In the multiparty communication game (CFL-game) of Chandra, Furst, and Lipton (Proc. 15th ACM STOC, 1983, 94-99) k players collaboratively evaluate a function f (x 0 , . . . , x k−1 ) in which player i knows all inputs except x i . The players have unlimited computational power. The objective is to minimize communication.In this paper, we study the Simultaneous Messages (SM) model of multiparty communication complexity. The SM model is a restricted version of the CFL-game in which the players are not allowed to communicate with each other. Instead, each of the k players simultaneously sends a message to a referee, who sees none of the inputs. The referee then announces the function value.We prove lower and upper bounds on the SM-complexity of several classes of explicit functions. Our lower bounds extend to randomized SM complexity via an entropy argument. A lemma establishing a tradeoff between average Hamming distance and range size for transformations of the Boolean cube might be of independent interest.Our lower bounds on SM-complexity imply an exponential gap between the SM-model and the CFL-model for up to (log n) 1− players, for any > 0. This separation is obtained by comparing the respective complexities of the generalized addressing function, GAF G,k , where G is a group of order n. We also combine our lower bounds on SM complexity with ideas of Håstad and Goldmann (Computational Complexity 1 (1991), 113-129) to derive superpolynomial lower bounds for certain depth-2 circuits computing a function related to the GAF function.We prove some counter-intuitive upper bounds on SM-complexity. We show that GAF Z t 2 ,3has SM-complexity O(n 0.92 ). When the number of players is at least c log n, for some constant c > 0, our SM protocol for GAF Z t 2 ,k has polylog(n) complexity. We also examine a class of functions defined by certain depth-2 circuits. This class includes the "Generalized Inner Product" function and "Majority of Majorities." When the number of players is at least 2+log n, we obtain polylog(n) upper bounds for this class of functions.

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Abstract. Despite considerable research efforts, no efficient reduction from the discrete log problem to forging a discrete log based signature (e.g. Schnorr) is currently known. In fact, negative results are known. Paillier and Vergnaud [PV05] show that the forgeability of several discrete log based signatures cannot be equivalent to solving the discrete log problem in the standard model, assuming the so-called one-more discrete log assumption and algebraic reductions. They also show, under the same assumptions, that, any security reduction in the Random Oracle Model (ROM) from discrete log to forging a Schnorr signature must lose a factor of at least √ q h in the success probability. Here q h is the number of queries the forger makes to the random oracle. The best known positive result, due to Pointcheval and Stern [PS00], also in the ROM, gives a reduction that loses a factor of q h . In this paper, we improve the negative result from [PV05]. In particular, we show that any algebraic reduction in the ROM from discrete log to forging a Schnorr signature must lose a factor of at least q 2/3 h , assuming the one-more discrete log assumption. We also hint at certain circumstances (by way of restrictions on the forger) under which this lower bound may be tight. These negative results indicate that huge loss factors may be inevitable in reductions from discrete log to discrete log based signatures.

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