{"id":477059,"date":"2023-08-09T09:06:59","date_gmt":"2023-08-09T09:06:59","guid":{"rendered":""},"modified":"2023-09-05T11:13:56","modified_gmt":"2023-09-05T11:13:56","slug":"elliptic-curve-cryptography","status":"publish","type":"wiki","link":"https:\/\/oneproxy.pro\/vn\/wiki\/elliptic-curve-cryptography\/","title":{"rendered":"M\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong elip"},"content":{"rendered":"<p>M\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic (ECC) l\u00e0 m\u1ed9t ph\u01b0\u01a1ng ph\u00e1p m\u00e3 h\u00f3a kh\u00f3a c\u00f4ng khai hi\u1ec7n \u0111\u1ea1i v\u00e0 hi\u1ec7u qu\u1ea3 cao \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 b\u1ea3o m\u1eadt vi\u1ec7c truy\u1ec1n d\u1eef li\u1ec7u, x\u00e1c th\u1ef1c v\u00e0 ch\u1eef k\u00fd s\u1ed1. N\u00f3 d\u1ef1a v\u00e0o c\u00e1c \u0111\u1eb7c t\u00ednh to\u00e1n h\u1ecdc c\u1ee7a \u0111\u01b0\u1eddng cong elip \u0111\u1ec3 th\u1ef1c hi\u1ec7n c\u00e1c ho\u1ea1t \u0111\u1ed9ng m\u00e3 h\u00f3a, cung c\u1ea5p gi\u1ea3i ph\u00e1p thay th\u1ebf m\u1ea1nh m\u1ebd v\u00e0 hi\u1ec7u qu\u1ea3 cho c\u00e1c thu\u1eadt to\u00e1n m\u00e3 h\u00f3a truy\u1ec1n th\u1ed1ng nh\u01b0 RSA v\u00e0 DSA. ECC \u0111\u00e3 \u0111\u01b0\u1ee3c \u00e1p d\u1ee5ng r\u1ed9ng r\u00e3i nh\u1edd c\u00e1c t\u00ednh n\u0103ng b\u1ea3o m\u1eadt m\u1ea1nh m\u1ebd v\u00e0 kh\u1ea3 n\u0103ng cung c\u1ea5p c\u00f9ng m\u1ee9c \u0111\u1ed9 b\u1ea3o m\u1eadt v\u1edbi \u0111\u1ed9 d\u00e0i kh\u00f3a ng\u1eafn h\u01a1n, khi\u1ebfn n\u00f3 \u0111\u1eb7c bi\u1ec7t ph\u00f9 h\u1ee3p v\u1edbi c\u00e1c m\u00f4i tr\u01b0\u1eddng h\u1ea1n ch\u1ebf v\u1ec1 t\u00e0i nguy\u00ean, nh\u01b0 thi\u1ebft b\u1ecb di \u0111\u1ed9ng v\u00e0 Internet of Things (IoT) .<\/p>\n<h2>L\u1ecbch s\u1eed ngu\u1ed3n g\u1ed1c c\u1ee7a m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic v\u00e0 l\u1ea7n \u0111\u1ea7u ti\u00ean \u0111\u1ec1 c\u1eadp \u0111\u1ebfn n\u00f3<\/h2>\n<p>L\u1ecbch s\u1eed c\u1ee7a c\u00e1c \u0111\u01b0\u1eddng cong elip b\u1eaft \u0111\u1ea7u t\u1eeb \u0111\u1ea7u th\u1ebf k\u1ef7 19 khi c\u00e1c nh\u00e0 to\u00e1n h\u1ecdc kh\u00e1m ph\u00e1 nh\u1eefng \u0111\u01b0\u1eddng cong h\u1ea5p d\u1eabn n\u00e0y v\u00ec nh\u1eefng \u0111\u1eb7c t\u00ednh h\u1ea5p d\u1eabn c\u1ee7a ch\u00fang. Tuy nhi\u00ean, ph\u1ea3i \u0111\u1ebfn nh\u1eefng n\u0103m 1980, Neal Koblitz v\u00e0 Victor Miller m\u1edbi \u0111\u1ec1 xu\u1ea5t \u0111\u1ed9c l\u1eadp kh\u00e1i ni\u1ec7m s\u1eed d\u1ee5ng \u0111\u01b0\u1eddng cong elip cho m\u1ee5c \u0111\u00edch m\u00e3 h\u00f3a. H\u1ecd nh\u1eadn ra r\u1eb1ng b\u00e0i to\u00e1n logarit r\u1eddi r\u1ea1c tr\u00ean c\u00e1c \u0111\u01b0\u1eddng cong elip c\u00f3 th\u1ec3 l\u00e0 n\u1ec1n t\u1ea3ng c\u1ee7a m\u1ed9t h\u1ec7 th\u1ed1ng m\u1eadt m\u00e3 kh\u00f3a c\u00f4ng khai m\u1ea1nh m\u1ebd.<\/p>\n<p>Ngay sau \u0111\u00f3, v\u00e0o n\u0103m 1985, Neal Koblitz v\u00e0 Alfred Menezes, c\u00f9ng v\u1edbi Scott Vanstone, \u0111\u00e3 gi\u1edbi thi\u1ec7u m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong elip nh\u01b0 m\u1ed9t s\u01a1 \u0111\u1ed3 m\u1eadt m\u00e3 kh\u1ea3 thi. Nghi\u00ean c\u1ee9u mang t\u00ednh \u0111\u1ed9t ph\u00e1 c\u1ee7a h\u1ecd \u0111\u00e3 \u0111\u1eb7t n\u1ec1n m\u00f3ng cho s\u1ef1 ph\u00e1t tri\u1ec3n v\u00e0 \u00e1p d\u1ee5ng r\u1ed9ng r\u00e3i c\u1ee7a ECC.