{"id":65,"date":"2005-04-15T18:10:00","date_gmt":"2005-04-15T11:10:00","guid":{"rendered":"http:\/\/www.khudri.com\/web\/?p=65"},"modified":"2022-02-12T17:09:50","modified_gmt":"2022-02-12T10:09:50","slug":"kriptografi-kurva-elliptic","status":"publish","type":"post","link":"https:\/\/www.khudri.com\/web\/kriptografi-kurva-elliptic-200504\/","title":{"rendered":"Kriptografi Kurva Elliptic"},"content":{"rendered":"<p style=\"text-align: justify;\">Tahun 1985, Koblitz dan Miller mengenalkan kriptografi kurva <em>elliptic<\/em> (<em>elliptic curve cryptography<\/em>) yang menggunakan masalah logaritma diskrit pada titik kurva <em>elliptic<\/em>. Kriptografi kurva <em>elliptic<\/em> dapat digunakan untuk beberapa keperluan seperti skema enkripsi (contohnya ElGamal ECC), tanda tangan digital (contohnya ECDSA) dan protokol pertukaran kunci (contohnya Diffie-Hielman ECC).<\/p>\n<p style=\"text-align: justify;\">Stallings [<a href=\"https:\/\/wan.khudri.com\/my_files\/skripsi\/VI.pdf\" target=\"_blank\" rel=\"noopener\">13<\/a>] mendefinisikan kurva <em>elliptic<\/em> sebagai suatu kurva yang dibentuk oleh persamaan kubik dan memiliki persamaan umum<br \/>\n<img decoding=\"async\" class=\"aligncenter\" title=\"\\large y^{2}+Axy\\: =\\: x^{3}+Cx^{2}+Dx+E\\; \\; \\; (4.1)\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;y^{2}+Axy\\:&amp;space;=\\:&amp;space;x^{3}+Cx^{2}+Dx+E\\;&amp;space;\\;&amp;space;\\;&amp;space;(4.1)\" alt=\"\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" title=\"\\large y^{2}+Axy\\: =\\: x^{3}+Cx^{2}+Dx+E\\; \\; \\; (4.1)\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;y^{2}+Axy\\:&amp;space;=\\:&amp;space;x^{3}+Cx^{2}+Dx+E\\;&amp;space;\\;&amp;space;\\;&amp;space;(4.1)\" alt=\"\"><\/p>\n<p style=\"text-align: justify;\">dengan <em>A,B,C,D<\/em> dan <em>E<\/em> adalah konstanta bilangan real. Domain <em>x<\/em> dan <em>y<\/em> adalah bilangan real (<img decoding=\"async\" title=\"\\large \\mathbb{R}\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;\\mathbb{R}\" alt=\"\">). Dalam penulisan ini, tidak dibahas mengenai persamaan (4.1). Untuk mendukung tujuan penulisan ini, penulis akan menjelaskan bentuk kurva yang lebih sederhana dari persamaan (4.1), yaitu<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" class=\"aligncenter\" title=\"\\large y^{2}\\: =\\: x^{3}+Ax+B\\; \\; \\; (4.2)\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;y^{2}\\:&amp;space;=\\:&amp;space;x^{3}+Ax+B\\;&amp;space;\\;&amp;space;\\;&amp;space;(4.2)\" alt=\"\"><\/p>\n<p style=\"text-align: justify;\">dengan <em>A<\/em> dan <em>B<\/em> dalam <img decoding=\"async\" title=\"\\large \\mathbb{R}\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;\\mathbb{R}\" alt=\"\"> serta <em>x,<\/em> <em>y<\/em> <img decoding=\"async\" title=\"\\large \\in\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;\\in\" alt=\"\"> <img decoding=\"async\" title=\"\\large \\mathbb{R}\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;\\mathbb{R}\" alt=\"\">.<\/p>\n<p style=\"text-align: justify;\">Gambar 4.1 merupakan salah satu contoh bentuk geometri dari kurva <em>elliptic<\/em> dengan persamaan <img decoding=\"async\" class=\"alignnone\" title=\"\\large y^{2}\\: =\\: x^{3}+x+1\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;y^{2}\\:&amp;space;=\\:&amp;space;x^{3}+x+1\" alt=\"\">. Setiap kurva <em>elliptic<\/em> akan berbentuk simetris terhadap sumbu <em>x<\/em> atau garis <em>y=0<\/em><em>.<\/em> Karena untuk setiap nilai <em>x <img decoding=\"async\" title=\"\\large \\in\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;\\in\" alt=\"\"> <img decoding=\"async\" title=\"\\large \\mathbb{R}\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;\\mathbb{R}\" alt=\"\"><\/em>, terdapat sepasang nilai <em>y<\/em> <img decoding=\"async\" title=\"\\large \\in\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;\\in\" alt=\"\"> <img decoding=\"async\" title=\"\\large \\mathbb{R}\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;\\mathbb{R}\" alt=\"\"> yang memenuhi persamaan (4.2), yaitu<\/p>\n<p style=\"text-align: left;\"><img loading=\"lazy\" decoding=\"async\" class=\"\" title=\"\\large y_{1}=+\\sqrt{x^{3}+Ax+B}\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;y_{1}=+\\sqrt{x^{3}+Ax+B}\" alt=\"\" width=\"180\" height=\"25\"> dan <img loading=\"lazy\" decoding=\"async\" class=\"\" title=\"\\large y_{2}=-\\sqrt{x^{3}+Ax+B}\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;y_{2}=-\\sqrt{x^{3}+Ax+B}\" alt=\"\" width=\"180\" height=\"25\">.