{"id":666,"date":"2025-07-01T08:20:00","date_gmt":"2025-07-01T06:20:00","guid":{"rendered":"http:\/\/ai-biztonsag.hu\/?p=666"},"modified":"2025-06-22T12:58:20","modified_gmt":"2025-06-22T10:58:20","slug":"mceliece-es-niederreiter-rendszer-mukodesi-elve","status":"publish","type":"post","link":"http:\/\/ai-biztonsag.hu\/?p=666","title":{"rendered":"McEliece- \u00e9s Niederreiter-rendszer m\u0171k\u00f6d\u00e9si elve"},"content":{"rendered":"<div class=\"seriesmeta\">Ez a bejegyz\u00e9s a 8 r\u00e9sze a 9 sorozatban <a href=\"http:\/\/ai-biztonsag.hu\/?series=kvantumkriptografia\" class=\"series-759\" title=\"Kvantumkriptogr\u00e1fia\">Kvantumkriptogr\u00e1fia<\/a> sorozatban<\/div>\n<p class=\"wp-block-paragraph\">A McEliece-kriptosziszt\u00e9ma l\u00e9nyege, hogy egy hibajav\u00edt\u00f3 k\u00f3dot haszn\u00e1lunk nyilv\u00e1nos kulcsnak \u00e1lc\u00e1zva. Ez a k\u00f3d bel\u00fcl j\u00f3l struktur\u00e1lt \u2013 p\u00e9ld\u00e1ul egy Goppa-k\u00f3d \u2013, de k\u00edv\u00fclr\u0151l v\u00e9letlenszer\u0171nek t\u0171nik. A felhaszn\u00e1l\u00f3 olyan transzform\u00e1ci\u00f3kat alkalmaz a k\u00f3drendszer gener\u00e1torm\u00e1trix\u00e1n, amelyek elrejtik annak val\u00f3di szerkezet\u00e9t. <strong>A t\u00e1mad\u00f3 nem l\u00e1t m\u00e1st, csak egy v\u00e9letlenszer\u0171 m\u00e1trixot \u2013 \u00e9s ez teszi a visszafejt\u00e9st gyakorlatilag lehetetlenn\u00e9<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>A kulcsgener\u00e1l\u00e1s h\u00e1rom r\u00e9szb\u0151l \u00e1ll<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>V\u00e1lasztunk egy k\u00f3dot<\/strong>, amely hat\u00e9konyan dek\u00f3dolhat\u00f3. Ez jellemz\u0151en egy bin\u00e1ris Goppa-k\u00f3d, amelyet a k\u00f3delm\u00e9let j\u00f3l ismer, \u00e9s amelyhez l\u00e9tezik gyors dek\u00f3dol\u00f3 algoritmus (pl. Patterson-algoritmus).<\/li>\n\n\n\n<li><strong>Gener\u00e1lunk egy S m\u00e1trixot<\/strong>, amely v\u00e9letlenszer\u0171, invert\u00e1lhat\u00f3 \u00e9s elrejti az eredeti k\u00f3d szerkezet\u00e9t soronk\u00e9nti kever\u00e9ssel.<\/li>\n\n\n\n<li><strong>Alkalmazunk egy P permut\u00e1ci\u00f3s m\u00e1trixot<\/strong>, amely az oszlopokat keveri \u00f6ssze.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">A nyilv\u00e1nos kulcs ezek ut\u00e1n a k\u00f6vetkez\u0151 lesz:<br><strong>G_pub = S \u00d7 G \u00d7 P<\/strong>,<br>ahol G az eredeti k\u00f3d gener\u00e1torm\u00e1trixa. Az \u00edgy keletkezett G_pub v\u00e9letlenszer\u0171nek t\u0171nik, \u00e9s alkalmas titkos\u00edt\u00e1sra.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>A titkos\u00edt\u00e1s menete egyszer\u0171, de hat\u00e9kony<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A k\u00fcld\u0151 (pl. Alice) el\u0151sz\u00f6r vesz egy \u00fczenetet m, amely egy bin\u00e1ris vektor. Ezt megszorozza a nyilv\u00e1nos kulccsal:<br><strong>c = m \u00d7 G_pub + e<\/strong>,<br>ahol e egy el\u0151re meghat\u00e1rozott sz\u00e1m\u00fa hib\u00e1t (p\u00e9ld\u00e1ul t darab 1-est) tartalmaz\u00f3 vektor. <strong>Ez a hiba biztos\u00edtja, hogy a titkos\u00edtott \u00fczenet ne