The technique of replicating frequently-accessed files to other nodes has been widely used in a high-performance distributed system to reduce the load of the nodes hosting these files. Traditional file replication algorithms rely on the analysis of client-access logs to determine the location of the replicated nodes. In this paper, we present LessLog, a logless file replication algorithm, developed for a peer-to-peer distributed system. We first construct a lookup tree for each node. LessLog uses bitwise operations to determine the location of the replicated node without any client-access history. In addition, each replication is guaranteed to reduce the workload of the replicating node by half. A fault-tolerant LessLog model is also presented. The experimental results show that LessLog successfully and efficiently reduces the load of overloaded nodes.