We introduce a new class of random Gottesman-Kitaev-Preskill (GKP) codes
derived from the cryptanalysis of the so-called NTRU cryptosystem. The derived
codes are good in that they exhibit constant rate and average distance scaling
Ξβnβ with high probability, where n is the number of
bosonic modes, which is a distance scaling equivalent to that of a GKP code
obtained by concatenating single mode GKP codes into a qubit-quantum error
correcting code with linear distance. The derived class of NTRU-GKP codes has
the additional property that decoding for a stochastic displacement noise model
is equivalent to decrypting the NTRU cryptosystem, such that every random
instance of the code naturally comes with an efficient decoder. This
construction highlights how the GKP code bridges aspects of classical error
correction, quantum error correction as well as post-quantum cryptography. We
underscore this connection by discussing the computational hardness of decoding
GKP codes and propose, as a new application, a simple public key quantum
communication protocol with security inherited from the NTRU cryptosystem.Comment: 23 pages, 10 figures, comments welcome! Version 2 has minor
correction