{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:21:47Z","timestamp":1764980507429,"version":"3.46.0"},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2017,10,18]],"date-time":"2017-10-18T00:00:00Z","timestamp":1508284800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Breaking RSA is one of the fundamental problems in cryptography. Due to its reliance on the difficulty of the integer factorization problem, no efficient solution has been found despite decades of extensive research.\nOne of the possible ways to break RSA is by finding the value\n                    <jats:inline-formula id=\"j_jmc-2016-0046_ineq_9998_w2aab3b7b2b1b6b1aab1c17b1b1Aa\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>p<\/m:mi>\n                            <m:mo>+<\/m:mo>\n                            <m:mi>q<\/m:mi>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2016-0046_eq_mi573.png\"\/>\n                        <jats:tex-math>{p+q}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , which requires searching the set of all even integers.\nIn this paper, we present a number of formulas for\n                    <jats:inline-formula id=\"j_jmc-2016-0046_ineq_9997_w2aab3b7b2b1b6b1aab1c17b1b3Aa\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>p<\/m:mi>\n                            <m:mo>+<\/m:mo>\n                            <m:mi>q<\/m:mi>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2016-0046_eq_mi573.png\"\/>\n                        <jats:tex-math>{p+q}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    that depend on the form of\n                    <jats:inline-formula id=\"j_jmc-2016-0046_ineq_9996_w2aab3b7b2b1b6b1aab1c17b1b5Aa\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>n<\/m:mi>\n                            <m:mo>=<\/m:mo>\n                            <m:mrow>\n                              <m:mi>p<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mi>q<\/m:mi>\n                            <\/m:mrow>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2016-0046_eq_mi570.png\"\/>\n                        <jats:tex-math>{n=pq}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . These formulas make the set to be searched much smaller than the set of all even integers.\n                  <\/jats:p>","DOI":"10.1515\/jmc-2016-0046","type":"journal-article","created":{"date-parts":[[2017,10,18]],"date-time":"2017-10-18T07:08:15Z","timestamp":1508310495000},"page":"195-203","source":"Crossref","is-referenced-by-count":1,"title":["RSA: A number of formulas to improve the search for p+qp+q"],"prefix":"10.1515","volume":"11","author":[{"given":"Ahmed","family":"Mohammed","sequence":"first","affiliation":[{"name":"King Abdulaziz City for Science and Technology , Riyadh , Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdulrahman","family":"Alkhelaifi","sequence":"additional","affiliation":[{"name":"King Abdulaziz City for Science and Technology , Riyadh , Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,10,18]]},"reference":[{"key":"2025120600200396743_j_jmc-2016-0046_ref_001_w2aab3b7b2b1b6b1ab1b5b1Aa","doi-asserted-by":"crossref","unstructured":"D.  Aggarwal and U.  Maurer,\nBreaking RSA generically is equivalent to factoring,\nAdvances in Cryptology \u2013 EUROCRYPT 2009.\nLecture Notes in Comput. Sci.,\nSpringer, Berlin (2009), 36\u201353.","DOI":"10.1007\/978-3-642-01001-9_2"},{"key":"2025120600200396743_j_jmc-2016-0046_ref_002_w2aab3b7b2b1b6b1ab1b5b2Aa","doi-asserted-by":"crossref","unstructured":"D. J.  Bernstein and A. K.  Lenstra,\nA general number field sieve implementation,\nThe Development of the Number Field Sieve,\nSpringer, Berlin (1993), 103\u2013126.","DOI":"10.1007\/BFb0091541"},{"key":"2025120600200396743_j_jmc-2016-0046_ref_003_w2aab3b7b2b1b6b1ab1b5b3Aa","unstructured":"D.  Boneh,\nTwenty years of attacks on the RSA cryptosystem,\nNotices Amer. Math. Soc. 46 (1999), 203\u2013213."},{"key":"2025120600200396743_j_jmc-2016-0046_ref_004_w2aab3b7b2b1b6b1ab1b5b4Aa","doi-asserted-by":"crossref","unstructured":"D.  Boneh and R.  Venkatesan,\nBreaking RSA may not be equivalent to factoring,\nAdvances in Cryptology \u2013 EUROCRYPT\u201998,\nLecture Notes in Comput. Sci.,\nSpringer, Berlin (1998), 59\u201371.","DOI":"10.1007\/BFb0054117"},{"key":"2025120600200396743_j_jmc-2016-0046_ref_005_w2aab3b7b2b1b6b1ab1b5b5Aa","unstructured":"D. R. L.  Brown,\nBreaking RSA may be as difficult as factoring,\nCryptology ePrint Archive 2008, https:\/\/eprint.iacr.org\/2005\/380.pdf."},{"key":"2025120600200396743_j_jmc-2016-0046_ref_006_w2aab3b7b2b1b6b1ab1b5b6Aa","doi-asserted-by":"crossref","unstructured":"B.  de Weger,\nCryptanalysis of RSA with small prime difference,\nAppl. Algebra Engrg. Comm. Comput. 13 (2002), 17\u201328.\n10.1007\/s002000100088","DOI":"10.1007\/s002000100088"},{"key":"2025120600200396743_j_jmc-2016-0046_ref_007_w2aab3b7b2b1b6b1ab1b5b7Aa","doi-asserted-by":"crossref","unstructured":"H. W.  Lenstra,\nFactoring integers with elliptic curves,\nAnn. of Math. (2) 126 (1987), 649\u2013673.\n10.2307\/1971363","DOI":"10.2307\/1971363"},{"key":"2025120600200396743_j_jmc-2016-0046_ref_008_w2aab3b7b2b1b6b1ab1b5b8Aa","doi-asserted-by":"crossref","unstructured":"A.  Odlyzko,\nDiscrete logarithms: The past and the future,\nDes. Codes Cryptogr. 19 (2000), 129\u2013145.\n10.1023\/A:1008350005447","DOI":"10.1023\/A:1008350005447"},{"key":"2025120600200396743_j_jmc-2016-0046_ref_009_w2aab3b7b2b1b6b1ab1b5b9Aa","doi-asserted-by":"crossref","unstructured":"R. L.  Rivest and B.  Kaliski,\nRSA problem,\nEncyclopedia of Cryptography and Security,\nSpringer US, Boston (2011), 1065\u20131069.","DOI":"10.1007\/978-1-4419-5906-5_475"},{"key":"2025120600200396743_j_jmc-2016-0046_ref_010_w2aab3b7b2b1b6b1ab1b5c10Aa","doi-asserted-by":"crossref","unstructured":"R. L.  Rivest, A.  Shamir and L.  Adleman,\nA method for obtaining digital signatures and public-key cryptosystems,\nCommun. 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