Lattice Cryptography

Visualisation of the lattice method of cryptography. Green vectors represent the "good basis", used as the private key. Yellow points represent the lattice of integer multiples of any basis of this vector space. Red vectors represent the "bad basis", used as the public key, and selected as a particular (secret) integer multiple of the good basis. Encryption exploits the computational difficulty of the "closest vector problem" for a bad basis of a given vector space, which is believed difficult for both classical and quantum computers. The magenta point represents a position not in the lattice to which the cyan vector is the closest point in the lattice to it. The red and green ghost vectors represent the closest lattice point found to the magenta point after 1,000,000 brute-force iterations for each basis (if the encryption is to have any value, no better approach can exist for the bad-basis). The message can be encoded in such magenta positions, encrypted with the bad basis and decrypted with the good basis.