A client device or other processing device separates a file into blocks and distributes the blocks across multiple servers for storage. In one aspect, subsets of the blocks are allocated to respective primary servers, a code of a first type is applied to the subsets of the blocks to expand the subsets by generating one or more additional blocks for each subset, and the expanded subsets of the blocks are stored on the respective primary servers. A code of a second type is applied to groups of blocks each including one block from each of the expanded subsets to expand the groups by generating one or more additional blocks for each group, and the one or more additional blocks for each expanded group are stored on respective secondary servers. The first and second codes are advantageously configured to provide security against an adversary that is able to corrupt all of the servers over multiple periods of time but fewer than all of the servers within any particular one of the periods of time.