<\/p>\n<h2>Th\u00f4ng tin chi ti\u1ebft v\u1ec1 m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic<\/h2>\n<p>M\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong elip, gi\u1ed1ng nh\u01b0 c\u00e1c h\u1ec7 th\u1ed1ng m\u1eadt m\u00e3 kh\u00f3a c\u00f4ng khai kh\u00e1c, s\u1eed d\u1ee5ng hai kh\u00f3a c\u00f3 li\u00ean quan v\u1ec1 m\u1eb7t to\u00e1n h\u1ecdc: kh\u00f3a chung, \u0111\u01b0\u1ee3c m\u1ecdi ng\u01b0\u1eddi bi\u1ebft v\u00e0 kh\u00f3a ri\u00eang, \u0111\u01b0\u1ee3c gi\u1eef b\u00ed m\u1eadt b\u1edfi ng\u01b0\u1eddi d\u00f9ng c\u00e1 nh\u00e2n. Qu\u00e1 tr\u00ecnh n\u00e0y bao g\u1ed3m vi\u1ec7c t\u1ea1o kh\u00f3a, m\u00e3 h\u00f3a v\u00e0 gi\u1ea3i m\u00e3:<\/p>\n<ol>\n<li>\n<p><strong>T\u1ea1o kh\u00f3a<\/strong>: M\u1ed7i ng\u01b0\u1eddi d\u00f9ng t\u1ea1o m\u1ed9t c\u1eb7p kh\u00f3a \u2013 kh\u00f3a ri\u00eang v\u00e0 kh\u00f3a chung t\u01b0\u01a1ng \u1ee9ng. Kh\u00f3a chung \u0111\u01b0\u1ee3c l\u1ea5y t\u1eeb kh\u00f3a ri\u00eang v\u00e0 c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c chia s\u1ebb c\u00f4ng khai.<\/p>\n<\/li>\n<li>\n<p><strong>M\u00e3 h\u00f3a<\/strong>: \u0110\u1ec3 m\u00e3 h\u00f3a tin nh\u1eafn cho ng\u01b0\u1eddi nh\u1eadn, ng\u01b0\u1eddi g\u1eedi s\u1eed d\u1ee5ng kh\u00f3a chung c\u1ee7a ng\u01b0\u1eddi nh\u1eadn \u0111\u1ec3 chuy\u1ec3n v\u0103n b\u1ea3n g\u1ed1c th\u00e0nh v\u0103n b\u1ea3n m\u00e3 h\u00f3a. Ch\u1ec9 ng\u01b0\u1eddi nh\u1eadn c\u00f3 kh\u00f3a ri\u00eang t\u01b0\u01a1ng \u1ee9ng m\u1edbi c\u00f3 th\u1ec3 gi\u1ea3i m\u00e3 \u0111\u01b0\u1ee3c b\u1ea3n m\u00e3 v\u00e0 kh\u00f4i ph\u1ee5c tin nh\u1eafn g\u1ed1c.<\/p>\n<\/li>\n<li>\n<p><strong>gi\u1ea3i m\u00e3<\/strong>: Ng\u01b0\u1eddi nh\u1eadn s\u1eed d\u1ee5ng kh\u00f3a ri\u00eang c\u1ee7a h\u1ecd \u0111\u1ec3 gi\u1ea3i m\u00e3 v\u0103n b\u1ea3n m\u00e3 h\u00f3a v\u00e0 truy c\u1eadp v\u00e0o tin nh\u1eafn g\u1ed1c.<\/p>\n<\/li>\n<\/ol>\n<h2>C\u1ea5u tr\u00fac b\u00ean trong c\u1ee7a m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic \u2013 C\u00e1ch th\u1ee9c ho\u1ea1t \u0111\u1ed9ng<\/h2>\n<p>C\u01a1 s\u1edf c\u01a1 b\u1ea3n c\u1ee7a ECC l\u00e0 c\u1ea5u tr\u00fac to\u00e1n h\u1ecdc c\u1ee7a c\u00e1c \u0111\u01b0\u1eddng cong elip. M\u1ed9t \u0111\u01b0\u1eddng cong elip \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh b\u1edfi m\u1ed9t ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 d\u1ea1ng:<\/p>\n<pre><div class=\"bg-black rounded-md mb-4\"><div class=\"flex items-center relative text-gray-200 bg-gray-800 px-4 py-2 text-xs font-sans justify-between rounded-t-md\"><span>css<\/span><button class=\"flex ml-auto gap-2\"><svg stroke=\"currentColor\" fill=\"none\" stroke-width=\"2\" viewbox=\"0 0 24 24\" stroke-linecap=\"round\" stroke-linejoin=\"round\" class=\"h-4 w-4\" height=\"1em\" width=\"1em\" ><path d=\"M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2\"><\/path><rect x=\"8\" y=\"2\" width=\"8\" height=\"4\" rx=\"1\" ry=\"1\"><\/rect><\/svg>Sao ch\u00e9p m\u00e3<\/button><\/div><div class=\"p-4 overflow-y-auto\"><code class=\"!whitespace-pre hljs language-css\" data-no-translation=\"\">y^<span class=\"hljs-number\">2<\/span> = x^<span class=\"hljs-number\">3<\/span> + ax + <span class=\"hljs-selector-tag\">b<\/span>\n<\/code><\/div><\/div><\/pre>\n<p>\u1ede \u0111\u00e2u <code data-no-translation=\"\">a<\/code> V\u00e0 <code data-no-translation=\"\">b<\/code> l\u00e0 c\u00e1c h\u1eb1ng s\u1ed1. \u0110\u01b0\u1eddng cong c\u00f3 c\u00e1c thu\u1ed9c t\u00ednh b\u1ed5 sung gi\u00fap n\u00f3 c\u00f3 th\u1ec3 tu\u00e2n theo c\u00e1c ho\u1ea1t \u0111\u1ed9ng m\u00e3 h\u00f3a.