In anonymous broadcast, one or more parties want to anonymously send messages to all parties. This problem is increasingly important as a black-box in many privacy-preserving applications such as anonymous communication, distributed auctions, and multi-party computation. In this paper, we design decentralized protocols for anonymous broadcast that require each party to send (and compute) a polylogarithmic number of bits (and operations) per anonymous bit delivered with $O(\log n)$ rounds of communication. Our protocol is provably secure against traffic analysis, does not require any trusted party, and is completely load-balanced. The protocol tolerates up to $n/6$ statically-scheduled Byzantine parties that are controlled by a computationally unbounded adversary. Our main strategy for achieving scalability is to perform local communications (and computations) among a logarithmic number of parties. We provide simulation results to show that our protocol improves significantly over previous work. We finally show that using a common cryptographic tool in our protocol one can achieve practical results for anonymous broadcast.