LWE with Quantum Amplitudes: Algorithm, Hardness, and Oblivious Sampling
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916423806222336 |
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| author | Chen, Yilei Hu, Zihan Liu, Qipeng Luo, Han Tu, Yaxin |
| author_facet | Chen, Yilei Hu, Zihan Liu, Qipeng Luo, Han Tu, Yaxin |
| contents | In this paper, we show new algorithms, hardness results and applications for $\sf{S|LWE\rangle}$ and $\sf{C|LWE\rangle}$ with real Gaussian, Gaussian with linear or quadratic phase terms, and other related amplitudes. Let $n$ be the dimension of LWE samples. Our main results are
1. There is a $2^{\tilde{O}(\sqrt{n})}$-time algorithm for $\sf{S|LWE\rangle}$ with Gaussian amplitude with \emph{known} phase, given $2^{\tilde{O}(\sqrt{n})}$ many quantum samples. The algorithm is modified from Kuperberg's sieve, and in fact works for more general amplitudes as long as the amplitudes and phases are completely \emph{known}.
2. There is a polynomial time quantum algorithm for solving $\sf{S|LWE\rangle}$ and $\sf{C|LWE\rangle}$ for Gaussian with quadratic phase amplitudes, where the sample complexity is as small as $\tilde{O}(n)$. As an application, we give a quantum oblivious LWE sampler where the core quantum sampler requires only quasi-linear sample complexity. This improves upon the previous oblivious LWE sampler given by Debris-Alazard, Fallahpour, Stehlé [STOC 2024], whose core quantum sampler requires $\tilde{O}(nr)$ sample complexity, where $r$ is the standard deviation of the error.
3. There exist polynomial time quantum reductions from standard LWE or worst-case GapSVP to $\sf{S|LWE\rangle}$ with Gaussian amplitude with small \emph{unknown} phase, and arbitrarily many samples. Compared to the first two items, the appearance of the unknown phase term places a barrier in designing efficient quantum algorithm for solving standard LWE via $\sf{S|LWE\rangle}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_00644 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | LWE with Quantum Amplitudes: Algorithm, Hardness, and Oblivious Sampling Chen, Yilei Hu, Zihan Liu, Qipeng Luo, Han Tu, Yaxin Quantum Physics Cryptography and Security In this paper, we show new algorithms, hardness results and applications for $\sf{S|LWE\rangle}$ and $\sf{C|LWE\rangle}$ with real Gaussian, Gaussian with linear or quadratic phase terms, and other related amplitudes. Let $n$ be the dimension of LWE samples. Our main results are 1. There is a $2^{\tilde{O}(\sqrt{n})}$-time algorithm for $\sf{S|LWE\rangle}$ with Gaussian amplitude with \emph{known} phase, given $2^{\tilde{O}(\sqrt{n})}$ many quantum samples. The algorithm is modified from Kuperberg's sieve, and in fact works for more general amplitudes as long as the amplitudes and phases are completely \emph{known}. 2. There is a polynomial time quantum algorithm for solving $\sf{S|LWE\rangle}$ and $\sf{C|LWE\rangle}$ for Gaussian with quadratic phase amplitudes, where the sample complexity is as small as $\tilde{O}(n)$. As an application, we give a quantum oblivious LWE sampler where the core quantum sampler requires only quasi-linear sample complexity. This improves upon the previous oblivious LWE sampler given by Debris-Alazard, Fallahpour, Stehlé [STOC 2024], whose core quantum sampler requires $\tilde{O}(nr)$ sample complexity, where $r$ is the standard deviation of the error. 3. There exist polynomial time quantum reductions from standard LWE or worst-case GapSVP to $\sf{S|LWE\rangle}$ with Gaussian amplitude with small \emph{unknown} phase, and arbitrarily many samples. Compared to the first two items, the appearance of the unknown phase term places a barrier in designing efficient quantum algorithm for solving standard LWE via $\sf{S|LWE\rangle}$. |
| title | LWE with Quantum Amplitudes: Algorithm, Hardness, and Oblivious Sampling |
| topic | Quantum Physics Cryptography and Security |
| url | https://arxiv.org/abs/2310.00644 |