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The derived codes are <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>g<\/mml:mi><mml:mi>o<\/mml:mi><mml:mi>o<\/mml:mi><mml:mi>d<\/mml:mi><\/mml:math> in that they exhibit constant rate and average distance scaling <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x0394;<\/mml:mi><mml:mo>&amp;#x221D;<\/mml:mo><mml:msqrt><mml:mi>n<\/mml:mi><\/mml:msqrt><\/mml:math> with high probability, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math> 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 <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><mml:mi>e<\/mml:mi><mml:mi>c<\/mml:mi><mml:mi>o<\/mml:mi><mml:mi>d<\/mml:mi><mml:mi>i<\/mml:mi><mml:mi>n<\/mml:mi><mml:mi>g<\/mml:mi><\/mml:math> for a stochastic displacement noise model is equivalent to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><mml:mi>e<\/mml:mi><mml:mi>c<\/mml:mi><mml:mi>r<\/mml:mi><mml:mi>y<\/mml:mi><mml:mi>p<\/mml:mi><mml:mi>t<\/mml:mi><mml:mi>i<\/mml:mi><mml:mi>n<\/mml:mi><mml:mi>g<\/mml:mi><\/mml:math> 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.<\/jats:p>","DOI":"10.22331\/q-2024-07-04-1398","type":"journal-article","created":{"date-parts":[[2024,7,4]],"date-time":"2024-07-04T14:49:43Z","timestamp":1720104583000},"page":"1398","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":7,"title":["Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6120-9930","authenticated-orcid":false,"given":"Jonathan","family":"Conrad","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universit\u00e4t Berlin, Arnimallee 14, 14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin f\u00fcr Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3033-1292","authenticated-orcid":false,"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universit\u00e4t Berlin, Arnimallee 14, 14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin f\u00fcr Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany"},{"name":"Fraunhofer Heinrich Hertz Institute, Einsteinufer 37, 10587 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5372-4825","authenticated-orcid":false,"given":"Jean-Pierre","family":"Seifert","sequence":"additional","affiliation":[{"name":"Electrical Engineering and Computer Science Department, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 135, 10587 Berlin, Germany"},{"name":"Fraunhofer Institute for Secure Information Technology, Rheinstra\u00dfe 75, 64295 Darmstadt, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2024,7,4]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"D. Gottesman, A. Kitaev, and J. Preskill. ``Encoding a qubit in an oscillator&apos;&apos;. Phys. Rev. A 64, 012310 (2001).","DOI":"10.1103\/PhysRevA.64.012310"},{"key":"1","doi-asserted-by":"publisher","unstructured":"J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su, B. Q. Baragiola, S. Guha, G. Dauphinais, and et al. ``Blueprint for a scalable photonic fault-tolerant quantum computer&apos;&apos;. Quantum 5, 392 (2021).","DOI":"10.22331\/q-2021-02-04-392"},{"key":"2","doi-asserted-by":"publisher","unstructured":"S. Bartolucci, P. Birchall, H. Bombin, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. 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