From Sphere Packings to Post-Quantum Cryptography

In 1611, Kepler made the following conjecture: In three dimensions, the density of the densest sphere packing is $\pi/\sqrt{18}$. Through the works of Newton, Gauss, Hilbert, Minkowski and others, sphere packings has been developed into an important mathematical discipline between number theory and geometry. In 1940s, the methods and results of sphere packings had been applied into information theory by Shannon, Hamming and others. Around 2000, lattice sphere packings surprisingly found applications in modern cryptography. In particular, Shor, Ajtai, Pipher and others applied it into post-quantum cryptography. In this talk, we will briefly introduce this dramatic development.