22afc2a0ca
Co-authored-by: Oliver Eyton-Williams <ojeytonwilliams@gmail.com> Co-authored-by: Kristofer Koishigawa <scissorsneedfoodtoo@gmail.com> Co-authored-by: Beau Carnes <beaucarnes@gmail.com>
73 lines
1.9 KiB
Markdown
73 lines
1.9 KiB
Markdown
---
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id: 5900f4231000cf542c50ff35
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challengeType: 5
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isHidden: false
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title: 'Problem 182: RSA encryption'
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forumTopicId: 301818
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---
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## Description
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<section id='description'>
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The RSA encryption is based on the following procedure:
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Generate two distinct primes p and q.Compute n=pq and φ=(p-1)(q-1).
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Find an integer e, 1<e<φ, such that gcd(e,φ)=1.
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A message in this system is a number in the interval [0,n-1].
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A text to be encrypted is then somehow converted to messages (numbers in the interval [0,n-1]).
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To encrypt the text, for each message, m, c=me mod n is calculated.
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To decrypt the text, the following procedure is needed: calculate d such that ed=1 mod φ, then for each encrypted message, c, calculate m=cd mod n.
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There exist values of e and m such that me mod n=m.We call messages m for which me mod n=m unconcealed messages.
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An issue when choosing e is that there should not be too many unconcealed messages. For instance, let p=19 and q=37.
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Then n=19*37=703 and φ=18*36=648.
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If we choose e=181, then, although gcd(181,648)=1 it turns out that all possible messagesm (0≤m≤n-1) are unconcealed when calculating me mod n.
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For any valid choice of e there exist some unconcealed messages.
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It's important that the number of unconcealed messages is at a minimum.
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Choose p=1009 and q=3643.
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Find the sum of all values of e, 1<e<φ(1009,3643) and gcd(e,φ)=1, so that the number of unconcealed messages for this value of e is at a minimum.
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</section>
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## Instructions
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<section id='instructions'>
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</section>
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## Tests
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<section id='tests'>
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```yml
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tests:
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- text: <code>euler182()</code> should return 399788195976.
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testString: assert.strictEqual(euler182(), 399788195976);
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```
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</section>
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## Challenge Seed
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<section id='challengeSeed'>
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<div id='js-seed'>
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```js
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function euler182() {
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// Good luck!
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return true;
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}
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euler182();
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```
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</div>
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</section>
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## Solution
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<section id='solution'>
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```js
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// solution required
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```
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</section>
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