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puzzles/project_euler/pe/p182.html
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puzzles/project_euler/pe/p182.html
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<meta charset="utf-8" />
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<meta name="author" content="Colin Hughes" />
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<meta name="description" content="A website dedicated to the fascinating world of mathematics and programming" />
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<meta name="keywords" content="programming,mathematics,problems,puzzles" />
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<title>Problem 182 - Project Euler</title>
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<h2>RSA encryption</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 182</h3><span style="width:300px;color:#666;">Published on Friday, 15th February 2008, 01:00 pm; Solved by 1338</span></div>
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<div class="problem_content" role="problem">
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<p>The RSA encryption is based on the following procedure:</p>
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<p>Generate two distinct primes <var>p</var> and <var>q</var>.<br />Compute <var>n=pq</var> and φ=(<var>p</var>-1)(<var>q</var>-1).<br />
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Find an integer <var>e</var>, 1<img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /><var>e</var><img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' />φ, such that gcd(<var>e</var>,φ)=1.</p>
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<p>A message in this system is a number in the interval [0,<var>n</var>-1].<br />
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A text to be encrypted is then somehow converted to messages (numbers in the interval [0,<var>n</var>-1]).<br />
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To encrypt the text, for each message, <var>m</var>, <var>c</var>=<var>m</var><sup><var>e</var></sup> mod <var>n</var> is calculated.</p>
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<p>To decrypt the text, the following procedure is needed: calculate <var>d</var> such that <var>ed</var>=1 mod φ, then for each encrypted message, <var>c</var>, calculate <var>m=c<sup>d</sup></var> mod <var>n</var>.</p>
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<p>There exist values of <var>e</var> and <var>m</var> such that <var>m<sup>e</sup></var> mod <var>n=m</var>.<br />We call messages <var>m</var> for which <var>m<sup>e</sup></var> mod <var>n=m</var> unconcealed messages.</p>
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<p>An issue when choosing <var>e</var> is that there should not be too many unconcealed messages. <br />For instance, let <var>p</var>=19 and <var>q</var>=37.<br />
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Then <var>n</var>=19*37=703 and φ=18*36=648.<br />
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If we choose <var>e</var>=181, then, although gcd(181,648)=1 it turns out that all possible messages<br />
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<var>m</var> (0<img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /><var>m</var><img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /><var>n</var>-1) are unconcealed when calculating <var>m<sup>e</sup></var> mod <var>n</var>.<br />
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For any valid choice of <var>e</var> there exist some unconcealed messages.<br />
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It's important that the number of unconcealed messages is at a minimum.</p>
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<p>Choose <var>p</var>=1009 and <var>q</var>=3643.<br />
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Find the sum of all values of <var>e</var>, 1<img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /><var>e</var><img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' />φ(1009,3643) and gcd(<var>e</var>,φ)=1, so that the number of unconcealed messages for this value of <var>e</var> is at a minimum.</p>
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