When NIST chose the algorithms meant to protect the internet from quantum computers, most of the winners came from the same unexpected place: geometry. Not the geometry of triangles, but of infinite grids of points in high-dimensional space, where one innocent-sounding question — what's the nearest grid point to here? — turns out to be brutally hard. That hardness is the new foundation of secure communication.
Today's public-key cryptography rests on two problems: factoring large numbers (RSA) and the discrete logarithm (elliptic-curve and classic Diffie-Hellman systems). Both are hard for ordinary computers. Both are easy for a sufficiently large quantum computer running Shor's algorithm. That's the entire reason post-quantum cryptography exists — and lattices are why we have an answer.
What is a lattice?
Start simple. Take two vectors in a plane and consider every point you can reach by adding integer multiples of them: zero of the first plus three of the second, minus-two of the first plus one of the second, and so on. The set of all those reachable points is a lattice — a regular, infinitely repeating grid. The two vectors you started with are a basis.
Now scale that picture up: instead of two vectors in a plane, use hundreds or a thousand vectors in a space of hundreds or a thousand dimensions. The lattice is still "all integer combinations of the basis vectors," but our visual intuition collapses completely. And that collapse is exactly where the cryptography lives.