<\/p>\n<p>ECC d\u1ef1a v\u00e0o \u0111\u1ed9 kh\u00f3 c\u1ee7a b\u00e0i to\u00e1n logarit r\u1eddi r\u1ea1c tr\u00ean \u0111\u01b0\u1eddng cong elip. Cho m\u1ed9t \u0111i\u1ec3m <code data-no-translation=\"\">P<\/code> tr\u00ean \u0111\u01b0\u1eddng cong v\u00e0 m\u1ed9t v\u00f4 h\u01b0\u1edbng <code data-no-translation=\"\">n<\/code>, tin h\u1ecdc <code data-no-translation=\"\">nP<\/code> t\u01b0\u01a1ng \u0111\u1ed1i \u0111\u01a1n gi\u1ea3n. Tuy nhi\u00ean, \u0111\u01b0a ra <code data-no-translation=\"\">P<\/code> V\u00e0 <code data-no-translation=\"\">nP<\/code>, t\u00ecm v\u00f4 h\u01b0\u1edbng <code data-no-translation=\"\">n<\/code> l\u00e0 kh\u00f4ng th\u1ec3 th\u1ef1c hi\u1ec7n \u0111\u01b0\u1ee3c v\u1ec1 m\u1eb7t t\u00ednh to\u00e1n. Thu\u1ed9c t\u00ednh n\u00e0y t\u1ea1o th\u00e0nh c\u01a1 s\u1edf cho s\u1ef1 an to\u00e0n c\u1ee7a ECC.<\/p>\n<p>T\u00ednh b\u1ea3o m\u1eadt c\u1ee7a ECC n\u1eb1m \u1edf kh\u00f3 kh\u0103n trong vi\u1ec7c gi\u1ea3i b\u00e0i to\u00e1n logarit r\u1eddi r\u1ea1c tr\u00ean \u0111\u01b0\u1eddng cong elip. Kh\u00f4ng gi\u1ed1ng nh\u01b0 RSA, d\u1ef1a tr\u00ean b\u00e0i to\u00e1n ph\u00e2n t\u00edch s\u1ed1 nguy\u00ean, t\u00ednh b\u1ea3o m\u1eadt c\u1ee7a ECC b\u1eaft ngu\u1ed3n t\u1eeb \u0111\u1ed9 kh\u00f3 c\u1ee7a b\u00e0i to\u00e1n c\u1ee5 th\u1ec3 n\u00e0y.<\/p>\n<h2>Ph\u00e2n t\u00edch c\u00e1c t\u00ednh n\u0103ng ch\u00ednh c\u1ee7a m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic<\/h2>\n<p>M\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong elip cung c\u1ea5p m\u1ed9t s\u1ed1 t\u00ednh n\u0103ng ch\u00ednh g\u00f3p ph\u1ea7n v\u00e0o s\u1ef1 ph\u1ed5 bi\u1ebfn v\u00e0 \u00e1p d\u1ee5ng c\u1ee7a n\u00f3:<\/p>\n<ol>\n<li>\n<p><strong>B\u1ea3o m\u1eadt m\u1ea1nh m\u1ebd<\/strong>: ECC cung c\u1ea5p m\u1ee9c \u0111\u1ed9 b\u1ea3o m\u1eadt cao v\u1edbi \u0111\u1ed9 d\u00e0i kh\u00f3a ng\u1eafn h\u01a1n so v\u1edbi c\u00e1c thu\u1eadt to\u00e1n m\u00e3 h\u00f3a kh\u00f3a c\u00f4ng khai kh\u00e1c. \u0110i\u1ec1u n\u00e0y d\u1eabn \u0111\u1ebfn gi\u1ea3m y\u00eau c\u1ea7u t\u00ednh to\u00e1n v\u00e0 hi\u1ec7u su\u1ea5t nhanh h\u01a1n.<\/p>\n<\/li>\n<li>\n<p><strong>Hi\u1ec7u qu\u1ea3<\/strong>: ECC ho\u1ea1t \u0111\u1ed9ng hi\u1ec7u qu\u1ea3, ph\u00f9 h\u1ee3p v\u1edbi c\u00e1c thi\u1ebft b\u1ecb c\u00f3 ngu\u1ed3n l\u1ef1c h\u1ea1n ch\u1ebf nh\u01b0 \u0111i\u1ec7n tho\u1ea1i th\u00f4ng minh v\u00e0 thi\u1ebft b\u1ecb IoT.<\/p>\n<\/li>\n<li>\n<p><strong>K\u00edch th\u01b0\u1edbc ph\u00edm nh\u1ecf h\u01a1n<\/strong>: K\u00edch th\u01b0\u1edbc kh\u00f3a nh\u1ecf h\u01a1n c\u00f3 ngh\u0129a l\u00e0 \u00edt kh\u00f4ng gian l\u01b0u tr\u1eef h\u01a1n v\u00e0 truy\u1ec1n d\u1eef li\u1ec7u nhanh h\u01a1n, \u0111i\u1ec1u n\u00e0y r\u1ea5t quan tr\u1ecdng trong c\u00e1c \u1ee9ng d\u1ee5ng hi\u1ec7n \u0111\u1ea1i.