Picture a huge grid of points in hundreds of dimensions. Finding the grid point closest to a target is extremely hard, even for quantum computers.
The idea: Picture a grid of points, tilted and skewed in hundreds of dimensions, described by a set of arrows. Starting near the grid, finding the closest grid point is easy with a "good" description (short arrows at nearly right angles) and hopeless with a bad one (long, nearly parallel arrows). Cryptography publishes a bad description and keeps a good one, or a small secret, as the trapdoor. Unlike factoring, quantum computers have no known trick for these grids.
Why it matters: Most of the new quantum-safe cryptography is built on this.
Good and bad descriptions of a grid
Both pictures show the same grid of points, described by two different pairs of arrows. To find the grid point nearest the orange target, a simple method is to measure the target in arrow-units and round. With the good arrows that lands on the nearest point. With the bad ones, rounding inside long, skinny cells lands far away. Real lattices have hundreds of dimensions, where no known method finds a good description from a bad one.
The details
On the map: best attack: BKZ lattice reduction
A lattice is a regular grid of points in a space with hundreds of dimensions. Finding a short vector in it (SVP), or the lattice point closest to a target (BDD), is believed hard even for quantum computers: the best attacks, based on BKZ lattice reduction[1, 2], take time exponential in the dimension.
LWE and SIS are provably at least as hard as worst-case lattice problems (Regev 2005, Ajtai 1996)[3, 4, 5], an unusually strong theoretical foundation. Most standardized post-quantum cryptography sits here.
C. P. Schnorr and M. Euchner. Lattice basis reduction: Improved practical algorithms and solving subset sum problems. Mathematical Programming, 1994. doi:10.1007/bf01581144
Yuanmi Chen and Phong Q. Nguyen. BKZ 2.0: Better Lattice Security Estimates. Advances in Cryptology – ASIACRYPT 2011. doi:10.1007/978-3-642-25385-0_1
Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, 2005. doi:10.1145/1060590.1060603
Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. Journal of the ACM, 2009. doi:10.1145/1568318.1568324
M. Ajtai. Generating hard instances of lattice problems (extended abstract). Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96, 1996. doi:10.1145/237814.237838