Abstract
Cryptography and channel coding rely both on the same mathematical tools, namely codes and lattices, but with different objectives and different criteria. Our aim in this talk is to present major results from coding theory on the construction of high-dimensional lattices and their decoding. We start by a quick comparison of the design criteria in channel coding versus cryptography. We recall the principal algebraic constructions of lattices from codes. Then we focus on LDA/GLD lattices built from p-ary codes on graphs. Iterative probabilistic decoding of these lattices is described and results are shown in dimension n as large as 10^6 with a very small gap to theoretical limits. LDA lattices are proven to achieve Shannon capacity under MMSE lattice decoding, with an alphabet size p superlinear in n. Finally, a new proof is given for the Gaussian goodness of GLD lattices and the asymptotic goodness of the underlying codes, where the alphabet size is polylogarithmic in n. We believe there exists a great potential for LDA/GLD lattices and their iterative decoding in building secure cryptosystems after tuning their parameters to fit cryptography constraints.