<\/p>\n<\/li>\n<li>\n<p><strong>Chuy\u1ec3n ti\u1ebfp b\u00ed m\u1eadt<\/strong>: ECC cung c\u1ea5p t\u00ednh b\u1ea3o m\u1eadt chuy\u1ec3n ti\u1ebfp, \u0111\u1ea3m b\u1ea3o r\u1eb1ng ngay c\u1ea3 khi kh\u00f3a ri\u00eang c\u1ee7a m\u1ed9t phi\u00ean b\u1ecb x\u00e2m ph\u1ea1m, c\u00e1c th\u00f4ng tin li\u00ean l\u1ea1c trong qu\u00e1 kh\u1ee9 v\u00e0 t\u01b0\u01a1ng lai v\u1eabn \u0111\u01b0\u1ee3c b\u1ea3o m\u1eadt.<\/p>\n<\/li>\n<li>\n<p><strong>Kh\u1ea3 n\u0103ng t\u01b0\u01a1ng th\u00edch<\/strong>: ECC c\u00f3 th\u1ec3 d\u1ec5 d\u00e0ng t\u00edch h\u1ee3p v\u00e0o c\u00e1c h\u1ec7 th\u1ed1ng v\u00e0 giao th\u1ee9c m\u1eadt m\u00e3 hi\u1ec7n c\u00f3.<\/p>\n<\/li>\n<\/ol>\n<h2>C\u00e1c lo\u1ea1i m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic<\/h2>\n<p>C\u00f3 nhi\u1ec1u bi\u1ebfn th\u1ec3 v\u00e0 tham s\u1ed1 kh\u00e1c nhau c\u1ee7a ECC, t\u00f9y thu\u1ed9c v\u00e0o vi\u1ec7c l\u1ef1a ch\u1ecdn \u0111\u01b0\u1eddng cong elip v\u00e0 tr\u01b0\u1eddng c\u01a1 b\u1ea3n c\u1ee7a n\u00f3. C\u00e1c bi\u1ebfn th\u1ec3 th\u01b0\u1eddng \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng bao g\u1ed3m:<\/p>\n<ol>\n<li>\n<p><strong>\u0110\u01b0\u1eddng cong Elliptic Diffie-Hellman (ECDH)<\/strong>: \u0110\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 trao \u0111\u1ed5i kh\u00f3a trong vi\u1ec7c thi\u1ebft l\u1eadp c\u00e1c k\u00eanh li\u00ean l\u1ea1c an to\u00e0n.<\/p>\n<\/li>\n<li>\n<p><strong>Thu\u1eadt to\u00e1n ch\u1eef k\u00fd s\u1ed1 \u0111\u01b0\u1eddng cong Elliptic (ECDSA)<\/strong>: \u0110\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 t\u1ea1o v\u00e0 x\u00e1c minh ch\u1eef k\u00fd s\u1ed1 \u0111\u1ec3 x\u00e1c th\u1ef1c d\u1eef li\u1ec7u v\u00e0 tin nh\u1eafn.<\/p>\n<\/li>\n<li>\n<p><strong>L\u01b0\u1ee3c \u0111\u1ed3 m\u00e3 h\u00f3a t\u00edch h\u1ee3p \u0111\u01b0\u1eddng cong Elliptic (ECIES)<\/strong>: S\u01a1 \u0111\u1ed3 m\u00e3 h\u00f3a lai k\u1ebft h\u1ee3p ECC v\u00e0 m\u00e3 h\u00f3a \u0111\u1ed1i x\u1ee9ng \u0111\u1ec3 truy\u1ec1n d\u1eef li\u1ec7u an to\u00e0n.<\/p>\n<\/li>\n<li>\n<p><strong>\u0110\u01b0\u1eddng cong Edwards v\u00e0 \u0110\u01b0\u1eddng cong Edwards xo\u1eafn<\/strong>: C\u00e1c d\u1ea1ng thay th\u1ebf c\u1ee7a \u0111\u01b0\u1eddng cong elip cung c\u1ea5p c\u00e1c t\u00ednh ch\u1ea5t to\u00e1n h\u1ecdc kh\u00e1c nhau.<\/p>\n<\/li>\n<\/ol>\n<p>D\u01b0\u1edbi \u0111\u00e2y l\u00e0 b\u1ea3ng so s\u00e1nh gi\u1edbi thi\u1ec7u m\u1ed9t s\u1ed1 bi\u1ebfn th\u1ec3 c\u1ee7a ECC:<\/p>\n<table>\n<thead>\n<tr>\n<th>Bi\u1ebfn th\u1ec3 ECC<\/th>\n<th>Tr\u01b0\u1eddng h\u1ee3p s\u1eed d\u1ee5ng<\/th>\n<th>\u0110\u1ed9 d\u00e0i ph\u00edm<\/th>\n<th>T\u00ednh n\u0103ng n\u1ed5i b\u1eadt<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>ECDH<\/td>\n<td>Trao \u0111\u1ed5i kh\u00f3a<\/td>\n<td>Ng\u1eafn h\u01a1n<\/td>\n<td>Cho ph\u00e9p c\u00e1c k\u00eanh li\u00ean l\u1ea1c an to\u00e0n<\/td>\n<\/tr>\n<tr>\n<td>ECDSA<\/td>\n<td>Ch\u1eef k\u00fd s\u1ed1<\/td>\n<td>Ng\u1eafn h\u01a1n<\/td>\n<td>Cung c\u1ea5p x\u00e1c th\u1ef1c d\u1eef li\u1ec7u v\u00e0 tin nh\u1eafn<\/td>\n<\/tr>\n<tr>\n<td>ECIES<\/td>\n<td>M\u00e3 h\u00f3a lai<\/td>\n<td>Ng\u1eafn h\u01a1n<\/td>\n<td>K\u1ebft h\u1ee3p ECC v\u1edbi m\u00e3 h\u00f3a \u0111\u1ed1i x\u1ee9ng<\/td>\n<\/tr>\n<tr>\n<td>\u0110\u01b0\u1eddng