Quantum secure non-malleable-extractors
Naresh Goud Boddu, Rahul Jain, Upendra Kapshikar
TL;DR
The work addresses extracting uniform randomness from weak quantum sources while remaining non-malleable against adaptive tampering and quantum side information. It develops explicit quantum-secure seeded and 2-source non-malleable extractors by integrating quantum-proof Trevisan extractors with alternating extraction, flip-flop, and correlation-breakers, all within the quantum-privacy amplification framework. The seeded NMExt achieves $(k, O(\varepsilon))$-security for seed length $d = O(\log^{7}(n/\varepsilon))$ and $k = Ω(d)$, enabling a two-round privacy amplification protocol with polylogarithmic communication, while the 2-source NMExt attains $(n-k, n-k, O(\varepsilon))$-security for $k = O(n^{1/4})$ and error $2^{-n^{Ω(1)}}$, along with $t$-tampering extensions for seeded and 2-source cases. The results also analyze several adversary models (e.g., $\mathsf{qia}$, $\mathsf{qMara}$) and connect to quantum-secure non-malleable codes and PA, highlighting significant advances in quantum-secure randomness extraction and cryptographic primitives under active quantum threats.
Abstract
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 $k \geq \textsf{poly}\left(\log \left( \frac{n}ε \right)\right)$ ($n$ is the length of the source and $ε$ 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 $\textsf{poly}\left(\log \left( \frac{n}ε \right)\right)$, 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 $k \geq n- n^{Ω(1)}$, with an output of size $n^{Ω(1)}$ and error $2^{- n^{Ω(1)}}$. 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 ($t=d^{Ω(1)}$) as well as $2$-source case ($t=n^{Ω(1)}$).