  Nonlinear cellular automata are extensively used in simulations, image
processing, cryptography, and so on. The determination of their fundamental
properties, injectivity and surjectivity, related to information loss during
the evolution, is necessary in various applications. Currently, people still
use Amoroso's algorithms for injectivity and surjectivity determinations, but
this incurs significant computational costs when applied to complex nonlinear
cellular automata. We have optimized Amoroso's surjectivity algorithm,
improving its operational efficiency greatly and extended its applicability to
various boundaries. Furthermore, we have introduced new theorems and algorithms
for determining injectivity, which offer substantial improvements over
Amoroso's algorithm in both time and space. With these new algorithms, we are
equipped to determine the properties of larger and more complex cellular
automata, thereby employing more advanced cellular automata to achieve
increasingly complex functionalities.
