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Commun."],"published-print":{"date-parts":[[2026,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In this paper we introduce a rank\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\varvec{2}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    lattice over a polynomial ring arising from the public key of the BIKE cryptosystem (2024). The secret key is a sparse vector in this lattice. We study properties of this lattice and generalize the recovery of weak keys from Bardet, Dragoi, Luque and Otmani (2016). In particular, we show that they implicitly solved a shortest vector problem in the lattice we constructed. Rather than finding only a shortest vector, we obtain a reduced basis of the lattice which makes it possible to check for more weak keys.\n                  <\/jats:p>","DOI":"10.1007\/s12095-026-00875-1","type":"journal-article","created":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T08:48:18Z","timestamp":1771836498000},"page":"1723-1733","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Polynomial lattices for the BIKE cryptosystem"],"prefix":"10.1007","volume":"18","author":[{"given":"Michael","family":"Schaller","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,23]]},"reference":[{"key":"875_CR1","unstructured":"Aragon, N., Barreto, P., Bettaieb, S., Bidoux, L., Blazy, O., Deneuville, J.-C., Gaborit, P., Ghosh, S., Gueron, S., G\u00fcneysu, T., Aguilar-Melchor, C., Misoczki, R., Persichetti, E., Richter-Brockmann, J., Sendrier, N., Tillich, J.-P., Vasseur, V., Z\u00e9mor, G.: BIKE: bit flipping key encapsulation (2024)"},{"key":"875_CR2","doi-asserted-by":"publisher","unstructured":"Bardet, M., Dragoi, V., Luque, J., Otmani, A.: Weak keys for the quasi-cyclic MDPC public key encryption scheme. 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