solving ECDLP, it is still uncertain whether it can pose a real threat in practice. From the perspective of the quantum circuits of the algorithm, this paper analyzes the feasibility of cracking ECDLP using an ion trap quantum computer with improved quantum circuits for the extended Shor’s algorithm. We give precise quantum circuits for extended Shor’s algorithm to calculate discrete logarithms on elliptic curves over prime fields, including modular subtraction, three different modular multiplication, and modular inverse. Additionally, we incorporate and improve upon windowed arithmetic in the circuits to reduce the CNOT-counts. Whereas previous studies mostly focused on minimizing the number of qubits or the depth of the circuit, we focus on minimizing the number of CNOT gates in the circuit, which greatly affects the running time of the algorithm on an ion trap quantum computer. Specifically, we begin by presenting implementations of basic arithmetic operations with the lowest known CNOT-counts, along with improved constructions for modular inverse, point addition, and windowed arithmetic. Next, we precisely estimate that, to execute the extended Shor’s algorithm with the improved circuits to factor an n-bit integer, the CNOT-count required is $$1237n^3/\log n+2n^2+n$$ 1237 n 3 / log n + 2 n 2 + n . Finally, we analyze the running time and feasibility of the extended Shor’s algorithm on an ion trap quantum computer.

DOI
10.1186/s42400-023-00181-w

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Abstract The elliptic curve discrete logarithm problem (ECDLP) is a popular choice for cryptosystems due to its high level of security. However, with the advent of the extended Shor’s algorithm, there is concern that ECDLP may soon be vulnerable. While the algorithm does offer hope in solving ECDLP, it is still uncertain whether it can pose a real threat in practice. From the perspective of the quantum circuits of the algorithm, this paper analyzes the feasibility of cracking ECDLP using an ion trap quantum computer with improved quantum circuits for the extended Shor’s algorithm. We give precise quantum circuits for extended Shor’s algorithm to calculate discrete logarithms on elliptic curves over prime fields, including modular subtraction, three different modular multiplication, and modular inverse. Additionally, we incorporate and improve upon windowed arithmetic in the circuits to reduce the CNOT-counts. Whereas previous studies mostly focused on minimizing the number of qubits or the depth of the circuit, we focus on minimizing the number of CNOT gates in the circuit, which greatly affects the running time of the algorithm on an ion trap quantum computer. Specifically, we begin by presenting implementations of basic arithmetic operations with the lowest known CNOT-counts, along with improved constructions for modular inverse, point addition, and windowed arithmetic. Next, we precisely estimate that, to execute the extended Shor’s algorithm with the improved circuits to factor an n-bit integer, the CNOT-count required is 1237n3/log⁡n+2n2+n1237n^3/\log n+2n^2+n1237n3/logn+2n2+n 1237 n 3 / log n + 2 n 2 + n . Finally, we analyze the running time and feasibility of the extended Shor’s algorithm on an ion trap quantum computer.
Solving the Elliptic Curve Discrete Logarithm Problem (ECDLP) is critical for evaluating the quantum security of widely deployed elliptic-curve cryptosystems. Consequently, minimizing the number of logical qubits required to execute this algorithm is a key object. In implementations of Shor's algorithm, the space complexity is largely dictated by the modular inversion operation during point addition. Starting from the extended Euclidean algorithm (EEA), we refine the register-sharing method of Proos and Zalka and propose a space-efficient reversible modular inversion algorithm. We use length registers together with location-controlled arithmetic to store the intermediate variables in a compact form throughout the computation. We then optimize the stepwise update rules and give concrete circuit constructions for the resulting controlled arithmetic components. This leads to a modular inversion circuit that uses 3n+4⌊log⁡2n⌋+O(1)3n + 4\lfloor \log_2 n \rfloor + O(1)3n+4⌊log2​n⌋+O(1) logical qubits and 204n2log⁡2n+O(n2)204n^2\log_2 n + O(n^2)204n2log2​n+O(n2) Toffoli gates. By inserting this modular inversion component into the controlled affine point-addition circuit, we obtain a space-efficient algorithm for the ECDLP with 5n+4⌊log⁡2n⌋+O(1)5n + 4\lfloor \log_2 n \rfloor + O(1)5n+4⌊log2​n⌋+O(1) qubits and O(n3)O(n^3)O(n3) Toffoli gates. In particular, for a 256-bit prime-field curve, our estimate reduces the logical-qubit count to 1333, compared with 2124 in the previous low-width implementation of Häner et al.