cong Edwards<\/td>\n<td>M\u1ee5c \u0111\u00edch chung<\/td>\n<td>Ng\u1eafn h\u01a1n<\/td>\n<td>Cung c\u1ea5p c\u00e1c thu\u1ed9c t\u00ednh to\u00e1n h\u1ecdc kh\u00e1c nhau<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>C\u00e1ch s\u1eed d\u1ee5ng m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic, b\u00e0i to\u00e1n v\u00e0 gi\u1ea3i ph\u00e1p<\/h2>\n<p>ECC t\u00ecm th\u1ea5y c\u00e1c \u1ee9ng d\u1ee5ng trong nhi\u1ec1u l\u0129nh v\u1ef1c kh\u00e1c nhau, bao g\u1ed3m:<\/p>\n<ol>\n<li>\n<p><strong>Truy\u1ec1n th\u00f4ng an to\u00e0n<\/strong>: ECC \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng trong c\u00e1c giao th\u1ee9c SSL\/TLS \u0111\u1ec3 b\u1ea3o m\u1eadt th\u00f4ng tin li\u00ean l\u1ea1c qua internet gi\u1eefa m\u00e1y ch\u1ee7 v\u00e0 m\u00e1y kh\u00e1ch.<\/p>\n<\/li>\n<li>\n<p><strong>Ch\u1eef k\u00fd s\u1ed1<\/strong>: ECC \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 t\u1ea1o v\u00e0 x\u00e1c minh ch\u1eef k\u00fd s\u1ed1, \u0111\u1ea3m b\u1ea3o t\u00ednh x\u00e1c th\u1ef1c v\u00e0 to\u00e0n v\u1eb9n c\u1ee7a d\u1eef li\u1ec7u.<\/p>\n<\/li>\n<li>\n<p><strong>Thi\u1ebft b\u1ecb di \u0111\u1ed9ng v\u00e0 IoT<\/strong>: Do t\u00ednh hi\u1ec7u qu\u1ea3 v\u00e0 k\u00edch th\u01b0\u1edbc kh\u00f3a nh\u1ecf, ECC \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng r\u1ed9ng r\u00e3i trong c\u00e1c \u1ee9ng d\u1ee5ng di \u0111\u1ed9ng v\u00e0 thi\u1ebft b\u1ecb IoT.<\/p>\n<\/li>\n<\/ol>\n<p>B\u00ean c\u1ea1nh nh\u1eefng th\u1ebf m\u1ea1nh c\u1ee7a m\u00ecnh, ECC c\u0169ng ph\u1ea3i \u0111\u1ed1i m\u1eb7t v\u1edbi nh\u1eefng th\u00e1ch th\u1ee9c:<\/p>\n<ol>\n<li>\n<p><strong>C\u00e1c v\u1ea5n \u0111\u1ec1 v\u1ec1 b\u1eb1ng s\u00e1ng ch\u1ebf v\u00e0 c\u1ea5p ph\u00e9p<\/strong>: M\u1ed9t s\u1ed1 thu\u1eadt to\u00e1n ECC ban \u0111\u1ea7u \u0111\u00e3 \u0111\u01b0\u1ee3c c\u1ea5p b\u1eb1ng s\u00e1ng ch\u1ebf, d\u1eabn \u0111\u1ebfn lo ng\u1ea1i v\u1ec1 quy\u1ec1n s\u1edf h\u1eefu tr\u00ed tu\u1ec7 v\u00e0 c\u1ea5p ph\u00e9p.<\/p>\n<\/li>\n<li>\n<p><strong>C\u00e1c m\u1ed1i \u0111e d\u1ecda t\u00ednh to\u00e1n l\u01b0\u1ee3ng t\u1eed<\/strong>: Gi\u1ed1ng nh\u01b0 c\u00e1c s\u01a1 \u0111\u1ed3 m\u00e3 h\u00f3a b\u1ea5t \u0111\u1ed1i x\u1ee9ng kh\u00e1c, ECC d\u1ec5 b\u1ecb t\u1ea5n c\u00f4ng b\u1eb1ng \u0111i\u1ec7n to\u00e1n l\u01b0\u1ee3ng t\u1eed. C\u00e1c bi\u1ebfn th\u1ec3 ECC kh\u00e1ng l\u01b0\u1ee3ng t\u1eed \u0111ang \u0111\u01b0\u1ee3c ph\u00e1t tri\u1ec3n \u0111\u1ec3 gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1 n\u00e0y.<\/p>\n<\/li>\n<\/ol>\n<h2>C\u00e1c \u0111\u1eb7c \u0111i\u1ec3m ch\u00ednh v\u00e0 so s\u00e1nh v\u1edbi c\u00e1c thu\u1eadt ng\u1eef t\u01b0\u01a1ng t\u1ef1<\/h2>\n<p>H\u00e3y so s\u00e1nh ECC v\u1edbi RSA, m\u1ed9t trong nh\u1eefng s\u01a1 \u0111\u1ed3 m\u00e3 h\u00f3a b\u1ea5t \u0111\u1ed1i x\u1ee9ng \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng r\u1ed9ng r\u00e3i nh\u1ea5t:<\/p>\n<table>\n<thead>\n<tr>\n<th>\u0111\u1eb7c tr\u01b0ng<\/th>\n<th>M\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic (ECC)<\/th>\n<th>RSA<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u0110\u1ed9 d\u00e0i kh\u00f3a \u0111\u1ec3 b\u1ea3o m\u1eadt t\u01b0\u01a1ng \u0111\u01b0\u01a1ng<\/td>\n<td>\u0110\u1ed9 