Erasure encoded fragments are generated by an erasure encoding scheme, represented by an erasure encoding matrix, operating on a data file. A new erasure encoded fragment may be generated from previously-generated erasure encoded fragments without reconstructing the original data file. Available and valid erasure encoded fragments are identified and a set of those fragments is selected. A composite encoding matrix is generated based upon the selected fragments and the fragment specified to be generated. The composite matrix is applied to the selected fragments to produce a plurality of partial sums. The partial sums are then combined to generate the specified fragment. The partial sums may be produced by different devices so as to distribute the computational workload and/or to reduce network traffic. The integrity of a generated fragment may be verified by generating the specified fragment twice, using two different sets of fragments, and then comparing the two results.