{"title":"满足双线性类型递推关系的序列的存在性","authors":"A. A. Illarionov","doi":"10.1134/s0032946023020072","DOIUrl":null,"url":null,"abstract":"<p>We study sequences <span>\\(\\left\\{A_n\\right\\}_{n=-\\infty}^{+\\infty}\\)</span> of elements of an arbitrary field <span>\\(\\mathbb{F}\\)</span> that satisfy decompositions of the form\n</p><span>$$\n\\begin{aligned}\nA_{m+n} A_{m-n}&=a_1(m) b_1(n)+a_2(m) b_2(n),\\\\ A_{m+n+1} A_{m-n}&=\\widetilde a_1(m)\n\\widetilde b_1(n)+\\widetilde a_2(m) \\widetilde b_2(n),\n\\end{aligned}\n$$</span><p>\nwhere <span>\\(a_1,a_2,b_1,b_2\\colon \\mathbb{Z}\\to\\mathbb{F}\\)</span>. We prove some results concerning the existence and uniqueness of such sequences. The results are used to construct analogs of the Diffie–Hellman and ElGamal cryptographic algorithms. The discrete logarithm problem is considered in the group <span>\\((S,+)\\)</span>, where the set <span>\\(S\\)</span> consists of quadruples <span>\\(S(n)=(A_{n-1},A_n, A_{n+1}, A_{n+2})\\)</span>, <span>\\(n\\in\\mathbb{Z}\\)</span>, and <span>\\(S(n)+S(m)=S(n+m)\\)</span>.</p>","PeriodicalId":54581,"journal":{"name":"Problems of Information Transmission","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence of Sequences Satisfying Bilinear Type Recurrence Relations\",\"authors\":\"A. A. Illarionov\",\"doi\":\"10.1134/s0032946023020072\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study sequences <span>\\\\(\\\\left\\\\{A_n\\\\right\\\\}_{n=-\\\\infty}^{+\\\\infty}\\\\)</span> of elements of an arbitrary field <span>\\\\(\\\\mathbb{F}\\\\)</span> that satisfy decompositions of the form\\n</p><span>$$\\n\\\\begin{aligned}\\nA_{m+n} A_{m-n}&=a_1(m) b_1(n)+a_2(m) b_2(n),\\\\\\\\ A_{m+n+1} A_{m-n}&=\\\\widetilde a_1(m)\\n\\\\widetilde b_1(n)+\\\\widetilde a_2(m) \\\\widetilde b_2(n),\\n\\\\end{aligned}\\n$$</span><p>\\nwhere <span>\\\\(a_1,a_2,b_1,b_2\\\\colon \\\\mathbb{Z}\\\\to\\\\mathbb{F}\\\\)</span>. We prove some results concerning the existence and uniqueness of such sequences. The results are used to construct analogs of the Diffie–Hellman and ElGamal cryptographic algorithms. The discrete logarithm problem is considered in the group <span>\\\\((S,+)\\\\)</span>, where the set <span>\\\\(S\\\\)</span> consists of quadruples <span>\\\\(S(n)=(A_{n-1},A_n, A_{n+1}, A_{n+2})\\\\)</span>, <span>\\\\(n\\\\in\\\\mathbb{Z}\\\\)</span>, and <span>\\\\(S(n)+S(m)=S(n+m)\\\\)</span>.</p>\",\"PeriodicalId\":54581,\"journal\":{\"name\":\"Problems of Information Transmission\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-12-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Problems of Information Transmission\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1134/s0032946023020072\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Problems of Information Transmission","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1134/s0032946023020072","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Existence of Sequences Satisfying Bilinear Type Recurrence Relations
We study sequences \(\left\{A_n\right\}_{n=-\infty}^{+\infty}\) of elements of an arbitrary field \(\mathbb{F}\) that satisfy decompositions of the form
where \(a_1,a_2,b_1,b_2\colon \mathbb{Z}\to\mathbb{F}\). We prove some results concerning the existence and uniqueness of such sequences. The results are used to construct analogs of the Diffie–Hellman and ElGamal cryptographic algorithms. The discrete logarithm problem is considered in the group \((S,+)\), where the set \(S\) consists of quadruples \(S(n)=(A_{n-1},A_n, A_{n+1}, A_{n+2})\), \(n\in\mathbb{Z}\), and \(S(n)+S(m)=S(n+m)\).
期刊介绍:
Problems of Information Transmission is of interest to researcher in all fields concerned with the research and development of communication systems. This quarterly journal features coverage of statistical information theory; coding theory and techniques; noisy channels; error detection and correction; signal detection, extraction, and analysis; analysis of communication networks; optimal processing and routing; the theory of random processes; and bionics.