{"id":24063,"date":"2022-11-30T05:26:28","date_gmt":"2022-11-29T23:56:28","guid":{"rendered":"https:\/\/jassweb.com\/solved\/solved-modular-exponentiation-power-in-modular-arithmetic-closed\/"},"modified":"2022-11-30T05:26:28","modified_gmt":"2022-11-29T23:56:28","slug":"solved-modular-exponentiation-power-in-modular-arithmetic-closed","status":"publish","type":"post","link":"https:\/\/jassweb.com\/solved\/solved-modular-exponentiation-power-in-modular-arithmetic-closed\/","title":{"rendered":"[Solved] Modular Exponentiation (Power in Modular Arithmetic) [closed]"},"content":{"rendered":"<p> [ad_1]<br \/>\n<\/p>\n<div id=\"answer-45567391\" class=\"answer js-answer accepted-answer js-accepted-answer\" data-answerid=\"45567391\" data-parentid=\"45566228\" data-score=\"5\" data-position-on-page=\"1\" data-highest-scored=\"1\" data-question-has-accepted-highest-score=\"1\" itemprop=\"acceptedAnswer\" itemscope itemtype=\"https:\/\/schema.org\/Answer\">\n<div class=\"post-layout\">\n<div class=\"votecell post-layout--left\"><\/div>\n<div class=\"answercell post-layout--right\">\n<div class=\"s-prose js-post-body\" itemprop=\"text\">\n<p>From the modulo laws on DAle&#8217;s linked Wikipedia page (on your previous question), we can obtain two formulas:<\/p>\n<p><a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/11\/Solved-Modular-Exponentiation-Power-in-Modular-Arithmetic-closed.gif\"><img decoding=\"async\" src=\"https:\/\/jassweb.com\/solved\/wp-content\/uploads\/2022\/11\/Solved-Modular-Exponentiation-Power-in-Modular-Arithmetic-closed.gif\" alt=\"enter image description here\"><\/a><\/p>\n<p>From the first formula it is clear that we can iteratively calculate the modulo for <code>n<\/code> from the result for <code>n \/ 2<\/code>. This is done by the lines<\/p>\n<pre><code>x = x * x;\nx = x % m;\n<\/code><\/pre>\n<p>There are thus <code>log n<\/code> steps in the algorithm, because each time the exponent of <code>x<\/code> doubles. The step counting is done by <code>n &gt;&gt;= 1<\/code> and <code>while (n &gt; 0)<\/code>, which counts <code>log n<\/code> steps.<\/p>\n<p>Now, you may be wondering 1) why doesn&#8217;t this part set the value of <code>res<\/code>, and 2) what is the purpose of these lines<\/p>\n<pre><code>if(n &amp; 1){\n   res = res * x;\n   res = res % m;\n}\n<\/code><\/pre>\n<p>This is necessary as at certain points in the iteration, be it the start or the end, the value of <code>n<\/code> may be odd. We can&#8217;t just ignore it and keep using formula 1, because that means we would <em>skip<\/em> a power of <code>x<\/code>! (Integer division rounds down, so e.g. <code>5 &gt;&gt; 1 = 2<\/code>, and we would have <code>x^4<\/code> instead of <code>x^5<\/code>). This if statement handles the case when <code>n<\/code> is odd, i.e. <code>n % 2 = n &amp; 1 = 1<\/code>. It simply uses formula 2 above to &#8220;add&#8221; a single power of <code>x<\/code> to the result.<\/p>\n<\/p><\/div>\n<div class=\"mt24\"><\/div>\n<\/div>\n<p>            <span class=\"d-none\" itemprop=\"commentCount\"><\/span> <\/p><\/div>\n<\/div>\n<p>[ad_2]<\/p>\n<p>solved Modular Exponentiation (Power in Modular Arithmetic) [closed] <\/p>\n","protected":false},"excerpt":{"rendered":"<p>[ad_1] From the modulo laws on DAle&#8217;s linked Wikipedia page (on your previous question), we can obtain two formulas: From the first formula it is clear that we can iteratively calculate the modulo for n from the result for n \/ 2. This is done by the lines x = x * x; x = &#8230; <a title=\"[Solved] Modular Exponentiation (Power in Modular Arithmetic) [closed]\" class=\"read-more\" href=\"https:\/\/jassweb.com\/solved\/solved-modular-exponentiation-power-in-modular-arithmetic-closed\/\" aria-label=\"More on [Solved] Modular Exponentiation (Power in Modular Arithmetic) [closed]\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[320],"tags":[457,324,357],"class_list":["post-24063","post","type-post","status-publish","format-standard","hentry","category-solved","tag-algorithm","tag-c","tag-math"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>[Solved] Modular Exponentiation (Power in Modular Arithmetic) [closed] - JassWeb<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/jassweb.com\/solved\/solved-modular-exponentiation-power-in-modular-arithmetic-closed\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"[Solved] Modular Exponentiation (Power in Modular Arithmetic) [closed] - JassWeb\" \/>\n<meta property=\"og:description\" content=\"[ad_1] From the modulo laws on DAle&#8217;s linked Wikipedia page (on your previous question), we can obtain two formulas: From the first formula it is clear that we can iteratively calculate the modulo for n from the result for n \/ 2. 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