In higher dimensions, such features are called hyperplanes. In the Ajtai-Dwork cryptosystem, a particular set of hidden hyperplanes constitutes the private key. The public key is a method of generating points that are guaranteed to be near one of those hyperplanes without revealing where any of the hyperplanes is. Even generating a large number of points isn't sufficient to unveil the hyperplanes they trace out. To send an encrypted message, the user provides the information as a string of 1s and 0s.
Software encrypts the message one bit at a time.
If the bit is 0, the computer uses the public key to find a point whose coordinates place it very near one of the hidden hyperplanes. If the bit is 1, the computer comes up with a random point somewhere in the high-dimensional space of the lattice. The private key allows the recipient to determine the distance of each point from the nearest hyperplane. If the distance is sufficiently small, the point is decrypted to represent 0. Otherwise, the point represents 1.