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The efficiency and security of PRNG are closely related to its output bits at each iteration. Especially, we have recently found that linear congruential generator (LCG) is commonly used as the underlying PRNG in short message service (SMS) app, fast knapsack generator (FKG), and programming languages such as Python, while the quadratic generator plays an important role in Monte Carlo method. Therefore, in this paper, we revisit the security of these two number\u2010theoretic pseudo\u2010random generators and obtain the best results for attacking these two kinds of PRNGs up to now. More precisely, we prove that when the mapping function of LCG and the quadratic generator is unknown, if during each iteration, generators only output the most significant bits of\n                    <jats:italic>v<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>i<\/jats:italic>\n                    <\/jats:sub>\n                    , one can also recover the seed of PRNG when enough consecutive or nonconsecutive outputs are obtained. The primary tool of our attack is the Coppersmith method which can find small roots on polynomial equations. Our advantage lies in applying the local linearization technique to the polynomial equations to make them simple and easy to solve and applying the analytic combinatorics method to simplify the calculation of solution conditions in the Coppersmith method. Experimental data validate the effectiveness of our work.\n                  <\/jats:p>","DOI":"10.1049\/ise2\/5569393","type":"journal-article","created":{"date-parts":[[2025,7,18]],"date-time":"2025-07-18T13:06:00Z","timestamp":1752843960000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Cryptanalysis on Two Kinds of Number Theoretic Pseudo\u2010Random Generators Using Coppersmith Method"],"prefix":"10.1049","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8857-6260","authenticated-orcid":false,"given":"Ran","family":"Zhang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6727-5485","authenticated-orcid":false,"given":"Jingguo","family":"Bi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8541-308X","authenticated-orcid":false,"given":"Lixiang","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4415-0126","authenticated-orcid":false,"given":"Haipeng","family":"Peng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"265","published-online":{"date-parts":[[2025,7,18]]},"reference":[{"key":"e_1_2_11_1_2","first-page":"36","article-title":"Various Techniques Used in Connection With Random Digits","volume":"3","author":"von Neumann J.","year":"1951","journal-title":"Applied Mathematics"},{"key":"e_1_2_11_2_2","doi-asserted-by":"publisher","DOI":"10.1145\/63039.63042"},{"key":"e_1_2_11_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-021-06346-x"},{"key":"e_1_2_11_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10470-020-01703-z"},{"key":"e_1_2_11_5_2","doi-asserted-by":"publisher","DOI":"10.3390\/app13137768"},{"key":"e_1_2_11_6_2","doi-asserted-by":"crossref","unstructured":"AltunkayaS. 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