We introduce a new information-theoretic measure, which we call Public Information Complexity (PIC), as a tool for the study of multi-party computation protocols, and of quantities such as their communication complexity, or the amount of randomness they require in the context of information-theoretic private computations. We are able to use this measure directly in the natural asynchronous message-passing peer-to-peer model and show a number of interesting properties and applications of our new notion: The Public Information Complexity is a lower bound on the Communication Complexity and an upper bound on the Information Complexity; the difference between the Public Information Complexity and the Information Complexity provides a lower bound on the amount of randomness used in a protocol; any communication protocol can be compressed to its Public Information Cost; and an explicit calculation of the zero-error Public Information Complexity of the k -party, n -bit Parity function, where a player outputs the bitwise parity of the inputs. The latter result also establishes that the amount of randomness needed by a private protocol that computes this function is Ω ( n ).
We introduce a new information-theoretic measure, which we call Public Information Complexity (PIC), as a tool for the study of multi-party computation protocols, and of quantities such as their communication complexity, or the amount of randomness they require in the context of information-theoretic private computations. We are able to use this measure directly in the natural asynchronous message-passing peer-to-peer model and show a number of interesting properties and applications of our new notion: The Public Information Complexity is a lower bound on the Communication Complexity and an upper bound on the Information Complexity; the difference between the Public Information Complexity and the Information Complexity provides a lower bound on the amount of randomness used in a protocol; any communication protocol can be compressed to its Public Information Cost; and an explicit calculation of the zero-error Public Information Complexity of the k -party, n -bit Parity function, where a player outputs the bitwise parity of the inputs. The latter result also establishes that the amount of randomness needed by a private protocol that computes this function is Ω ( n ).
In the field of information-theoretic cryptography, randomness complexity is a key metric for protocols for private computation, that is, the number of random bits needed to realize the protocol. Although some general bounds are known, even for the relatively simple example of 1-private computation of n-party AND, the exact complexity is unknown. We study two settings. First, we consider the model of Goyal, Ishai, and Song (Crypto '22) where helper parties without any inputs are allowed to assist in the computation. In this setting, we show that two random bits always suffice to compute an arbitrary Boolean circuit C 1-privately: a single designated inputless helper flips the two bits and privately distributes the derived one-time bits to the other helper parties and the input parties as they are needed. We give an explicit construction using seven helper parties per AND gate and three helper parties per XOR gate (plus the single global randomness dealer). Moreover, two random bits are necessary already for the AND functionality (by a reduction to the standard no-helper model together with the lower bound of Kushilevitz, Ostrovsky, Prouff, Rosén, Thillard and Vergnaud (TCC '19), and therefore the worst-case helper-party randomness complexity is exactly 2 bits. Second, in the setting without helper parties, we improve the upper bound from Couteau and Rosén (Asiacrypt '22) on the (asymptotic) randomness complexity of n-party AND from 6 to 5 bits. That is, we give a 1-private protocol for computing the AND of n parties' inputs requiring 5 bits of randomness, for all n ≥ 6. Our construction, like that of Couteau and Rosén, uses a single party to flip the 5 bits and distribute the required derived values during the execution. Our approach to both problems is built around a more systematic exploration of techniques for recycling randomness across sub-computations. As part of resolving the second problem, we isolate an exact local-independence combinatorial object called a Sliding-Window Independence Generator, or a SWIG. A (k,m)-SWIG is a linear generator from a k-bit seed to m ≥ k output bits, where every cyclic length-k sliding window chosen from m output bits is perfectly uniform. We give an explicit (k,m)-SWIG for every k ≥ 1 and every m ≥ k and use a (5,n-1)-SWIG in our no-helper AND protocol.
In secure multiparty computation (MPC), mutually distrusting users collaborate to compute a function of their private data without revealing any additional information about their data to the other users. While it is known that information theoretically secure MPC is possible among n users having access to private randomness and are pairwise connected by secure, noiseless, and bidirectional links against the collusion of less than n/2 users (in the honest-but-curious model; the threshold is n/3 in the malicious model), relatively little is known about the communication and randomness complexity of secure computation, i.e., the amount of communication and randomness required to compute securely. In this paper, we employ information theoretic techniques to obtain lower bounds on communication and randomness complexity of secure MPC. We restrict ourselves to a concrete interactive setting involving three users under which all functions are securely computable against corruption of individual users in the honest-but-curious model. We derive lower bounds for both the perfect security case (i.e., zero-error and no leakage of information) and asymptotic security (where the probability of error and information leakage vanish as block-length goes to ∞). Our techniques include the use of a data processing inequality for residual information (i.e., the gap between mutual information and Gács-Körner common information), a new information inequality for three-user protocols, and the idea of distribution switching by which lower bounds computed under certain worst case scenarios can be shown to apply for the general case. Our lower bounds are shown to be tight for various functions of interest. In particular, we show concrete functions which have communication-ideal protocols, i.e., which achieve the minimum communication simultaneously on all links in the network. Also, we obtain the first explicit example of a function that incurs a higher communication cost than the input length, in the secure computation model of Feige et al. (26th Annual ACM Symposium on Theory of Computing, 1994), who had shown that such functions exist. We also show that our communication bounds imply tight lower bounds on the amount of randomness required by MPC protocols for many interesting functions.