d\u00e0i kh\u00f3a ng\u1eafn h\u01a1n (v\u00ed d\u1ee5: 256 bit)<\/td>\n<td>\u0110\u1ed9 d\u00e0i kh\u00f3a d\u00e0i h\u01a1n (v\u00ed d\u1ee5: 2048 bit)<\/td>\n<\/tr>\n<tr>\n<td>Hi\u1ec7u qu\u1ea3 t\u00ednh to\u00e1n<\/td>\n<td>Hi\u1ec7u qu\u1ea3 h\u01a1n, \u0111\u1eb7c bi\u1ec7t \u0111\u1ed1i v\u1edbi c\u00e1c ph\u00edm nh\u1ecf h\u01a1n<\/td>\n<td>K\u00e9m hi\u1ec7u qu\u1ea3 h\u01a1n \u0111\u1ed1i v\u1edbi c\u00e1c ph\u00edm l\u1edbn h\u01a1n<\/td>\n<\/tr>\n<tr>\n<td>B\u1ea3o v\u1ec7<\/td>\n<td>B\u1ea3o m\u1eadt m\u1ea1nh m\u1ebd d\u1ef1a tr\u00ean \u0111\u01b0\u1eddng cong elip<\/td>\n<td>B\u1ea3o m\u1eadt m\u1ea1nh m\u1ebd d\u1ef1a tr\u00ean s\u1ed1 nguy\u00ean t\u1ed1<\/td>\n<\/tr>\n<tr>\n<td>T\u1ed1c \u0111\u1ed9 t\u1ea1o kh\u00f3a<\/td>\n<td>T\u1ea1o kh\u00f3a nhanh h\u01a1n<\/td>\n<td>T\u1ea1o kh\u00f3a ch\u1eadm h\u01a1n<\/td>\n<\/tr>\n<tr>\n<td>T\u1ea1o\/X\u00e1c minh ch\u1eef k\u00fd<\/td>\n<td>N\u00f3i chung nhanh h\u01a1n<\/td>\n<td>Ch\u1eadm h\u01a1n, \u0111\u1eb7c bi\u1ec7t l\u00e0 \u0111\u1ec3 x\u00e1c minh<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>C\u00e1c quan \u0111i\u1ec3m v\u00e0 c\u00f4ng ngh\u1ec7 c\u1ee7a t\u01b0\u01a1ng lai li\u00ean quan \u0111\u1ebfn m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic<\/h2>\n<p>T\u01b0\u01a1ng lai c\u1ee7a ECC c\u00f3 v\u1ebb \u0111\u1ea7y h\u1ee9a h\u1eb9n. Khi nhu c\u1ea7u li\u00ean l\u1ea1c an to\u00e0n ti\u1ebfp t\u1ee5c t\u0103ng l\u00ean, ECC s\u1ebd \u0111\u00f3ng m\u1ed9t vai tr\u00f2 quan tr\u1ecdng, \u0111\u1eb7c bi\u1ec7t l\u00e0 trong m\u00f4i tr\u01b0\u1eddng h\u1ea1n ch\u1ebf v\u1ec1 t\u00e0i nguy\u00ean. C\u00e1c n\u1ed7 l\u1ef1c nghi\u00ean c\u1ee9u \u0111ang \u0111\u01b0\u1ee3c ti\u1ebfn h\u00e0nh \u0111\u1ec3 ph\u00e1t tri\u1ec3n c\u00e1c bi\u1ebfn th\u1ec3 ECC kh\u00e1ng l\u01b0\u1ee3ng t\u1eed, \u0111\u1ea3m b\u1ea3o kh\u1ea3 n\u0103ng t\u1ed3n t\u1ea1i l\u00e2u d\u00e0i c\u1ee7a n\u00f3 trong th\u1ebf gi\u1edbi \u0111i\u1ec7n to\u00e1n h\u1eadu l\u01b0\u1ee3ng t\u1eed.<\/p>\n<h2>C\u00e1ch s\u1eed d\u1ee5ng ho\u1eb7c li\u00ean k\u1ebft m\u00e1y ch\u1ee7 proxy v\u1edbi m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic<\/h2>\n<p>M\u00e1y ch\u1ee7 proxy \u0111\u00f3ng vai tr\u00f2 trung gian gi\u1eefa m\u00e1y kh\u00e1ch v\u00e0 m\u00e1y ch\u1ee7, chuy\u1ec3n ti\u1ebfp y\u00eau c\u1ea7u c\u1ee7a m\u00e1y kh\u00e1ch v\u00e0 nh\u1eadn ph\u1ea3n h\u1ed3i c\u1ee7a m\u00e1y ch\u1ee7. M\u1eb7c d\u00f9 ECC ch\u1ee7 y\u1ebfu \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 li\u00ean l\u1ea1c an to\u00e0n gi\u1eefa ng\u01b0\u1eddi d\u00f9ng cu\u1ed1i v\u00e0 m\u00e1y ch\u1ee7, nh\u01b0ng m\u00e1y ch\u1ee7 proxy c\u00f3 th\u1ec3 t\u0103ng c\u01b0\u1eddng b\u1ea3o m\u1eadt b\u1eb1ng c\u00e1ch tri\u1ec3n khai c\u00e1c giao th\u1ee9c x\u00e1c th\u1ef1c v\u00e0 m\u00e3 h\u00f3a d\u1ef1a tr\u00ean ECC trong giao ti\u1ebfp v\u1edbi c\u1ea3 m\u00e1y kh\u00e1ch v\u00e0 m\u00e1y ch\u1ee7.<\/p>\n<p>B\u1eb1ng c\u00e1ch s\u1eed d\u1ee5ng ECC trong m\u00e1y ch\u1ee7 proxy, vi\u1ec7c truy\u1ec1n d\u1eef li\u1ec7u gi\u1eefa m\u00e1y kh\u00e1ch v\u00e0 m\u00e1y ch\u1ee7 proxy, c\u0169ng nh\u01b0 gi\u1eefa m\u00e1y ch\u1ee7 proxy v\u00e0 m\u00e1y ch\u1ee7 \u0111\u00edch, c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c b\u1ea3o m\u1eadt b\u1eb1ng c\u00e1ch s\u1eed d\u1ee5ng \u0111\u1ed9 d\u00e0i kh\u00f3a ng\u1eafn h\u01a1n, gi\u1ea3m chi ph\u00ed t\u00ednh to\u00e1n v\u00e0 c\u1ea3i thi\u1ec7n hi\u1ec7u su\u1ea5t t\u1ed5ng th\u1ec3.