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Quantum Information Processing

Concrete quantum cryptanalysis of binary elliptic curves via addition chain

Ren Taguchi, Atsushi Takayasu

Published Mar 25, 2024• 2 citations
Publisher

Abstract

Abstract Thus far, several papers reported concrete resource estimates of Shor’s quantum algorithm for solving the elliptic curve discrete logarithm problem. In this paper, we study quantum FLT-based inversion algorithms over binary elliptic curves. There are two major algorithms proposed by Banegas et al. and Putranto et al., where the former and latter algorithms achieve fewer numbers of qubits and smaller depths of circuits, respectively. We propose two quantum FLT-based inversion algorithms that essentially outperform previous FLT-based algorithms and compare the performance for NIST curves of the degreen. Specifically, for alln, our first algorithm achieves fewer qubits than Putranto et al.’s one without sacrificing the number of Toffoli gates and the depth of circuits, while our second algorithm achieves smaller depths of circuits without sacrificing the number of qubits and Toffoli gates. For example, when $$n = 571$$ n=571 , the number of qubits of our first algorithm is 74 % of that of Putranto et al.’s one, while the depth of our second algorithm is 83 % of that of Banegas et al.’s one. The improvements stem from the fact that FLT-based inversions can be performed with arbitrary sequences of addition chains for $$n - 1$$ n-1 although both Banegas et al. and Putranto et al. follow fixed sequences that were introduced by Itoh and Tsujii’s classical FLT-based inversion. In particular, we analyze how several properties of addition chains, which do not affect the computational resources of classical FLT-based inversions, affect the computational resources of quantum FLT-based inversions and find appropriate sequences.

DOI
10.1007/s11128-024-04323-y

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Abstract Thus far, several papers reported concrete resource estimates of Shor’s quantum algorithm for solving the elliptic curve discrete logarithm problem. In this paper, we study quantum FLT-based inversion algorithms over binary elliptic curves. There are two major algorithms proposed by Banegas et al. and Putranto et al., where the former and latter algorithms achieve fewer numbers of qubits and smaller depths of circuits, respectively. We propose two quantum FLT-based inversion algorithms that essentially outperform previous FLT-based algorithms and compare the performance for NIST curves of the degreen. Specifically, for alln, our first algorithm achieves fewer qubits than Putranto et al.’s one without sacrificing the number of Toffoli gates and the depth of circuits, while our second algorithm achieves smaller depths of circuits without sacrificing the number of qubits and Toffoli gates. For example, when n=571n = 571n=571 n=571 , the number of qubits of our first algorithm is 74 % of that of Putranto et al.’s one, while the depth of our second algorithm is 83 % of that of Banegas et al.’s one. The improvements stem from the fact that FLT-based inversions can be performed with arbitrary sequences of addition chains for n−1n - 1n−1 n-1 although both Banegas et al. and Putranto et al. follow fixed sequences that were introduced by Itoh and Tsujii’s classical FLT-based inversion. In particular, we analyze how several properties of addition chains, which do not affect the computational resources of classical FLT-based inversions, affect the computational resources of quantum FLT-based inversions and find appropriate sequences.
Works on quantum computing and cryptanalysis have increased significantly in the past few years. Various constructions of quantum arithmetic circuits as one of the primary elements in the field have also been proposed. However, there have only been a few studies on finite field inversion despite its essential use in realizing quantum algorithms, such as in Shor’s algorithm for Elliptic Curve Discrete Logarithm Problem (ECDLP). In this study, we propose to reduce the depth of the existing quantum Fermat’s Little Theorem (FLT)-based inversion circuit for the binary finite field. In particular, we propose to follow a complete waterfall approach to translate the Itoh-Tsujii’s variant of FLT to the corresponding quantum circuit and remove the inverse squaring operations employed in the previous work by Banegas et al., lowering the number of CNOT gates (i.e., CNOT count) as well as slightly reducing theTTTdepth, which contributes to a reduced overall depth and gate count. Furthermore, we concretely verify our method and compare it with the previous work in Qiskit, a quantum computer simulation environment, by constructing both our method and the previous work from scratch and performing the resource analysis. Additionally, we propose employing the relative-phase Toffoli gate by Gidney as opposed to the standard Toffoli implementation, which yields a significantly lowerTTTdepth while further reducing the overall depth. Our approach can serve as an alternative for a time-efficient implementation.
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.
Works on quantum computing and cryptanalysis has increased significantly in the past few years. Various constructions of quantum arithmetic circuits, as one of the essential components in the field, has also been proposed. However, there has only been a few studies on finite field inversion despite its essential use in realizing quantum algorithms, such as in Shor's algorithm for Elliptic Curve Discrete Logarith Problem (ECDLP). In this study, we propose to reduce the depth of the existing quantum Fermat's Little Theorem (FLT)-based inversion circuit for binary finite field. In particular, we propose follow a complete waterfall approach to translate the Itoh-Tsujii's variant of FLT to the corresponding quantum circuit and remove the inverse squaring operations employed in the previous work by Banegas et al., lowering the number of CNOT gates (CNOT count), which contributes to reduced overall depth and gate count. Furthermore, compare the cost by firstly constructing our method and previous work's in Qiskit quantum computer simulator and perform the resource analysis. Our approach can serve as an alternative for a time-efficient implementation.
This paper improves upon the quantum circuits required for the Shor’s attack on binary elliptic curves. We present two types of quantum point addition, taking both qubit count and circuit depth into consideration.In summary, we propose an in-place point addition that improves upon the work of Banegas et al. from CHES’21, reducing the qubit count – depth product by more than 73% – 81% depending on the variant. Furthermore, we develop an out-of-place point addition by using additional qubits. This method achieves the lowest circuit depth and offers an improvement of over 92% in the qubit count – quantum depth product (for a single step).To the best of our knowledge, our work improves from all previous works (including the CHES’21 paper by Banegas et al., the IEEE Access’22 paper by Putranto et al., and the CT-RSA’23 paper by Taguchi and Takayasu) in terms of circuit depth and qubit count – depth product.Equipped with the implementations, we discuss the post-quantum security of the binary elliptic curve cryptography. Under the MAXDEPTH metric (proposed by the US government’s NIST), the quantum circuit with the highest depth in our work is 224, which is significantly lower than the MAXDEPTH limit of 240. For the gate count – full depth product, a metric for estimating quantum attack cost (proposed by NIST), the highest complexity in our work is 260 for the curve having degree 571 (which is comparable to AES-256 in terms of classical security), considerably below the post-quantum security level 1 threshold (of the order of 2156).
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.
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.