In secure summation, $K$ parties each hold an input, and the sum of the inputs is computed at one party, securely against at most $T$ colluding parties. Communication typically takes one hop: each party sends one message directly to the party that computes the sum, using correlated randomness that is independent of the inputs and held in advance. We consider interactive secure summation, where every pair of parties is connected by a secure link, so that each party can send messages to every other party, each party has only independent local randomness, and the messages may depend on the inputs and on earlier messages over multiple rounds. We characterize the capacity region, the set of achievable rates of communication, counting all messages exchanged among the parties over all rounds, and of the randomness of every party, counted in symbols per symbol of the sum. For $0\le T\le K-2$, the optimal communication rate is $K(K-1)/(K-T)$, and the randomness rates form a polyhedron with, up to a relabeling of the parties, a single extreme point: $T$ parties have the randomness rates $(K+T-2t)/(K-T)$, $t=1,\ldots,T$, and the others none. In particular, the optimal total randomness rate is $T(K-1)/(K-T)$. Linear schemes achieve every extreme point together with the optimal communication rate. The same techniques give the capacity region when every party recovers the sum and when the sum is computed at a server without input.
In secure summation, K parties each hold an input, and the sum of the inputs is computed at one party, securely against at most T colluding parties. Communication typically takes one hop: each party sends one message directly to the party that computes the sum, using correlated randomness that is independent of the inputs and held in advance. We consider interactive secure summation, where every pair of parties is connected by a secure link, so that each party can send messages to every other party, each party has only independent local randomness, and the messages may depend on the inputs and on earlier messages over multiple rounds. We characterize the capacity region, the set of achievable rates of communication, counting all messages exchanged among the parties over all rounds, and of the randomness of every party, counted in symbols per symbol of the sum. For 0≤T≤K−2, the optimal communication rate is K(K−1)/(K−T), and the randomness rates form a polyhedron with, up to a relabeling of the parties, a single extreme point: T parties have the randomness rates (K+T−2t)/(K−T), t=1,…,T, and the others none. In particular, the optimal total randomness rate is T(K−1)/(K−T). Linear schemes achieve every extreme point together with the optimal communication rate. The same techniques give the capacity region when every party recovers the sum and when the sum is computed at a server without input.
In secure summation,Kusers, each holds an input, wish to compute the sum of the inputs at a server without revealing any information aboutall the inputseven if the server may collude withan arbitrary subset of users. In this work, we relax the security and colluding constraints, where the set of inputs whose information is prohibited from leakage is from a predetermined collection of sets (e.g., any set of up toSinputs) and the set of colluding users is from another predetermined collection of sets (e.g., any set of up toTusers). For arbitrary collection of security input sets and colluding user sets, we characterize the optimal randomness assumption, i.e., the minimum number of key bits that need to be held by the users, per input bit, for weakly secure summation to be feasible, which generally involves solving a linear program.
The problem of entanglement-assisted summation over a quantum multiple access channel (Σ-QMAC) is intro-duced, involvingSservers,Kclassical(Fd)data streams that are replicated arbitrarily across various subsets of servers, and a receiver who wishes to compute the sum of theKdata streams. Independent of the data, entangled quantum systemsQ1,Q2,⋯,QSare prepared in advance and distributed to the corresponding servers. Each servers,s∈[S]locally manipulates its quantum systemQsaccording to its classical data and sendsQsto the receiver. The total communication cost islogd∣Q1∣+logd∣Q2∣+⋯+logd∣QS∣qudits, where∣Qsdenotes the dimension ofQs. Based on a measurement of the composite systemQ1Q2⋯QS, the receiver must recover the desired sum. The rate thus achieved is defined as the number of dits(Fdsymbols) of the desired sum computed by the receiver per qudit (d-dimsional quantum system) of download. The capacityCis the supremum of the set of all achievable rates. As the main result of this work, the precise capacity ofΣ-QMAC is obtained, from which it follows that quantum entanglements allow a factor of 2 gain in capacity (superdense coding gain) relative to capacity with no entanglements, in all cases (anyS,K,Fdand any data replication pattern) provided that the entanglement-assisted capacity does not exceed 1 dit/qudit (Holevo bound). Coding schemes based on a recentN-sum box abstraction are sufficient to achieve capacity.
