We show that some widely accepted criteria for cryptographic functions, including the strict avalanche criterion (SAC) and the propagation criterion, have various limitations in capturing properties of vital importance to cryptographic algorithms, and propose a new criterion called GAC to measure the global avalanche characteristics of cryptographic functions. We also introduce two indicators related to the new criterion, one forecasts the sum-of-squares while the other the absolute avalanche characterist- ics of a function. Lower and upper bounds on the two indicators are derived, and two methods are presented to construct cryptographic functions that achieve nearly optimal global avalanche characteristics.