This short video introduces the concept of a lattice, why they are being considered as the basis for the next generation of public key cryptography, and a short walkthrough of a specific encryption algorithm. For a very thorough paper designed to be readable for undergraduates I highly recommend https://eprint.iacr.org/2015/938.pdf.
*One technical note: At 1:30 I claim that lattices are composed only of integers. This is not true. Some lattices only contain integers, but in general, any rational number will do.
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