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 $\Delta \propto \sqrt{n}$ 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.

翻译：我们提出了一类新型随机Gottesman-Kitaev-Preskill（GKP）码，该码源自对所谓NTRU密码系统的密码分析。所构造的码具有优良特性：以高概率保持恒定码率与平均距离缩放$\Delta \propto \sqrt{n}$（其中$n$为玻色子模数），该距离缩放等价于将单模GKP码级联为具有线性距离的量子比特纠错码所获得的GKP码。此类NTRU-GKP码还具备额外性质：对于随机位移噪声模型的译码过程等价于对NTRU密码系统的解密，使得每个随机码实例自然配备高效译码器。该构造揭示了GKP码如何连接经典纠错、量子纠错以及后量子密码学。我们通过讨论GKP码译码的计算难度来强调这一联系，并提出一项新应用：基于NTRU密码系统安全性的简易公钥量子通信协议。