Abstract
Owing to smaller key size, hyperelliptic curve cryptosystem (HCC) has attracted much attention in modern cryptography, which is generally based on the discrete logarithm problem on the hyperelliptic curves of genus 2 (HCDLP). Unfortunately, quantum computation may threaten this widely applied cryptosystem, yet the exact quantum cost of HCDLP is still unexploited because of complicated divisor addition formulae. In this work, we present the concrete quantum resource estimate for Shor’s algorithm to compute HCDLP over the Mersenne prime fields. For this aim, we first modify basic modular operations for quantum computation. Then, we realize the quantum circuit from the reversible transforms of divisor additions. As the core of our work, the transforms have been decomposed into the straight-line program of basic modular operations with minimal auxiliary registers. Finally, we expound that the HCDLP over an n-bit Mersenne prime field can be computed on a quantum computer with \(3344n^{3}-72n^{2}-1360n\) Toffoli gates using \(20n+2\lceil \log n\rceil +10\) qubits. In particular, under the 128-bit security level, the quantum circuit for HCDLP over the Mersenne prime field \(\mathbb {F}_{2^{127}-1}\) requires more quantum resources than that of ECDLP over the generic prime fields.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Nos. 61972429 and 62272491) and the Guangdong Major Project of Basic and Applied Basic Research (2019B030302008) and the National R &D Key Program of China under Grant (2022YFB2701500) and General Project of Hunan Provincial Department of Education (No. 21C0326) and Hunan Provincial Natural Science Foundation of China (No. 2022JJ40154).
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Chen, C., Guan, P., Huang, Y. et al. Quantum circuits for hyperelliptic curve discrete logarithms over the Mersenne prime fields. Quantum Inf Process 22, 274 (2023). https://doi.org/10.1007/s11128-023-04017-x
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DOI: https://doi.org/10.1007/s11128-023-04017-x