Since the RSA public key cryptosystem was proposed, it has been widely used because of its strong security. Although the proposal of the Shor’s algorithm offers hope for cracking RSA, it is debatable whether the algorithm can actually pose a threat in practice. From the perspective of the quantum circuit of the Shor’s algorithm, we analyse the feasibility of cracking RSA with improved quantum circuits using an ion-trap quantum computer. We present an improved quantum circuit for the modular exponentiation of a constant, which is the most expensive operation in Shor’s algorithm for integer factorization. Whereas previous studies mostly focused on minimizing the number of qubits or the depth of the circuit, we minimize the number of CNOTs, which greatly affects the time to run the algorithm on an ion-trap quantum computer. First, we give the implementation of the basic arithmetic with the lowest known number of CNOTs and the construction of an improved modular exponentiation of a constant by accumulating intermediate data and using a windowing technique. Then, we precisely estimate the number of improved quantum circuits needed to perform the Shor’s algorithm for factoring an n -bit integer, which is 217 n 3 / log 2 n + 4 n 2 + n . We analyse the running time and feasibility of the Shor’s algorithm on an ion-trap quantum computer according to the number of CNOTs. Finally, we discussed the lower bound of the number of CNOTs needed to implement the Shor’s algorithm.
The Elliptic Curve Discrete Logarithm Problem (ECDLP) is a fundamental problem in cryptography, and reducing the resource requirements of quantum algorithms for solving ECDLP is an important goal. In this work, we present a space-efficient quantum algorithm for solving the ECDLP over prime fields, achieving an implementation with only 3n+6⌊log⁡2n⌋+O(1)3n+6\lfloor \log_2 n \rfloor+O(1)3n+6⌊log2​n⌋+O(1) logical qubits and 1056n3/log⁡2n+O(n2)1056n^3/\log_2 n+O(n^2)1056n3/log2​n+O(n2) Toffoli gates, where nnn is the bit-length of the prime. For a 256-bit prime-field curve, our construction requires only 835 logical qubits, reducing the previous best estimates of 1098 and 1175 logical qubits by Chevignard et al. [EUROCRYPT 2026] and Babbush et al. [ArXiv Preprint 2026], respectively. The key to our improvement is a new space-efficient reversible modular inversion circuit, which addresses the dominant space bottleneck in affine-coordinate point addition. Starting from the extended Euclidean algorithm (EEA), we refine the register-sharing technique of Proos and Zalka by introducing length registers and location-controlled arithmetic to compactly store and update intermediate variables. We further optimize the reversible update procedures and construct the corresponding controlled arithmetic circuits, resulting in a modular inversion circuit implemented by only 2n+6⌊log⁡2n⌋+O(1)2n+6\lfloor \log_2 n \rfloor+O(1)2n+6⌊log2​n⌋+O(1) logical qubits and 229n2+O(nlog⁡2n)229n^2+O(n\log_2 n)229n2+O(nlog2​n) Toffoli gates. This modular inversion circuit together with mid-circuit measurements and classical feed-forward operations provides a space-efficient controlled affine point-addition circuit and a complete implementation of Shor's algorithm for ECDLP.
