---
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>