Secure multiparty computation refers to the problem where a number of users wish to securely compute a function on their inputs without revealing any unnecessary information. This dissertation focuses on the fundamental limits of secure summation under different constraints. We first focus on the minimal model of secure computation, in which two users each hold an input and wish to securely compute a function of their inputs at the server. We propose a novel scheme base on the algebraic structure of finite field and modulo ring of integers. Then we extend the minimal model of secure computation, in which K users wish to securely compute the sum of their inputs at the server. We prove a folklore result on the limits of communication cost and randomness cost. Then we characterized the optimal communication cost with user dropouts constraint, when some users may lose connection to the server and the server wishes to compute the sum of remaining inputs. Next, we characterize the optimal communication and randomness cost for symmetric groupwise keys and find the feasibility condition for arbitrary groupwise keys. Last, we study the secure summation with user selection, such that the server may select any subset of users to compute the sum of their inputs. This leads us to the MDS variable generation problem. We characterize the optimal individual key rate and the result is interestingly the harmonic number.
Direct-sum questions in (two-party) communication complexity ask whether two parties, Alice and Bob, can compute the value of a function f on l inputs (x_1,y_1),...,(x_l,y_l) more efficiently than by applying the best protocol for f, independently on each input (x_i,y_i). In spite of significant efforts to understand these questions (under various communication-complexity measures), the general question is still far from being well understood. In this paper, we offer a multiparty view of these questions: The direct-sum setting is just a two-player system with Alice having inputs x_1,...,x_l, Bob having inputs y_1,...,y_l and the desired output is f(x_1,y_1),...,f(x_l,y_l). The naive solution of solving the l problems independently, is modeled by a network with l (disconnected) pairs of players Alice i and Bob i, with inputs x_i,y_i respectively, and communication only within each pair. Then, we consider an intermediate ("star") model, where there is one Alice having l inputs x_1,...,x_l and l players Bob_1,...,Bob_l holding y_1,...,y_l, respectively (in fact, we consider few variants of this intermediate model, depending on whether communication between each Bob i and Alice is point-to-point or whether we allow broadcast). Our goal is to get a better understanding of the relation between the two extreme models (i.e., of the two-party direct-sum question). If, for instance, Alice and Bob can do better (for some complexity measure) than solving the l problems independently, we wish to understand what intermediate model already allows to do so (hereby understanding the "source" of such savings). If, on the other hand, we wish to prove that there is no better solution than solving the l problems independently, then our approach gives a way of breaking the task of proving such a statement into few (hopefully, easier) steps. We present several results of both types. Namely, for certain complexity measures, communication problems f and certain pairs of models, we can show gaps between the complexity of solving f on l instances in the two models in question; while, for certain other complexity measures and pairs of models, we can show that such gaps do not exist (for any communication problem f). For example, we prove that if only point-to-point communication is allowed in the intermediate "star" model, then significant savings are impossible in the public-coin randomized setting. On the other hand, in the private-coin randomized setting, if Alice is allowed to broadcast messages to all Bobs in the "star" network, then some savings are possible. While this approach does not lead yet to new results on the original two-party direct-sum question, we believe that our work gives new insights on the already-known direct-sum results, and may potentially lead to more such results in the future.
We consider secure computation of randomized functions by two users, where both the users (Alice and Bob) have inputs, Alice sends a message to Bob over a rate-limited, noise-free link, and then Bob produces the output. We study this problem when privacy is required only against Bob, i.e., from the message, Bob must not learn any information about Alice's input other than what can be inferred by his own input and output. We give a single-letter expression for the optimal rate. We also explicitly characterize securely computable randomized functions when input has full support, which leads to a much simpler expression for the optimal rate. Recently, Data (ISIT 2016) studied the other two cases (first, when privacy is required against both the users; and second, when privacy is required only against Alice) and obtained single-letter expressions for optimal rates in both the scenarios. Yassaee, Gohari, and Aref (IEEE Transactions on Information Theory 2015) studied the case when there is no privacy requirement and obtained a single-letter expression for the optimal rate, when Alice and Bob interact for arbitrary but finite number of rounds, and both of them may produce potentially different outputs.
We consider the communication scenario where K transmitters are each connected to a common receiver with an orthogonal noiseless link. One of the transmitters has a message for the receiver, who is prohibited from learning anything in the information theoretic sense about which transmitter sends the message (transmitter anonymity is guaranteed). The capacity of anonymous communications is the maximum number of bits of desired information that can be anonymously communicated per bit of total communication. For this anonymous communication problem over a parallel channel with K transmitters and one receiver, we show that the capacity is 1/K, i.e., to communicate 1 bit anonymously, each transmitter must send a 1 bit signal. Furthermore, it is required that each transmitter has at least 1 bit correlated randomness (that is independent of the messages and is not available to the receiver) per message bit and the size of correlated randomness at all K transmitters is at least K - 1 bits per message bit.