<\/p>\n<h2>Li\u00ean k\u1ebft li\u00ean quan<\/h2>\n<p>\u0110\u1ec3 bi\u1ebft th\u00eam th\u00f4ng tin v\u1ec1 m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic, b\u1ea1n c\u00f3 th\u1ec3 kh\u00e1m ph\u00e1 c\u00e1c t\u00e0i nguy\u00ean sau:<\/p>\n<ol>\n<li><a href=\"https:\/\/csrc.nist.gov\/projects\/elliptic-curve-cryptography\" target=\"_new\" rel=\"noopener nofollow\">Vi\u1ec7n Ti\u00eau chu\u1ea9n v\u00e0 C\u00f4ng ngh\u1ec7 Qu\u1ed1c gia (NIST) - M\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic<\/a><\/li>\n<li><a href=\"https:\/\/en.wikipedia.org\/wiki\/Elliptic-curve_cryptography\" target=\"_new\" rel=\"noopener nofollow\">M\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic tr\u00ean Wikipedia<\/a><\/li>\n<li><a href=\"https:\/\/www.khanacademy.org\/computing\/computer-science\/cryptography\/modern-crypt\/v\/elliptic-curve-cryptography-part-1\" target=\"_new\" rel=\"noopener nofollow\">Gi\u1edbi thi\u1ec7u v\u1ec1 M\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic \u2013 Khan Academy<\/a><\/li>\n<\/ol>\n<p>T\u00f3m l\u1ea1i, m\u1eadt m\u00e3 \u0111\u01b0\u1eddng cong Elliptic \u0111\u00e3 n\u1ed5i l\u00ean nh\u01b0 m\u1ed9t k\u1ef9 thu\u1eadt m\u00e3 h\u00f3a m\u1ea1nh m\u1ebd v\u00e0 hi\u1ec7u qu\u1ea3, gi\u1ea3i quy\u1ebft c\u00e1c th\u00e1ch th\u1ee9c b\u1ea3o m\u1eadt c\u1ee7a truy\u1ec1n th\u00f4ng k\u1ef9 thu\u1eadt s\u1ed1 hi\u1ec7n \u0111\u1ea1i. V\u1edbi c\u00e1c t\u00ednh n\u0103ng b\u1ea3o m\u1eadt m\u1ea1nh m\u1ebd, k\u00edch th\u01b0\u1edbc kh\u00f3a nh\u1ecf h\u01a1n v\u00e0 kh\u1ea3 n\u0103ng t\u01b0\u01a1ng th\u00edch v\u1edbi nhi\u1ec1u \u1ee9ng d\u1ee5ng kh\u00e1c nhau, ECC \u0111\u01b0\u1ee3c k\u1ef3 v\u1ecdng s\u1ebd v\u1eabn l\u00e0 c\u00f4ng c\u1ee5 c\u01a1 b\u1ea3n trong vi\u1ec7c \u0111\u1ea3m b\u1ea3o quy\u1ec1n ri\u00eang t\u01b0 v\u00e0 t\u00ednh to\u00e0n v\u1eb9n c\u1ee7a d\u1eef li\u1ec7u trong th\u1ebf gi\u1edbi k\u1ef9 thu\u1eadt s\u1ed1. B\u1eb1ng c\u00e1ch t\u1eadn d\u1ee5ng nh\u1eefng l\u1ee3i th\u1ebf c\u1ee7a ECC, c\u00e1c nh\u00e0 cung c\u1ea5p m\u00e1y ch\u1ee7 proxy, ch\u1eb3ng h\u1ea1n nh\u01b0 OneProxy, c\u00f3 th\u1ec3 t\u0103ng c\u01b0\u1eddng h\u01a1n n\u1eefa t\u00ednh b\u1ea3o m\u1eadt cho d\u1ecbch v\u1ee5 c\u1ee7a h\u1ecd v\u00e0 g\u00f3p ph\u1ea7n x\u00e2y d\u1ef1ng m\u1ed9t m\u00f4i tr\u01b0\u1eddng tr\u1ef1c tuy\u1ebfn an to\u00e0n h\u01a1n.<\/p>","protected":false},"featured_media":477060,"menu_order":0,"template":"","meta":{"_acf_changed":false,"content-type":"","inline_featured_image":false,"footnotes":""},"class_list":["post-477059","wiki","type-wiki","status-publish","has-post-thumbnail","hentry"],"acf":{"faq_title":"Frequently Asked Questions about <mark>Elliptic-curve cryptography: Securing the Digital World<\/mark>","faq_items":[{"question":"What is Elliptic-curve cryptography (ECC) and how does it work?","answer":"<p><strong>Elliptic-curve cryptography (ECC)<\/strong> is a modern cryptographic method that uses mathematical properties of elliptic curves to secure data transmission, authentication, and digital signatures. It involves two mathematically related keys - a public key and a private key. The public key is openly shared and used for encryption, while the private key, kept secret, is used for decryption.