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，这些码源自对所谓的NTRU密码系统的密码分析。所推导的码在性能上表现优良，因为它们以高概率具有恒定速率和平均距离标度 $\Delta \propto \sqrt{n}$，其中 $n$ 是玻色子模式的数量，这种距离标度等价于将单模GKP码级联成具有线性距离的量子比特-量子纠错码所获得的GKP码。所推导的NTRU-GKP码类还具有一个附加性质：对于随机位移噪声模型的解码等价于对NTRU密码系统进行解密，因此每个随机实例的码自然配备了一个高效解码器。这一构造凸显了GKP码如何桥接经典纠错、量子纠错以及后量子密码学的各个方面。我们通过讨论解码GKP码的计算难度来强调这一联系，并提出一种新的应用：基于NTRU密码系统安全性的简单公钥量子通信协议。