The symmetry of 1's complement arithmetic also means that after modifying a FITS file, the checksum may be incrementally updated using simple arithmetic without accumulating the checksum for portions of the file that have not changed.
An incremental update provides the mechanism for end-to-end checksum verification through any number of intermediate processing steps. By calculating rather than accumulating the intermediate checksums, the original checksum test is propagated through to the final data file. On the other hand, if a new checksum is accumulated with each change to the file, no information is preserved about the file's original state.
The recipe for updating the CHECKSUM keyword following some change to the file is: C' = C - m + m', where C and C' represent the file's checksum (that is, the complement of the CHECKSUM keyword) before and after the modification and m and m' are the corresponding checksums for the modified FITS records or keywords only. Since the CHECKSUM keyword contains the complement of the checksum, the correspondingly complemented form of the recipe is more directly useful: , where (tilde) denotes the (1's) complement operation. (See ref. 12-14.) Note that the tilde on the right hand side of the equation cannot be distributed over the contents of the parentheses due to the dual nature of zero in 1's complement arithmetic (ref. 14).