We construct several explicit quantum secure non-malleable-extractors. All the quantum secure non-malleable-extractors we construct are based on the constructions by Chattopadhyay, Goyal and Li [2015] and Cohen [2015]. 1) We construct the first explicit quantum secure non-malleable-extractor for (source) min-entropy $k \geq \textsf{poly}\left(\log \left( \frac{n}{\epsilon} \right)\right)$ ($n$ is the length of the source and $\epsilon$ is the error parameter). Previously Aggarwal, Chung, Lin, and Vidick [2019] have shown that the inner-product based non-malleable-extractor proposed by Li [2012] is quantum secure, however it required linear (in $n$) min-entropy and seed length. Using the connection between non-malleable-extractors and privacy amplification (established first in the quantum setting by Cohen and Vidick [2017]), we get a $2$-round privacy amplification protocol that is secure against active quantum adversaries with communication $\textsf{poly}\left(\log \left( \frac{n}{\epsilon} \right)\right)$, exponentially improving upon the linear communication required by the protocol due to [2019]. 2) We construct an explicit quantum secure $2$-source non-malleable-extractor for min-entropy $k \geq n- n^{\Omega(1)}$, with an output of size $n^{\Omega(1)}$ and error $2^{- n^{\Omega(1)}}$. 3) We also study their natural extensions when the tampering of the inputs is performed $t$-times. We construct explicit quantum secure $t$-non-malleable-extractors for both seeded ($t=d^{\Omega(1)}$) as well as $2$-source case ($t=n^{\Omega(1)}$).

翻译：我们构造了若干显式的量子安全非 malleable 提取器。所有我们构造的量子安全非 malleable 提取器均基于 Chattopadhyay、Goyal 和 Li [2015] 以及 Cohen [2015] 的构造。 1) 我们构造了首个针对（源）最小熵 $k \geq \textsf{poly}\left(\log \left( \frac{n}{\epsilon} \right)\right)$（$n$ 为源长度，$\epsilon$ 为误差参数）的显式量子安全非 malleable 提取器。此前，Aggarwal、Chung、Lin 和 Vidick [2019] 已证明 Li [2012] 提出的基于内积的非 malleable 提取器是量子安全的，但该构造要求线性（关于 $n$）的最小熵和种子长度。利用非 malleable 提取器与隐私放大之间的联系（由 Cohen 和 Vidick [2017] 在量子场景中首次建立），我们得到了一个 $2$ 轮隐私放大协议，该协议可抵抗主动量子攻击，通信复杂度为 $\textsf{poly}\left(\log \left( \frac{n}{\epsilon} \right)\right)$，与 [2019] 协议所需的线性通信量相比，实现了指数级改进。 2) 我们构造了一个显式的量子安全 $2$ 源非 malleable 提取器，适用于最小熵 $k \geq n- n^{\Omega(1)}$，输出大小为 $n^{\Omega(1)}$，误差为 $2^{- n^{\Omega(1)}}$。 3) 我们还研究了当输入篡改执行 $t$ 次时的自然扩展。我们构造了显式的量子安全 $t$ 非 malleable 提取器，涵盖种子情况（$t=d^{\Omega(1)}$）和 $2$ 源情况（$t=n^{\Omega(1)}$）。