{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T10:49:16Z","timestamp":1785408556193,"version":"3.56.0"},"reference-count":22,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T00:00:00Z","timestamp":1779840000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61602143"],"award-info":[{"award-number":["61602143"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"award":["61602143"],"award-info":[{"award-number":["61602143"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Random integer lattices are fundamental to lattice-based cryptography and algorithmic number theory. A new random integer lattice model, free of any restrictions on the Hermite Normal Form (HNF), was introduced by in 2016. It was also observed that the probability of such a lattice being in a simple HNF form is approximately 44%. In this paper, the gap between general random integer lattices and those in a simple HNF is bridged by introducing the concept of the (n,k)-simple random integer lattice, where the first k diagonal entries of the HNF are fixed to 1. We derive the asymptotic counting formula for such lattices and compute their density among all integer lattices. Furthermore, a generation algorithm for the (n,k)-simple random integer lattice based on rejection sampling and inverse sampling methods are proposed, with the analysis showing that it achieves O(n2) expected running time. This work provides a theoretical foundation and practical toolkit for constructing structured random lattices with controlled HNF forms.<\/jats:p>","DOI":"10.3390\/e28060600","type":"journal-article","created":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T14:36:56Z","timestamp":1779892616000},"page":"600","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On (n,k)-Simple Random Integer Lattices"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3061-2388","authenticated-orcid":false,"given":"Gengran","family":"Hu","sequence":"first","affiliation":[{"name":"School of Cyberspace, Hangzhou Dianzi University, Hangzhou 310018, China"},{"name":"Zhejiang Provincial Key Laboratory of Sensitive Data Security and Confidentiality Governance, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,5,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hoffstein, J., Pipher, J., and Silverman, J.H. 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