dek\u00f3dolhat\u00f3 legyen egy ismeretlen t\u00e1mad\u00f3 sz\u00e1m\u00e1ra<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A dek\u00f3dol\u00e1s kulcsa az, hogy a c\u00edmzett (pl. Bob) ismeri az S, G \u00e9s P m\u00e1trixokat. Ezek seg\u00edts\u00e9g\u00e9vel l\u00e9p\u00e9sr\u0151l l\u00e9p\u00e9sre:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>El\u0151sz\u00f6r kisz\u00e1molja <strong>c\u2019 = c \u00d7 P\u207b\u00b9<\/strong>, \u00edgy vissza\u00e1ll\u00edtja a hib\u00e1s k\u00f3dolt \u00fczenetet az eredeti k\u00f3drendszerhez igaz\u00edtva.<\/li>\n\n\n\n<li>Ezut\u00e1n a hibajav\u00edt\u00f3 dek\u00f3dol\u00f3 algoritmussal (pl. Patterson) visszafejti az m\u2019 \u00fczenetet.<\/li>\n\n\n\n<li>V\u00e9g\u00fcl <strong>m = m\u2019 \u00d7 S\u207b\u00b9<\/strong>, \u00e9s ezzel megkapja az eredeti titkos\u00edtatlan \u00fczenetet.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>A Niederreiter-vari\u00e1ns: ugyanaz m\u00e1sk\u00e9pp<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A Niederreiter-verzi\u00f3 tulajdonk\u00e9ppen ugyanazon alapelven m\u0171k\u00f6dik, de m\u00e1trix-transzform\u00e1ci\u00f3 helyett a szindr\u00f3ma-alap\u00fa dek\u00f3dol\u00e1st haszn\u00e1lja. Itt a nyilv\u00e1nos kulcs egy parit\u00e1sellen\u0151rz\u0151 m\u00e1trix, \u00e9s a titkos\u00edtott \u00fczenet a hibavektor szindr\u00f3m\u00e1ja:<br><strong>c = H \u00d7 e\u1d57<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A dek\u00f3dol\u00e1s sor\u00e1n a titkos kulcs seg\u00edts\u00e9g\u00e9vel vissza\u00e1ll\u00edthat\u00f3 a hibavektor, \u00e9s ezzel az eredeti \u00fczenet. <strong>A k\u00e9t rendszer matematikailag ekvivalens, de k\u00fcl\u00f6nb\u00f6z\u0151 optimaliz\u00e1l\u00e1si lehet\u0151s\u00e9geket k\u00edn\u00e1lnak a gyakorlatban<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Mi\u00e9rt neh\u00e9z felt\u00f6rni ezeket a rendszereket?<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A t\u00e1mad\u00f3 a nyilv\u00e1nos kulcsot \u00e9s a titkos\u00edtott \u00fczenetet ismerheti, de <strong>a hibajav\u00edt\u00f3 k\u00f3d struktur\u00e1latlannak t\u0171nik sz\u00e1m\u00e1ra<\/strong>, \u00edgy nincs kapaszkod\u00f3ja az \u00e9rtelmez\u00e9shez. Az egyetlen lehet\u0151s\u00e9ge, hogy pr\u00f3b\u00e1l megoldani egy szindr\u00f3madek\u00f3dol\u00e1si probl\u00e9m\u00e1t, amely ismert NP-neh\u00e9z, m\u00e9g akkor is, ha kvantumalgoritmusokat is alkalmaz.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Sebess\u00e9g \u00e9s hat\u00e9konys\u00e1g: nem csak biztons\u00e1gr\u00f3l van sz\u00f3<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A McEliece-rendszer nemcsak biztons\u00e1gos, hanem <strong>gyors is<\/strong>: a kulcsgener\u00e1l\u00e1s \u00e9s a dek\u00f3dol\u00e1s kis sz\u00e1m\u00edt\u00e1si er\u0151forr\u00e1ssal v\u00e9grehajthat\u00f3. Ez k\u00fcl\u00f6n\u00f6sen vonz\u00f3 lehet alacsony fogyaszt\u00e1s\u00fa eszk\u00f6z\u00f6k\u00f6n, p\u00e9ld\u00e1ul okosk\u00e1rty\u00e1kon vagy IoT-rendszerekben. Az egyetlen akad\u00e1ly a <strong>kulcsm\u00e9ret<\/strong>, amely tov\u00e1bbra is jelent\u0151s \u2013 de a hardver fejl\u0151d\u00e9s\u00e9vel ez egyre kev\u00e9sb\u00e9 jelent probl\u00e9m\u00e1t.