Shor's algorithm represents the main threat of quantum computers to cryptography. In order to precisely understand its feasibility, many authors have worked towards reducing its costs, either at the logical level (assuming a fault-tolerant architecture), or at the physical level (taking into account the constraints of envisioned hardware). In particular, recent works by Chevignard et al. (CRYPTO 2024) and Gidney (arXiv 2025) used improved arithmetic to significantly reduce the qubit cost of factoring RSA public keys. Even more recently, Babbush et al. (arXiv 2026) improved the cost of computing elliptic curve discrete logarithms, with a reduction of a factor 2 to 3 in gate count and qubit count compared to a previous work by Litinski (arXiv 2023). Their result relies on optimized point addition circuits on elliptic curves over prime fields. However they did not reveal their logical quantum circuits, relying instead on a zero-knowledge proof. In this paper, we detail a quantum logical circuit architecture which gives similar results as Babbush et al., with a slightly higher number of qubits (around 1.5% increase) and a slightly smaller Toffoli gate count (between 6.5% and 10% reduction) for the curve secp256k1. We also give gate counts for a generic variant of the circuit, which is valid for any prime field.
One of the strengths of our recently proposed Walking Cat Architecture for a trapped-ion quantum computer is that it is straightforward to extend and optimize for a specific application. As a proof-of-concept, here we present such optimizations for solving the 256256256-bit elliptic curve discrete logarithm problem (ECDLP) on secp256k1\mathtt{secp256k1}secp256k1, which is the elliptic curve used by blockchain technologies such as Bitcoin, using Shor's algorithm. We optimize the circuits from Schrottenloher's recent work and arrive at a logical quantum circuit for solving the ECDLP using about 145014501450 qubits and 40⋅10640\cdot 10^640⋅106 Toffoli gates, with a rigorous lower bound on the logical-level success probability that holds with confidence at least 1−2−1281-2^{-128}1−2−128. Using our compilation toolchain with manual optimization of the logical layout and integrated routing, we produce estimates for the logical measurement depth and the required number of physical qubits by compiling all components to measurement schedules that obey the architectural constraints. A key ingredient is a fast CCZ magic-state factory and a depth-one CCZ state injection, reducing the execution time of CCZ gates by a factor of 313131. We increase the logical-measurement parallelism using non-overlapping cat-based measurements in parallel, and we leverage the recently proposed logical CliNR protocol to speed up Clifford operations. To reduce the qubit overhead, we introduce a more efficient loss correction protocol, design a layout that allows us to recycle the CliNR ancilla qubits, and provision reusable cat-state resources according to the circuit's peak measurement parallelism. All results and optimizations combined, we conclude that a trapped-ion quantum computer based on our architecture can solve the ECDLP on secp256k1\mathtt{secp256k1}secp256k1 in approximately 25.7 days using 19,397 physical qubits with an estimated success probability of 63%63\%63%.
This paper presents quantum cryptanalysis for binary elliptic curves from a time-efficient implementation perspective (i.e., reducing the circuit depth), complementing the previous research that focuses on the space-efficiency perspective (i.e., reducing the circuit width). To achieve depth optimization, we propose an improvement to the existing circuit implementation of the Karatsuba multiplier and FLT-based inversion, then construct and analyze the resource in the Qiskit quantum computer simulator. The proposed multiplier architecture, which improves the quantum Karatsuba multiplier from the previous study, reduces the depth and yields a lower number of CNOT gates that bound toO(nlog2(3))O(n^{log_{2}(3)})O(nlog2​(3))while maintaining a similar number of Toffoli gates and qubits. Furthermore, our improved FLT-based inversion reduces CNOT count and overall depth, with a tradeoff of higher qubit size. Finally, we employ the proposed multiplier and FLT-based inversion for performing quantum cryptanalysis of binary point addition as well as the complete Shor’s algorithm for the elliptic curve discrete logarithm problem (ECDLP). As a result, apart from depth reduction, we are also able to reduce up to 90% of the Toffoli gates required in a single-step point addition compared to prior work, leading to significant improvements and giving new insights on quantum cryptanalysis for a depth-optimized implementation.