79d9012432
* fix(curriculum): tests quotes * fix(curriculum): fill seed-teardown * fix(curriculum): fix tests and remove unneeded seed-teardown
70 lines
1.9 KiB
Markdown
70 lines
1.9 KiB
Markdown
---
|
|
id: 5900f4231000cf542c50ff35
|
|
challengeType: 5
|
|
title: 'Problem 182: RSA encryption'
|
|
---
|
|
|
|
## Description
|
|
<section id='description'>
|
|
The RSA encryption is based on the following procedure:
|
|
Generate two distinct primes p and q.Compute n=pq and φ=(p-1)(q-1).
|
|
Find an integer e, 1<e<φ, such that gcd(e,φ)=1.
|
|
A message in this system is a number in the interval [0,n-1].
|
|
A text to be encrypted is then somehow converted to messages (numbers in the interval [0,n-1]).
|
|
To encrypt the text, for each message, m, c=me mod n is calculated.
|
|
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.
|
|
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.
|
|
An issue when choosing e is that there should not be too many unconcealed messages. For instance, let p=19 and q=37.
|
|
Then n=19*37=703 and φ=18*36=648.
|
|
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.
|
|
For any valid choice of e there exist some unconcealed messages.
|
|
It's important that the number of unconcealed messages is at a minimum.
|
|
Choose p=1009 and q=3643.
|
|
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.
|
|
</section>
|
|
|
|
## Instructions
|
|
<section id='instructions'>
|
|
|
|
</section>
|
|
|
|
## Tests
|
|
<section id='tests'>
|
|
|
|
```yml
|
|
tests:
|
|
- text: <code>euler182()</code> should return 399788195976.
|
|
testString: assert.strictEqual(euler182(), 399788195976, '<code>euler182()</code> should return 399788195976.');
|
|
|
|
```
|
|
|
|
</section>
|
|
|
|
## Challenge Seed
|
|
<section id='challengeSeed'>
|
|
|
|
<div id='js-seed'>
|
|
|
|
```js
|
|
function euler182() {
|
|
// Good luck!
|
|
return true;
|
|
}
|
|
|
|
euler182();
|
|
```
|
|
|
|
</div>
|
|
|
|
|
|
|
|
</section>
|
|
|
|
## Solution
|
|
<section id='solution'>
|
|
|
|
```js
|
|
// solution required
|
|
```
|
|
</section>
|