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u00d6sszegz\u00e9s<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A McEliece- \u00e9s Niederreiter-rendszerek val\u00f3di p\u00e9ld\u00e1i annak, hogyan lehet <strong>egy klasszikus matematikai strukt\u00far\u00e1t (hibajav\u00edt\u00f3 k\u00f3d) titkos\u00edt\u00e1si c\u00e9lra alkalmazni<\/strong>. A rendszer egyszer\u0171, gyors, \u00e9s az egyik legink\u00e1bb kvantumbiztos megk\u00f6zel\u00edt\u00e9s, amit ismer\u00fcnk. B\u00e1r a kulcsm\u00e9ret kih\u00edv\u00e1st jelent, a rendszer stabilit\u00e1sa \u00e9s id\u0151t\u00e1ll\u00f3s\u00e1ga miatt <strong>az egyik legkomolyabb jel\u00f6lt a kvantum ut\u00e1ni korszakban<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"<div class=\"seriesmeta\">Ez a bejegyz\u00e9s a 8 r\u00e9sze a 9 sorozatban <a href=\"http:\/\/ai-biztonsag.hu\/?series=kvantumkriptografia\" class=\"series-759\" title=\"Kvantumkriptogr\u00e1fia\">Kvantumkriptogr\u00e1fia<\/a> sorozatban<\/div><p>A McEliece-kriptosziszt\u00e9ma l\u00e9nyege, hogy egy hibajav\u00edt\u00f3 k\u00f3dot haszn\u00e1lunk nyilv\u00e1nos kulcsnak \u00e1lc\u00e1zva. Ez a k\u00f3d bel\u00fcl j\u00f3l struktur\u00e1lt \u2013 p\u00e9ld\u00e1ul egy Goppa-k\u00f3d \u2013, de k\u00edv\u00fclr\u0151l v\u00e9letlenszer\u0171nek t\u0171nik. A felhaszn\u00e1l\u00f3 olyan transzform\u00e1ci\u00f3kat alkalmaz a k\u00f3drendszer gener\u00e1torm\u00e1trix\u00e1n, amelyek elrejtik annak val\u00f3di szerkezet\u00e9t. A t\u00e1mad\u00f3 nem l\u00e1t m\u00e1st, csak egy v\u00e9letlenszer\u0171 m\u00e1trixot \u2013 \u00e9s ez teszi a visszafejt\u00e9st gyakorlatilag &#8230; <a title=\"McEliece- \u00e9s Niederreiter-rendszer m\u0171k\u00f6d\u00e9si elve\" class=\"read-more\" href=\"http:\/\/ai-biztonsag.hu\/?p=666\" aria-label=\"Read more about McEliece- \u00e9s Niederreiter-rendszer m\u0171k\u00f6d\u00e9si elve\">Olvass tov\u00e1bb<\/a><\/p>\n","protected":false},"author":1,"featured_media":673,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_sas_skip_auto_schedule":false,"_sas_force_auto_schedule":false,"footnotes":""},"categories":[760],"tags":[],"series":[759],"class_list":["post-666","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-kvantumszamitastechnika","series-kvantumkriptografia"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>McEliece- \u00e9s Niederreiter-rendszer m\u0171k\u00f6d\u00e9si elve - AI &amp; biztons\u00e1g<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/ai-biztonsag.hu\/?p=666\" \/>\n<meta property=\"og:locale\" content=\"hu_HU\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"McEliece- \u00e9s Niederreiter-rendszer m\u0171k\u00f6d\u00e9si elve - AI &amp; biztons\u00e1g\" \/>\n<meta property=\"og:description\" content=\"A McEliece-kriptosziszt\u00e9ma l\u00e9nyege, hogy egy hibajav\u00edt\u00f3 k\u00f3dot haszn\u00e1lunk nyilv\u00e1nos kulcsnak \u00e1lc\u00e1zva. 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