A distributed data storage system breaks data into n slices and k checksums using at least one matrix-based erasure code based on matrices with invertible submatrices, stores the slices and checksums on a plurality of storage elements, retrieves the slices from the storage elements, and, when slices have been lost or corrupted, retrieves the checksums from the storage elements and restores the data using the at least one matrix-based erasure code and the checksums. In a method for ensuring restoration and integrity of data in computer-related applications, data is broken into n pieces, k checksums are calculated using at least one matrix-based erasure code based on matrices with invertible submatrices, and the n data pieces and k checksums are stored on n+k storage elements or transmitted over a network. If, upon retrieving the n pieces from the storage elements or network, pieces have been lost or corrupted, the checksums are retrieved and the data is restored using the matrix-based erasure code and the checksums.