The problem of graph-based X-secure T-private linear computation (GXSTPLC) is to allow a user to retrieve a linear combination of K messages from a set of N distributed servers that store the messages in a graph-based fashion, i.e., each message is restricted to be distributed among a subset of servers. T-privacy requires that the coefficients of the linear combination are not revealed to any group of up to T colluding servers, and X-security guarantees that any set of up to X colluding servers learns nothing about the messages. In this paper, we propose an achievability scheme for GXSTPLC that enables a storage–communication trade-off by exploiting non-replicated storage codes. Novel aspects of our achievability scheme include the usage of the idea of cross-subspace alignment null shaper that addresses various challenges posed by the graph-based storage structure. In addition, unlike previous works, our scheme allows a direct transformation into a quantum one to achieve a superdense coding gain by leveraging the idea of N-Sum Box abstraction of quantum “over-the-air” computing.
The problem of graph-based X-secure T-private linear computation (GXSTPLC) is to allow a user to retrieve a linear combination of K messages from a set of N distributed servers that store the messages in a graph-based fashion, i.e., each message is restricted to be distributed among a subset of servers. T-privacy requires that the coefficients of the linear combination are not revealed to any group of up to T colluding servers, and X-security guarantees that any set of up to X colluding servers learns nothing about the messages. In this paper, we propose an achievability scheme for GXSTPLC that enables a storage–communication trade-off by exploiting non-replicated storage codes. Novel aspects of our achievability scheme include the usage of the idea of cross-subspace alignment null shaper that addresses various challenges posed by the graph-based storage structure. In addition, unlike previous works, our scheme allows a direct transformation into a quantum one to achieve a superdense coding gain by leveraging the idea of N-Sum Box abstraction of quantum “over-the-air” computing.
A communication-efficient protocol is introduced over a many-to-one quantum network for Q-E-B-MDS-X-TPIR, i.e., quantum private information retrieval with MDS-X-secure storage andT-private queries. The protocol is resilient to any set of up toEunresponsive servers (erased servers or stragglers) and any set of up toBByzantine servers. The underlying coding scheme incorporates an enhanced version of a Cross Subspace Alignment (CSA) code, namely a Modified CSA (MCSA) code, into the framework of CSS codes. The error-correcting capabilities of CSS codes are leveraged to encode the dimensions that carry desired computation results from the MCSA code into the error space of the CSS code, while the undesired interference terms are aligned into the stabilized code space. The challenge is to do this efficiently while also correcting quantum erasures and Byzantine errors. The protocol achieves superdense coding gain over comparable classical baselines for Q-E-B-MDS-X-TPIR, recovers as special cases the state of art results for various other quantum PIR settings previously studied in the literature, and paves the way for applications in quantum coded distributed computation, where CSA code structures are important for communication efficiency, while security and resilience to stragglers and Byzantine servers are critical.
Building on recent constructions of Quantum Cross Subspace Alignment (QCSA) codes, this work develops a coding scheme for QEXSTPIR, i.e., classical private information retrieval with X -secure storage andT-private queries, over a quantum multiple access channel, that is resilient to any set of up toEerased servers (equivalently known as unresponsive servers, or stragglers). The scheme is accordingly labeled QECSA, with the ‘E,indicating resilience to erased servers. The novelty of QECSA lies in achieving efficient E-straggler resilience on top of existing QCSA codes that already achieve X-secure storage, T- private queries, and distributed superdense coding gains for communication efficient decoding. The QECSA code structure may be broadly useful for problems such as quantum coded secure distributed computation, where security, straggler resilience, and distributed superdense coding gains are simultaneously required.
Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an “N-sum box”, a black box generalization of a two-sum protocol of Song et al. with recent applications to N-server private information retrieval. The N-sum box has a communication cost of N qudits and classical output of a vector ofNq-ary digits linearly dependent (via anN×2Ntransfer matrix) on2Nclassical inputs distributed among N transmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers. Furthermore, we provide a sample application to Cross-Subspace Alignment (CSA) schemes to obtain efficient instances of Quantum Private Information Retrieval (QPIR) and Quantum Secure Distributed Batch Matrix Multiplication (QSDBMM). We first describe N-sum boxes based on maximal stabilizers and we then consider non-maximal-stabilizer-based constructions to obtain an instance of Quantum Symmetric Private Information Retrieval.
N-Sum Box: An Abstraction for Linear Computation over Many-to-one Quantum Networks
Linear computations over quantum many-to-one communication networks offer opportunities for communication cost improvements through schemes that exploit quantum entanglement among transmitters to achieve superdense coding gains, combined with classical techniques such as interference alignment. The problem becomes much more broadly accessible if suitable abstractions can be found for the underlying quantum functionality via classical black box models. This work formalizes such an abstraction in the form of an "N-sum box", a black box generalization of a two-sum protocol of Song \emph{et al.} with recent applications to N-server private information retrieval. The N-sum box has a communication cost of N qudits and classical output of a vector of Nq-ary digits linearly dependent (via an N×2N transfer matrix) on 2N classical inputs distributed among N transmitters. We characterize which transfer matrices are feasible by our construction, both with and without the possibility of additional locally invertible classical operations at the transmitters and receivers. Furthermore, we provide a sample application to Cross-Subspace Alignment (CSA) schemes to obtain efficient instances of Quantum Private Information Retrieval (QPIR) and Quantum Secure Distributed Batch Matrix Multiplication (QSDBMM). We first describe N-sum boxes based on maximal stabilizers and we then consider non-maximal-stabilizer-based constructions to obtain an instance of Quantum Symmetric Private Information Retrieval.
Cross-subspace alignment (CSA) codes are used in various private information retrieval (PIR) schemes (e.g., with secure storage) and in secure distributed batch matrix multiplication (SDBMM). Using a recently developedN-sum box abstraction of a quantum multiple-access channel (QMAC), we translate CSA schemes over classical multiple-access channels into efficient quantum CSA schemes over a QMAC, achieving maximal superdense coding gain. Because of theN-sum box abstraction, the underlying problem of coding to exploit quantum entanglements for CSA schemes, becomes conceptually equivalent to that of designing a channel matrix for a MIMO MAC subject to given structural constraints imposed by theN-sum box abstraction, such that the resulting MIMO MAC is able to implement the functionality of a CSA scheme (encoding/decoding) over-the-air. Applications include Quantum PIR with secure and MDS-coded storage, as well as Quantum SDBMM.
We consider both classical and quantum variations ofX-secure,E-eavesdropped andT-colluding symmetric private information retrieval (SPIR). This is the first work to study SPIR withX-security in classical or quantum variations. We first develop a scheme for classicalX-secure,E-eavesdropped andT-colluding SPIR (XSETSPIR) based on a modified version of cross subspace alignment (CSA), which achieves a rate ofR=1−NX+max(T,E). The modified scheme achieves the same rate as the scheme used forX-secure PIR with the extra benefit of symmetric privacy, i.e., user-privacy as well as database-privacy. Next, we extend this scheme to its quantum counterpart based on theN-sum box abstraction. This is the first work to consider the presence of eavesdroppers in quantum private information retrieval (QPIR). In the quantum variation, the eavesdroppers have better access to information over the quantum channel compared to the classical channel due to the over-the-air decodability. To that end, we develop two different schemes for quantumX-secure,E-eavesdropped andT-colluding SPIR (QXSETSPIR) with secure over-the-air decoding. The first scheme achieves the highest possible super-dense coding gain, i.e.,RQ=min{1,2(1−NX+max(T,E))}, which requires additional uploads from the user. The second scheme on the other hand requires no extra uploads. However, it does not achieve the super-dense coding gain in some cases based on the relation between the number of eavesdropped links and the number of interference terms. The second scheme is based on the idea that there exist some special entanglement states that can be used to hide the contents of the user-required messages from the eavesdroppers using the interference symbols.