A sum of the digits in a given transmission modulo some number. The simplest form of checksum is a parity bit appended on to 7-bit numbers (e.g., ASCII characters) such that the total number of 1s is always Even (``even parity'') or Odd (``odd parity''). A significantly more sophisticated checksum is the Cyclic Redundancy Check (or CRC), which is based on the algebra of polynomials over the integers (mod 2). It is substantially more reliable in detecting transmission errors, and is one common error-checking protocol used in modems.
See also Cyclic Redundancy Check, Error-Correcting Code
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Cyclic Redundancy and Other Checksums.''
Ch. 20.3 in
Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 888-895, 1992.