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：我们引入了一类新的随机Gottesman-Kitaev-Preskill（GKP）码，其源于对所谓NTRU密码体制的密码分析。所推导的码是良好的，因为它们以高概率呈现恒定码率与平均距离缩放$\Delta \propto \sqrt{n}$，其中$n$为玻色子模式数，该距离缩放等效于通过将单模GKP码级联为具有线性距离的量子比特-量子纠错码所获得的GKP码的距离缩放。此类推导出的NTRU-GKP码还具有以下特性：针对随机位移噪声模型的解码等价于解密NTRU密码体制，因此每个随机码实例天然配备一个高效解码器。此构造凸显了GKP码如何连接经典纠错、量子纠错以及后量子密码学的多个方面。我们通过讨论解码GKP码的计算复杂性来强调这一关联，并提出一种新的应用：一种简单的公钥量子通信协议，其安全性继承自NTRU密码体制。