Recently, Peer-to-Peer (P2P) technology has become important in designing (desktop) grids for large-scale distributed computing over the Internet. We present a middleware for distributed computing based on Peer-to-Peer systems. When combining public-resource computation ideas with concepts of P2P networks, new challenges occur due to the lack of global knowledge as there is no central administration possible. Our Peer-to-Peer desktop grid (P2P Grid) framework includes an efficient and fault-tolerant communication scheme for job distribution combining epidemic algorithms with chord-style multicasts. We show that this hybrid scheme is more efficient than both epidemic algorithms and chord-style multicasting alone.