{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T05:00:00Z","timestamp":1764997200586,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,12,20]],"date-time":"2018-12-20T00:00:00Z","timestamp":1545264000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cryptography"],"abstract":"<jats:p>This paper presents new short decryption exponent attacks on RSA, which successfully leads to the factorization of RSA modulus     N = p q     in polynomial time. The paper has two parts. In the first part, we report the usage of the small prime difference method of the form      |   b 2  p \u2212  a 2   q | &lt;   N \u03b3      where the ratio of     q p     is close to      b 2   a 2     , which yields a bound     d &lt;   3   2    N   3 4  \u2212 \u03b3       from the convergents of the continued fraction expansion of     e  N \u2212 \u2308    a 2  +  b 2    a b    N  \u2309 + 1     . The second part of the paper reports four cryptanalytic attacks on t instances of RSA moduli      N s  =  p s   q s      for     s = 1 , 2 , \u2026 , t     where we use     N \u2212 \u2308    a 2  +  b 2    a b    N  \u2309 + 1     as an approximation of     \u03d5 ( N )     satisfying generalized key equations of the shape      e s  d \u2212  k s  \u03d5  (  N s  )  = 1    ,      e s   d s  \u2212 k \u03d5  (  N s  )  = 1    ,      e s  d \u2212  k s  \u03d5  (  N s  )  =  z s     , and      e s   d s  \u2212 k \u03d5  (  N s  )  =  z s      for unknown positive integers     d ,   k s  ,   d s  ,   k s     , and     z s    , where we establish that t RSA moduli can be simultaneously factored in polynomial time using combinations of simultaneous Diophantine approximations and lattice basis reduction methods. In all the reported attacks, we have found an improved short secret exponent bound, which is considered to be better than some bounds as reported in the literature.<\/jats:p>","DOI":"10.3390\/cryptography3010002","type":"journal-article","created":{"date-parts":[[2018,12,20]],"date-time":"2018-12-20T12:54:36Z","timestamp":1545310476000},"page":"2","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["New Cryptanalytic Attack on RSA Modulus N = pq Using Small Prime Difference Method"],"prefix":"10.3390","volume":"3","author":[{"given":"Muhammad Rezal","family":"Kamel Ariffin","sequence":"first","affiliation":[{"name":"Al-Kindi Cryptography Research Laboratory, Institute for Mathematical Research, Universiti Putra Malaysia, Selangor 43400, Malaysia"},{"name":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Selangor 43400, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0201-0064","authenticated-orcid":false,"given":"Saidu Isah","family":"Abubakar","sequence":"additional","affiliation":[{"name":"Al-Kindi Cryptography Research Laboratory, Institute for Mathematical Research, Universiti Putra Malaysia, Selangor 43400, Malaysia"}]},{"given":"Faridah","family":"Yunos","sequence":"additional","affiliation":[{"name":"Al-Kindi Cryptography Research Laboratory, Institute for Mathematical Research, Universiti Putra Malaysia, Selangor 43400, Malaysia"},{"name":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Selangor 43400, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0778-4456","authenticated-orcid":false,"given":"Muhammad Asyraf","family":"Asbullah","sequence":"additional","affiliation":[{"name":"Al-Kindi Cryptography Research Laboratory, Institute for Mathematical Research, Universiti Putra Malaysia, Selangor 43400, Malaysia"},{"name":"Centre of Foundation Studies for Agriculture Science, Universiti Putra Malaysia, Selangor 43400, Malaysia"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1145\/359340.359342","article-title":"A method for obtaining digital signatures and public-key cryptosystems","volume":"21","author":"Rivest","year":"1978","journal-title":"Commun. 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