Split-State Non-Malleable Codes and Secret Sharing Schemes for Quantum Messages
Naresh Goud Boddu, Vipul Goyal, Rahul Jain, João Ribeiro
TL;DR
The paper advances quantum cryptography by introducing and constructing split-state non-malleable codes and secret sharing schemes for quantum messages that are secure even when adversaries share entanglement. It combines quantum-resistant tools—such as Clifford-based masking, Pauli/Clifford twirling, transpose techniques, and quantum-secure 2-source non-malleable extractors—with augmented leakage-resilient secret sharing to realize both 2-out-of-2 and general threshold NMSS schemes. The authors prove average-case non-malleability with rate near 1/11 and, via a reduction, obtain worst-case non-malleability for quantum messages with sub-polynomial rate or polynomial-time encodings, depending on parameter choices; they also achieve threshold NMSS schemes with low privacy and non-malleability errors for quantum data. The constructions extend to classical quantum-secure NMSS as well, and the work opens the path to constant-rate, worst-case, quantum-secure NMMC/NMSS and more complex access structures, with significant implications for secure quantum communication and distributed quantum information processing.
Abstract
Non-malleable codes are fundamental objects at the intersection of cryptography and coding theory. These codes provide security guarantees even in settings where error correction and detection are impossible, and have found applications to several other cryptographic tasks. One of the strongest and most well-studied adversarial tampering models is $2$-split-state tampering. Here, a codeword is split into two parts and the adversary can then independently tamper with each part using arbitrary functions. This model can be naturally extended to the secret sharing setting with several parties by having the adversary independently tamper with each share. Previous works on non-malleable coding and secret sharing in the split-state tampering model only considered the encoding of \emph{classical} messages. Furthermore, until recent work by Aggarwal, Boddu, and Jain (IEEE Trans.\ Inf.\ Theory 2024), adversaries with quantum capabilities and \emph{shared entanglement} had not been considered, and it is a priori not clear whether previous schemes remain secure in this model. In this work, we introduce the notions of split-state non-malleable codes and secret sharing schemes for quantum messages secure against quantum adversaries with shared entanglement. Then, we present explicit constructions of such schemes that achieve low-error non-malleability. More precisely, we construct efficiently encodable and decodable split-state non-malleable codes and secret sharing schemes for quantum messages preserving entanglement with external systems and achieving security against quantum adversaries having shared entanglement with codeword length $n$, any message length at most $n^{Ω(1)}$, and error $ε=2^{-{n^{Ω(1)}}}$. In the easier setting of \emph{average-case} non-malleability, we achieve efficient non-malleable coding with rate close to $1/11$.