Sublinear Communication Complexity Research Articles

Adaptively Secure MPC with Sublinear Communication Complexity

Adaptively Secure MPC with Sublinear Communication Complexity

Communication Complexity and Graph Families

Given a graph G and a pair ( F 1 , F 2 ) of graph families, the function GDISJ G , F 1 , F 2 takes as input, two induced subgraphs G 1 and G 2 of G , such that G 1 ∈ F 1 and G 2 ∈ F 2 and returns 1 if V ( G 1 )∩ V ( G 2 )=∅ and 0 otherwise. We study the communication complexity of this problem in the two-party model. In particular, we look at pairs of hereditary graph families. We show that the communication complexity of this function, when the two graph families are hereditary, is sublinear if and only if there are finitely many graphs in the intersection of these two families. Then, using concepts from parameterized complexity, we obtain nuanced upper bounds on the communication complexity of GDISJ G , F 1 , F 2 . A concept related to communication protocols is that of a ( F 1 , F 2 )-separating family of a graph G . A collection F of subsets of V ( G ) is called a ( F 1 , F 2 )- separating family for G , if for any two vertex disjoint induced subgraphs G 1 ∈ F 1 , G 2 ∈ F 2 , there is a set F ∈ F with V ( G 1 ) ⊆ F and V ( G 2 ) ∩ F = ∅. Given a graph G on n vertices, for any pair ( F 1 , F 2 ) of hereditary graph families with sublinear communication complexity for GDISJ G , F 1 , F 2 , we give an enumeration algorithm that finds a subexponential sized ( F 1 , F 2 )-separating family. In fact, we give an enumeration algorithm that finds a 2 o ( k ) n O (1) sized ( F 1 , F 2 )-separating family, where k denotes the size of a minimum sized set S of vertices such that V ( G )\ S has a bipartition ( V 1 , V 2 ) with G [ V 1 ] ∈ F 1 and G [ V 2 ]∈ F 2 . We exhibit a wide range of applications for these separating families, to obtain combinatorial bounds, enumeration algorithms, as well as exact and FPT algorithms for several problems.

Quantum Private Information Retrieval has Linear Communication Complexity

In Private Information Retrieval (PIR), a client queries an n-bit database in order to retrieve an entry of her choice, while maintaining privacy of her query value. Chor, Goldreich, Kushilevitz, and Sudan showed that, in the information-theoretical setting, a linear amount of communication is required for classical PIR protocols (and thus that the trivial protocol is optimal). This linear lower bound was shown by Nayak to hold also in the quantum setting. Here, we extend Nayak's result by considering approximate privacy, and requiring security only against "specious" adversaries, which are, in analogy to classical honest-but-curious adversaries, the weakest reasonable quantum adversaries. We show that, even in this weakened scenario, Quantum Private Information Retrieval (QPIR) requires n qubits of communication. From this follows that Le Gall's recent QPIR protocol with sublinear communication complexity is not information-theoretically private, against the weakest reasonable cryptographic adversary.

Protecting Data Privacy in Private Information Retrieval Schemes

Private information retrieval (PIR) schemes allow a user to retrieve the ith bit of an n-bit data string x, replicated in k⩾2 databases (in the information-theoretic setting) or in k⩾1 databases (in the computational setting), while keeping the value of i private. The main cost measure for such a scheme is its communication complexity. In this paper we introduce a model of symmetrically-private information retrieval (SPIR), where the privacy of the data, as well as the privacy of the user, is guaranteed. That is, in every invocation of a SPIR protocol, the user learns only a single physical bit of x and no other information about the data. Previously known PIR schemes severely fail to meet this goal. We show how to transform PIR schemes into SPIR schemes (with information-theoretic privacy), paying a constant factor in communication complexity. To this end, we introduce and utilize a new cryptographic primitive, called conditional disclosure of secrets, which we believe may be a useful building block for the design of other cryptographic protocols. In particular, we get a k-database SPIR scheme of complexity O(n1/(2k−1)) for every constant k⩾2 and an O(logn)-database SPIR scheme of complexity O(log2n·loglogn). All our schemes require only a single round of interaction, and are resilient to any dishonest behavior of the user. These results also yield the first implementation of a distributed version of (n1)-OT (1-out-of-n oblivious transfer) with information-theoretic security and sublinear communication complexity.