“Non-Malleable Randomness Encoder” (NMRE) was introduced by Kanukurthi et al. (2018) as a useful cryptographic primitive helpful in the construction of non-malleable codes. To the best of our knowledge, their construction is not known to be quantum secure. We provide a construction of a first rate-$1/2$, 2-split, quantum secure NMRE and use this in a black-box manner, to construct the following: 1) rate$1/11$, 3-split, quantum non-malleable code; 2) rate$1/3$, 3-split, quantum secure non-malleable code; and 3) rate$1/5$, 2-split, average case quantum secure non-malleable code.

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 which are stored in physically distant servers, 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 classical messages. Furthermore, until recent work by Aggarwal, Boddu, and Jain (IEEE Trans. Inf. Theory 2024 & arXiv 2022), adversaries with quantum capabilities and 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, for some constant$c\gt 0$, 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$n^{c}$, and error$\varepsilon =2^{-{n^{c}}}$. In the easier setting of average-case non-malleability, we achieve efficient non-malleable coding with rate close to$1/11$.

Non-malleable codes introduced by Dziembowski, Pietrzak and Wichs [1] encode a classical messageSin a manner such that the tampered codeword either decodes to the original messageSor a message that is unrelated/independent ofS. Constructing non-malleable codes for various tampering function families has received significant attention in the recent years. We consider the well studied (2-part)split-statemodel, in which the messageSis encoded into two partsXandY, and the adversary is allowed to arbitrarily tamper with eachXandYindividually. Non-malleable codes in the split-state model have found applications in other important security notions likenon-malleable commitmentsandnon-malleable secret sharing. Thus, it is vital to understand if such non-malleable codes are secure against quantum adversaries. We consider the security of non-malleable codes in the split-state model when the adversary is allowed to make use of arbitrary entanglement to tamper the partsXandY. We construct explicit quantum secure non-malleable codes in the split-state model. Our construction of quantum secure non-malleable codes is based on the recent construction of quantum secure 2-source non-malleable extractorsby Boddu, Jain and Kapshikar [2]. • We extend the connection of Cheraghchi and Guruswami [3] between 2-source non-malleable extractors and non-malleable codes in the split-state model in the classical setting to the quantum setting, i.e. we show that explicit quantum secure 2-source non-malleable extractors in (k1,k2)-qpa-state framework of [2] give rise to explicit quantum secure non-malleable codes in the split-state model. • We construct the first quantum secure non-malleable code with efficient encoding and decoding procedures for message lengthm=nΩ(1), error ε = 2-nΩ(1)and codeword of size 2n. Prior to this work, it remained open to provide such quantum secure non-malleable code even for a single bit message in the split-state model. • We also study its natural extension when the tampering of the codeword is performedt-times. We construct quantum secure one-many non-malleable code with efficient encoding and decoding procedures fort=nΩ(1), message lengthm=nΩ(1), error ε = 2-nΩ(1)and codeword of size 2n. • As an application, we also construct the first quantum secure 2-out-of-2 non-malleable secret sharing scheme for message/secret lengthm=nΩ(1), error ε = 2-nΩ(1)and share of sizen.

Multi-source extractors are functions that extract uniform randomness from multiple (weak) sources of randomness. Quantum multi-source extractors were considered by Kasher and Kempe (2010) (for the quantum independent adversary and the quantum bounded storage adversary), Chung et al. (2014) (for the general entangled adversary) and Arnon-Friedman et al. (2016) (for the quantum Markov adversary). One of the main objectives of this work is to unify all the existing quantum multi-source adversary models. We propose two new models of adversaries: 1) the quantum measurement adversary ($\mathsf {qma}$), which generates side information using entanglement and on post-measurement; and 2) the quantum communication adversary ($\mathsf {qca}$), which generates side information using entanglement and communication between multiple sources. We show that: 1)$\mathsf {qma}$is the strongest adversary among all the known adversaries, in the sense that the side information of all other adversaries can be generated by$\mathsf {qma}$; 2) The (generalized) inner-product function (in fact a general class of two-wise independent functions) continues to work as a good extractor with matching parameters as that of Chor and Goldreich (1985) against classical adversaries; 3) A non-malleable extractor proposed by Li (2012) (against classical adversaries) continues to be secure against quantum side information. This result implies a non-malleable extractor result of Aggarwal et al. (2019) with uniform seed. We strengthen their result via a completely different proof to make the non-malleable extractor of Li secure against quantum side information even when the seed is not uniform; 4) A modification (working with weak local randomness instead of uniform local randomness) of the Dodis and Wichs (2009) protocol for privacy-amplification is secure against active quantum adversaries (those who arbitrarily modify the messages exchanged in the protocol). This strengthens on a recent result due to Aggarwal et al. (2019) which uses uniform local randomness; 5) A tight efficiency lower bound for the (generalized) inner-product function (in fact a general class of two-wise independent functions).

In a recent breakthrough result, Chattopadhyay, Mande and Sherif [ECCC TR18-17] showed an exponential separation between the log approximate rank and randomized communication complexity of a total function f, hence refuting the log approximate rank conjecture of Lee and Shraibman [2009]. We provide an alternate proof of their randomized communication complexity lower bound using the information complexity approach. Using the intuition developed there, we derive a polynomially-related quantum communication complexity lower bound using the quantum information complexity approach, thus providing an exponential separation between the log approximate rank and quantum communication complexity of f. Previously, the best known separation between these two measures was (almost) quadratic, due to Anshu, Ben-David, Garg, Jain, Kothari and Lee [CCC, 2017]. This settles one of the main question left open by Chattopadhyay, Mande and Sherif, and refutes the quantum log approximate rank conjecture of Lee and Shraibman [2009]. Along the way, we develop a Shearer-type protocol embedding for product input distributions that might be of independent interest.