Each of two parties $P_\mathcal{X} $ and $P_\matcal{Y} $ holds an n-bit input, x and y, respectively. They wish to privately compute the value of f(x,y). That is, $P_\mathcal{X} $ should not learn any additional information about y (in the information-theoretic sense) other than what follows from its input x and the function value f(x,y), and similarly, $P_\mathcal{Y} $ should not learn any additional information about x. In this paper, the two following basic questions in the theory of private computations are considered: 1. Which functions can be privately computed? 2. What is the communication complexity of protocols that privately compute a function f (in the case that such protocols exist)? A complete combinatorial characterization of privately computable functions is given. This characterization is used to derive tight bounds on the rounds complexity of any privately computable function and to design optimal private protocols that compute these functions. It is shown that for every 1≤g(n)≤2⋅(2n−1) there are functions that can be privately computed with g(n)-rounds of communication, but not with (g(n)−1)-rounds of communication. This implies that the communication costs of private protocols can be exponentially higher than the communication costs of nonprivate protocols. Interestingly, randomization helps neither to increase the set of privately computable functions, nor to improve the rounds complexity of these functions.
Information complexity is the extension of classical information theory to the interactive setting, where instead of one-way transmission we are interested in back-and-forth communication. This approach has been very influential in communication complexity, where it enables us to prove powerful lower bounds by quantifying the amount of information the participants in the computation must reveal about their inputs. In this paper we study information complexity in the classical broadcast model: k parties with private inputs wish to compute some function of their inputs, and they communicate by sending messages (one at a time) over a broadcast channel. We measure how much information the players reveal about their inputs to an external observer. This is called external information cost.
Communication complexity investigates the amount of communication needed for two or more players to determine some joint function of their private inputs. For many interesting functions, the communication complexity can be much smaller than basic information theoretic measures associated with the players' inputs such as the input length, the entropy, or even the conditional entropy. Communication complexity of many functions reduces further when the players share randomness. Classical works studied the communication complexity of functions when the interacting players share randomness perfectly, i.e., they get identical copies of randomness from a common source. This paper considers the variant of this question when the players share randomness imperfectly, i.e., when they get noisy copies of the randomness produced by some common source. Our main result shows that any function that can be computed by a k-bit protocol in the perfect sharing model has a 2k-bit protocol in the setting of imperfectly shared randomness and such an exponential growth is necessary. Our upper bound relies on ideas from locality sensitive hashing, while lower bounds rely on hypercontractivity and a new invariance principle tailored for communication protocols.
We prove two classes of lower bounds on the communication complexity of information-theoretically secure multiparty computation. The first lower bound applies to perfect passive secure multiparty computation in the standard model with n = 2t+1 parties of which t are corrupted. We show a lower bound that applies to secure evaluation of any function, assuming that each party can choose to learn or not learn the output. Specifically, we show that there is a function H^* such that for any protocol that evaluates y_i = b_i ⋅ f(x₁,...,x_n) with perfect passive security (where b_i is a private boolean input), the total communication must be at least 1/2 ∑_{i = 1}ⁿ H_f^*(x_i) bits of information. The second lower bound applies to the perfect maliciously secure setting with n = 3t+1 parties. We show that for any n and all large enough S, there exists a reactive functionality F_S taking an S-bit string as input (and with short output) such that any protocol implementing F_S with perfect malicious security must communicate Ω(nS) bits. Since the functionalities we study can be implemented with linear size circuits, the result can equivalently be stated as follows: for any n and all large enough g ∈ ℕ there exists a reactive functionality F_C doing computation specified by a Boolean circuit C with g gates, where any perfectly secure protocol implementing F_C must communicate Ω(n g) bits. The results easily extends to constructing similar functionalities defined over any fixed finite field. Using known techniques, we also show an upper bound that matches the lower bound up to a constant factor (existing upper bounds are a factor lg n off for Boolean circuits). Both results also extend to the case where the threshold t is suboptimal. Namely if n = kt+s the bound is weakened by a factor O(s), which corresponds to known optimizations via packed secret-sharing.