{"title":"一类类rsa密码系统的一种新的广义格攻击","authors":"Michel Seck, Abdoul Aziz Ciss","doi":"10.1016/j.tcs.2025.115548","DOIUrl":null,"url":null,"abstract":"<div><div>Recently, Cotan and Teşeleanu (NordSec 2023) published an RSA-like cryptosystem. While in RSA, the public exponent <span><math><mi>e</mi></math></span> and the private exponent <span><math><mi>d</mi></math></span> are related by the equation <span><math><mrow><mi>e</mi><mi>d</mi><mo>−</mo><mi>k</mi><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>1</mn></mrow></math></span>, in their scheme, <span><math><mi>e</mi></math></span> and <span><math><mi>d</mi></math></span> are related to the equation <span><math><mrow><mi>e</mi><mi>d</mi><mo>−</mo><mi>k</mi><mrow><mo>(</mo><msup><mi>p</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mi>q</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for some positive integer <span><math><mi>n</mi></math></span>. Teşeleanu (CSCML 2024) showed that one can factor the modulus <span><math><mi>N</mi></math></span> using a lattice attack if <span><math><mi>d</mi></math></span> is small. In this paper, we extend his attack by showing that if the private exponent is either too small or too large, one can factor <span><math><mi>N</mi></math></span> in polynomial time by solving the generalized equation <span><math><mrow><mi>e</mi><mi>u</mi><mo>−</mo><mrow><mo>(</mo><msup><mi>p</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mi>q</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>v</mi><mo>=</mo><mo>±</mo><mn>1</mn></mrow></math></span> using lattice reduction techniques.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1055 ","pages":"Article 115548"},"PeriodicalIF":1.0000,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A new generalized lattice attack against a family of RSA-like cryptosystems\",\"authors\":\"Michel Seck, Abdoul Aziz Ciss\",\"doi\":\"10.1016/j.tcs.2025.115548\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Recently, Cotan and Teşeleanu (NordSec 2023) published an RSA-like cryptosystem. While in RSA, the public exponent <span><math><mi>e</mi></math></span> and the private exponent <span><math><mi>d</mi></math></span> are related by the equation <span><math><mrow><mi>e</mi><mi>d</mi><mo>−</mo><mi>k</mi><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>1</mn></mrow></math></span>, in their scheme, <span><math><mi>e</mi></math></span> and <span><math><mi>d</mi></math></span> are related to the equation <span><math><mrow><mi>e</mi><mi>d</mi><mo>−</mo><mi>k</mi><mrow><mo>(</mo><msup><mi>p</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mi>q</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for some positive integer <span><math><mi>n</mi></math></span>. Teşeleanu (CSCML 2024) showed that one can factor the modulus <span><math><mi>N</mi></math></span> using a lattice attack if <span><math><mi>d</mi></math></span> is small. In this paper, we extend his attack by showing that if the private exponent is either too small or too large, one can factor <span><math><mi>N</mi></math></span> in polynomial time by solving the generalized equation <span><math><mrow><mi>e</mi><mi>u</mi><mo>−</mo><mrow><mo>(</mo><msup><mi>p</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mi>q</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>v</mi><mo>=</mo><mo>±</mo><mn>1</mn></mrow></math></span> using lattice reduction techniques.</div></div>\",\"PeriodicalId\":49438,\"journal\":{\"name\":\"Theoretical Computer Science\",\"volume\":\"1055 \",\"pages\":\"Article 115548\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theoretical Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0304397525004864\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397525004864","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
A new generalized lattice attack against a family of RSA-like cryptosystems
Recently, Cotan and Teşeleanu (NordSec 2023) published an RSA-like cryptosystem. While in RSA, the public exponent and the private exponent are related by the equation , in their scheme, and are related to the equation for some positive integer . Teşeleanu (CSCML 2024) showed that one can factor the modulus using a lattice attack if is small. In this paper, we extend his attack by showing that if the private exponent is either too small or too large, one can factor in polynomial time by solving the generalized equation using lattice reduction techniques.
期刊介绍：
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.
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