{"id":363,"date":"2018-04-21T20:59:19","date_gmt":"2018-04-21T11:59:19","guid":{"rendered":"http:\/\/tamatoyaku.com\/b\/?p=363"},"modified":"2018-04-21T20:59:19","modified_gmt":"2018-04-21T11:59:19","slug":"363","status":"publish","type":"post","link":"https:\/\/p-0.me\/b\/p\/363\/","title":{"rendered":"Montgomery curves\u3067y\u5ea7\u6a19\u3092\u5fa9\u5143\u3059\u308b\u3068\u304d\u306e\u8a71"},"content":{"rendered":"<p>[mathjax]\u3061\u3087\u3063\u3068\u30cf\u30de\u3063\u305f\u306e\u3067\u30e1\u30e2\uff0e<br \/>\n<!--more--><br \/>\n<a href=\"https:\/\/tamatoyaku.com\/b\/p\/361\">\u524d\u56de<\/a>\uff0c\u3042\u308b\u516c\u5f0f\u3092\u7528\u3044\u3066y\u5ea7\u6a19\u306e\u5fa9\u5143\u3092\u884c\u3063\u305f\uff0e\u305d\u306e\u516c\u5f0f\u306fHandbook of Elliptic and Hyperelliptic Curve Cryptography\u306b\u8a18\u8f09\u3057\u3066\u3042\u308a\uff0c\u305d\u306e\u5143\u3068\u306a\u3063\u305f\u306e\u306f<a href=\"https:\/\/link.springer.com\/content\/pdf\/10.1007\/3-540-44709-1_12.pdf\">Efficient Elliptic Curve Cryptosystems from a Scalar Multiplication Algorithm with Recovery of the y-Coordinate on a Montgomery-Form Elliptic Curve<\/a>\u306eTheorem2\u3067\u3042\u308b\uff0e\u8ad6\u6587\u306e\u65b9\u306b\u306f\\(y\\neq0\\)\u3068\u3059\u308b\u3088\u3046\u306b\u66f8\u304b\u308c\u3066\u3044\u308b\u304c\uff0c\u7121\u9650\u9060\u70b9\u306b\u95a2\u3059\u308b\u8a18\u8ff0\u306f\u898b\u3064\u3051\u3089\u308c\u306a\u304b\u3063\u305f\uff0e\u4ee5\u4e0b\u306e\u5f0f\u3067\\(P_{n+1}=(x_{n+1},y_{n+1})=\\omicron\\)\uff0c\u3064\u307e\u308a\\(P_{n+1}\\)\u304c\u7121\u9650\u9060\u70b9\u3068\u306a\u308b\u5834\u5408\u306f\uff0c\u7121\u9650\u9060\u70b9\u306b\u5bfe\u5fdc\u3059\u308b\u30a2\u30d5\u30a3\u30f3\u5ea7\u6a19\u304c\u5b58\u5728\u3057\u306a\u3044\u305f\u3081\uff0c\u6b63\u3057\u304f\u6c42\u3081\u3089\u308c\u306a\u3044\uff0e<br \/>\n$$y_n = \\frac{(x_1 x_n +1)(x_1 + x_n +2A)-2A-(x_1-x_n)^2x_{n+1}}{2By_1}$$<br \/>\n\\(P_{n+1}=\\omicron\\)\u3068\u306a\u308b\u3068\u304d\uff0c\u4ee5\u4e0b\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064\uff0e<br \/>\n$$P_{n+1}=P_1+P_n=\\omicron$$<br \/>\n$$P_n=-P_1$$<br \/>\n$$(x_n,y_n)=(x_1,-y_1)$$<br \/>\n\u3088\u3063\u3066\uff0c\\(P_{n+1}\\)\u304c\u7121\u9650\u9060\u70b9\u3068\u306a\u308b\u5834\u5408\u306f\\(y_n=-y_1\\)\u3068\u306a\u308b\uff0e<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[mathjax]\u3061\u3087\u3063\u3068\u30cf\u30de\u3063\u305f\u306e\u3067\u30e1\u30e2\uff0e<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-363","post","type-post","status-publish","format-standard","hentry","category-4"],"_links":{"self":[{"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/posts\/363","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/comments?post=363"}],"version-history":[{"count":0,"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/posts\/363\/revisions"}],"wp:attachment":[{"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/media?parent=363"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/categories?post=363"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/tags?post=363"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}