In the past few years, successive improvements of the asymptotic complexity of square matrix multiplication have been obtained by developing novel methods to analyze the powers of the Coppersmith-Winograd tensor, a basic construction introduced thirty years ago. In this paper we show how to generalize this approach to make progress on the complexity of rectangular matrix multiplication as well, by developing a framework to analyze powers of tensors in an asymmetric way. By applying this methodology to the fourth power of the Coppersmith-Winograd tensor, we succeed in improving the complexity of rectangular matrix multiplication.Let α denote the maximum value such that the product of an n × n α matrix by an n α × n matrix can be computed with O(n 2+ ) arithmetic operations for any > 0. By analyzing the fourth power of the Coppersmith-Winograd tensor using our methods, we obtain the new lower bound α > 0.31389, which improves the previous lower bound α > 0.30298 obtained by Le Gall (FOCS'12) from the analysis of the second power of the Coppersmith-Winograd tensor. More generally, we give faster algorithms computing the product of an n × n k matrix by an n k × n matrix for any value k = 1. (In the case k = 1, we recover the bounds recently obtained for square matrix multiplication).These improvements immediately lead to improvements in the complexity of a multitude of fundamental problems for which the bottleneck is rectangular matrix multiplication, such as computing the all-pair shortest paths in directed graphs with bounded weights.

We introduce a new information theoretic measure that 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 messagepassing 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; an explicit calculation of the zero-error Public Information Complexity of the k-party, n-bit Parity function, where a player outputs the bit-wise 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).Our main goal is to introduce novel information-theoretical measures for the study of number-in-hand, message-passing multi-party protocols, coupled with a natural model that, among other things, allows private protocols (which is not the case for, e.g., the coordinator model).We define the new measure of Public Information Complexity (PIC), as a tool for the study of multiparty 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. Intuitively, our new measure captures a combination of the amount of information about the inputs that the players leak to other players, and the amount of randomness that the protocol uses. By proving lower bounds on PIC for a given multi-party function f , we are able to give lower bounds on the multi-party communication complexity of f and on the amount of randomness needed to privately compute f . The crucial point is that the PIC of functions, in our multi-party model, is not always 0, unlike their IC.Our new measure works in a model which is a slight restriction of the most general asynchronous model, where, for a given player at a given time, the set of players from which that player waits for a message can be determined by that player's own local view. This allows us to have the property that for any protocol, the information which is leaked during the execution of the protocol is at most the communication cost of the protocol. Note that in the multi-party case, the information cost of a protocol may be higher than its communication cost, because the identity of the player from which one receives a message might carry some information. We are able to define our measure and use it directly in a natural asynchronous peer-to-peer model (and not, e.g., in the coordinator model used in most works studying...

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We introduce new models and new information theoretic measures for the study of communication complexity in the natural peer-to-peer, multi-party, number-in-hand setting. We prove a number of properties of our new models and measures, and then, in order to exemplify their effectiveness, we use them to prove two lower bounds. The more elaborate one is a tight lower bound of Ω(kn) on the multi-party peer-to-peer randomized communication complexity of the k-player, n-bit function Disjointness, Disj n k . The other one is a tight lower bound of Ω(kn) on the multi-party peer-to-peer randomized communication complexity of the k-player, n-bit bitwise parity function, Par n k . Both lower bounds hold when n = Ω(k). The lower bound for Disj n k improves over the lower bound that can be inferred from the result of Braverman et al. (FOCS 2013), which was proved in the coordinator model and can yield a lower bound of Ω(kn/ log k) in the peer-to-peer model.To the best of our knowledge, our lower bounds are the first tight (non-trivial) lower bounds on communication complexity in the natural peer-to-peer multi-party setting.In addition to the above results for communication complexity, we also prove, using the same tools, an Ω(n) lower bound on the number of random bits necessary for the (information theoretic) private computation of the function Disj n k .

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