<\/p>\n<p style=\"text-align: justify;\">Kurva <em>elliptic<\/em> juga dapat dipandang sebagai suatu himpunan yang terdiri dari titik-titik (<em>x,y<\/em>) yang memenuhi persamaan (4.2). Himpunan tersebut dinotasikan dengan <em>E(A,B). <\/em>Untuk setiap nilai <em>A<\/em> dan <em>B<\/em> yang berbeda, dihasilkan himpunan <em>E(A,B)<\/em> yang berbeda pula. Sebagai contoh, kurva dalam Gambar 4.1 dan Gambar 4.2.<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"wp-image-86 alignnone\" src=\"https:\/\/www.khudri.com\/web\/wp-content\/uploads\/gbr-4-1.png\" alt=\"gbr-4-1\" width=\"350\" srcset=\"https:\/\/www.khudri.com\/web\/wp-content\/uploads\/gbr-4-1.png 800w, https:\/\/www.khudri.com\/web\/wp-content\/uploads\/gbr-4-1-300x197.png 300w\" sizes=\"(max-width: 709px) 85vw, (max-width: 909px) 67vw, (max-width: 984px) 61vw, (max-width: 1362px) 45vw, 600px\" \/><br \/>\n<em>Gambar 4.1. Kurva Elliptic <img decoding=\"async\" class=\"alignnone\" title=\"\\large y^{2}\\: =\\: x^{3}+x+1\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;y^{2}\\:&amp;space;=\\:&amp;space;x^{3}+x+1\" alt=\"\"><\/em><\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"wp-image-87 alignnone\" src=\"https:\/\/www.khudri.com\/web\/wp-content\/uploads\/gbr-4-2-300x197.png\" alt=\"gbr-4-2\" width=\"350\" srcset=\"https:\/\/www.khudri.com\/web\/wp-content\/uploads\/gbr-4-2-300x197.png 300w, https:\/\/www.khudri.com\/web\/wp-content\/uploads\/gbr-4-2.png 800w\" sizes=\"(max-width: 300px) 85vw, 300px\" \/><br \/>\n<em>Gambar 4.2. Kurva Elliptic <img decoding=\"async\" class=\"alignnone\" title=\"\\large y^{2}\\: =\\: x^{3}-x\" src=\"https:\/\/latex.codecogs.com\/png.latex?\\large&amp;space;y^{2}\\:&amp;space;=\\:&amp;space;x^{3}-x\" alt=\"\"><\/em><\/p>\n<p style=\"text-align: justify;\">Source:<br \/>\n<a href=\"https:\/\/wan.khudri.com\/my_files\/skripsi\/IV.pdf\" target=\"_blank\" rel=\"noopener\">wan.khudri.com<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Tahun 1985, Koblitz dan Miller mengenalkan kriptografi kurva elliptic (elliptic curve cryptography) yang menggunakan masalah logaritma diskrit pada titik kurva elliptic. Kriptografi kurva elliptic dapat digunakan untuk beberapa keperluan seperti skema enkripsi (contohnya ElGamal ECC), tanda tangan digital (contohnya ECDSA) dan protokol pertukaran kunci (contohnya Diffie-Hielman ECC). Stallings [13] mendefinisikan kurva elliptic sebagai suatu kurva &hellip; <a href=\"https:\/\/www.khudri.com\/web\/kriptografi-kurva-elliptic-200504\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Kriptografi Kurva Elliptic&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,5,6],"tags":[48,53,49,54,44,56,51,50,52,55],"class_list":["post-65","post","type-post","status-publish","format-standard","hentry","category-cryptography","category-mathemathics","category-security","tag-ecc","tag-ecdsa","tag-elgamal-ecc","tag-elliptic","tag-elliptic-curve-cryptography","tag-elliptic-curve-digital-signature","tag-koblitz","tag-kriptografi-kurva-elliptic","tag-miller","tag-stalling"],"_links":{"self":[{"href":"https:\/\/www.khudri.com\/web\/wp-json\/wp\/v2\/posts\/65"}],"collection":[{"href":"https:\/\/www.khudri.com\/web\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.khudri.com\/web\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.khudri.com\/web\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.khudri.com\/web\/wp-json\/wp\/v2\/comments?post=65"}],"version-history":[{"count":25,"href":"https:\/\/www.khudri.com\/web\/wp-json\/wp\/v2\/posts\/65\/revisions"}],"predecessor-version":[{"id":177,"href":"https:\/\/www.khudri.com\/web\/wp-json\/wp\/v2\/posts\/65\/revisions\/177"}],"wp:attachment":[{"href":"https:\/\/www.khudri.com\/web\/wp-json\/wp\/v2\/media?parent=65"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.khudri.com\/web\/wp-json\/wp\/v2\/categories?post=65"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.khudri.com\/web\/wp-json\/wp\/v2\/tags?post=65"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}