<\/p>"},{"question":"What makes Elliptic-curve cryptography superior to traditional encryption algorithms?","answer":"<p>ECC offers several advantages over traditional encryption algorithms like RSA. It provides strong security with shorter key lengths, making it more efficient in terms of computation and faster in performance. Additionally, ECC's smaller key sizes enable better resource utilization, making it suitable for devices with limited computing power, such as mobile devices and IoT gadgets.<\/p>"},{"question":"How does Elliptic-curve cryptography ensure the security of data?","answer":"<p>The security of ECC is based on the difficulty of the elliptic curve discrete logarithm problem. While it is relatively easy to compute <code>nP<\/code> given a point <code>P<\/code> on the curve and a scalar <code>n<\/code>, calculating the scalar <code>n<\/code> given <code>P<\/code> and <code>nP<\/code> is computationally infeasible. This property forms the foundation of ECC's security, making it highly resistant to attacks.<\/p>"},{"question":"What are the different types of Elliptic-curve cryptography?","answer":"<p>There are various variations of ECC, each serving specific cryptographic purposes. Some common types include:<\/p><ul><li><strong>Elliptic Curve Diffie-Hellman (ECDH)<\/strong>: Used for key exchange in secure communication channels.<\/li><li><strong>Elliptic Curve Digital Signature Algorithm (ECDSA)<\/strong>: Employed for generating and verifying digital signatures.<\/li><li><strong>Elliptic Curve Integrated Encryption Scheme (ECIES)<\/strong>: A hybrid encryption scheme combining ECC and symmetric encryption.<\/li><\/ul>"},{"question":"Can Elliptic-curve cryptography be used with proxy servers?","answer":"<p>Yes, absolutely! Elliptic-curve cryptography can be implemented in proxy servers to enhance the security of data transmission between clients and servers. By using ECC, proxy servers can establish secure channels and authenticate data, contributing to a safer online environment.<\/p>"},{"question":"Is Elliptic-curve cryptography immune to all threats?","answer":"<p>While Elliptic-curve cryptography provides robust security, it is not entirely invulnerable. Like any cryptographic system, ECC is subject to potential threats. However, its strong security features and ongoing research for quantum-resistant variants make it a reliable and future-proof option in today's digital landscape.<\/p>"}]},"_links":{"self":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/wiki\/477059","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/wiki"}],"about":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/types\/wiki"}],"version-history":[{"count":0,"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/wiki\/477059\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/media\/477060"}],"wp:attachment":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/media?parent=477059"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}