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As the best-known scheme,  (Ducas et al., TCHES 2018) is SUF-CMA in QROM and has a relatively fast efficiency with many untrivial optimizations. The goal of this paper is to propose some techniques that can promote signing speed without sacrificing security. We first propose the pre-rejection sampling technique in  stage to reduce the rejections of the fourth condition, consequently resulting in some speedup in  stage. To prove security, we propose the <jats:italic>c<\/jats:italic>-selected MLWE problem, a variant of MLWE that can offer the equivalent security as original MLWE. Applying these two techniques to , we obtain an advanced signature scheme with better efficiency, and without any other losses except some pre-computations. Security reduction demonstrates that our scheme is also SUF-CMA in QROM. The experimental results show that pre-rejection sampling achieves a <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$47\\%$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>47<\/mml:mn>\n                    <mml:mo>%<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$22\\%$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>22<\/mml:mn>\n                    <mml:mo>%<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$17\\%$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>17<\/mml:mn>\n                    <mml:mo>%<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> reduction in the rejection probability of the fourth condition over  scheme when the parameter set corresponds to NIST\u2019s security levels 2, 3 and 5, respectively. This type of reduction increases signing speed by approximately <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$1\\%$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>%<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> under the parameter set 2 of .<\/jats:p>","DOI":"10.1186\/s42400-024-00325-6","type":"journal-article","created":{"date-parts":[[2025,2,8]],"date-time":"2025-02-08T02:02:16Z","timestamp":1738980136000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Speedup signing: pre-rejection sampling towards dilithium"],"prefix":"10.1186","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-8378-9446","authenticated-orcid":false,"given":"Lianglin","family":"Yan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ming","family":"Luo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingsheng","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,2,8]]},"reference":[{"key":"325_CR1","doi-asserted-by":"publisher","first-page":"418","DOI":"10.1007\/3-540-46035-7_28","volume-title":"EUROCRYPT 2002, LNCS","author":"M Abdalla","year":"2002","unstructured":"Abdalla M, An JH, Bellare M, Namprempre C (2002) From identification to signatures via the Fiat-Shamir transform: Minimizing assumptions for security and forward-security. 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