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，其源于对所谓的NTRU密码系统的密码分析。所导出的码是优良的，因为它们以高概率表现出恒定速率和平均距离缩放$\Delta \propto \sqrt{n}$，其中$n$是玻色子模式数，这一距离缩放等价于将单模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。所导出的NTRU-GKP码类具有附加性质：对于随机位移噪声模型的解码等价于解密NTRU密码系统，因此每个随机码实例自然附带一个高效解码器。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。我们通过讨论解码GKP码的计算难度来强调这一联系，并提出一个简单的公钥量子通信协议作为新应用，该协议的安全性继承自NTRU密码系统。