The principle of lattice cryptography is that a given single geometric structure can be described by two vastly different sets of vectors.
The Private Key (red vectors) is the secret, simple set of vectors. The vectors are short and nearly at right angles, making it easy to find any point on the grid.
The Public Key (blue vectors) is a deliberately complex set of vectors on the exact same grid.
The security of this system comes when you give someone the Public Key or bad basis vectors and ask from this what is the Private Key? Because bad basis vectors are long and skewed, it's extremely difficult to derive what simple set of vectors was used as the private key on the same lattice.
Higher Dimensions
We can only visualize lattices in 2D or 3D, but the cryptographic strength found in standards like CRYSTALS-Kyber emerges in hundreds of dimensions. The security relies on a trapdoor where it's easy to create a complex public key from a vvery simple private key, but computationally impossible for both logical and quantum systems to reverse the process to find the secret.
The math behind this is a simple unimodular transformation, applied thousands of times:
v'Ȉ_i = vȈ_i + k ⋅ vȈ_j
Let's break down this equation:
v'Ȉ_i: The resulting vector that becomes part of the bad basis or public key.
vȈ_i: An existing vector being modified.
vȈ_j: A random vector from the good basis (where i ≠ j).
k: A small, non-zero random number.
The sevveral applications of this rule obscure out original structure and after thousands of iterations, the simple private key is transformed into a complex public key that appears random and is unreasonably complex to try and reverse. While the security of the entire system doesn't solely rely on the computational difficulty of reversing this process it's one of the main protections against quantum computing.