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 $n^{\Omega(1)}$, and error $\epsilon=2^{-{n^{\Omega(1)}}}$. In the easier setting of \emph{average-case} non-malleability, we achieve efficient non-malleable coding with rate close to $1/11$.

翻译：非可延展编码是密码学与编码理论交叉领域中的基本对象。这类编码即使在错误校正和检测不可行的场景中也能提供安全保证，并已应用于多项其他密码学任务。最强大且研究最深入的对抗篡改模型之一是2-分裂态篡改模型。在此模型中，码字被分为两部分，攻击者可独立对每部分使用任意函数进行篡改。该模型可自然地扩展到包含多个参与方的秘密共享场景，即攻击者独立篡改每个份额。以往关于分裂态篡改模型下非可延展编码和秘密共享的研究仅考虑了对经典消息的编码。此外，在Aggarwal、Boddu和Jain（IEEE Trans. Inf. Theory 2024）近期工作之前，尚未考虑具备量子能力和共享纠缠的攻击者，且先前方案在此模型中是否依然安全尚不明确。在本工作中，我们提出了针对量子消息的分裂态非可延展编码和秘密共享方案的概念，该方案能抵御具备共享纠缠的量子攻击者。随后，我们给出了实现低误差非可延展性的显式构造方案。具体而言，我们构造了可高效编码与解码的分裂态非可延展编码和秘密共享方案，其量子消息能保持与外部系统的纠缠，并可抵御具有共享纠缠的量子攻击者，码字长度为$n$，消息长度最多为$n^{\Omega(1)}$，误差为$\epsilon=2^{-{n^{\Omega(1)}}}$。在更简单的平均情况非可延展性设定下，我们实现了速率接近$1/11$的高效非可延展编码。