114 lines
2.8 KiB
Markdown
114 lines
2.8 KiB
Markdown
---
|
|
id: 5900f4231000cf542c50ff35
|
|
title: 'Problem 182: RSA encryption'
|
|
challengeType: 5
|
|
forumTopicId: 301818
|
|
dashedName: problem-182-rsa-encryption
|
|
---
|
|
|
|
# --description--
|
|
|
|
The RSA encryption is based on the following procedure:
|
|
|
|
Generate two distinct primes `p` and `q`.
|
|
Compute `n=p*q` 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=m<sup>e</sup> 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=c<sup>d</sup> mod n.
|
|
|
|
There exist values of `e` and `m` such that m<sup>e</sup> mod n = m.
|
|
We call messages `m` for which m<sup>e</sup> 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 messages
|
|
m `(0≤m≤n-1)` are unconcealed when calculating m<sup>e</sup> 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.
|
|
|
|
For any given `p` and `q`, find the sum of all values of `e`, `1 < e < φ(p,q)` and `gcd(e,φ)=1`, so that the number of unconcealed messages for this value of `e` is at a minimum.
|
|
|
|
# --hints--
|
|
|
|
`RSAEncryption` should be a function.
|
|
|
|
```js
|
|
assert(typeof RSAEncryption === 'function')
|
|
```
|
|
|
|
`RSAEncryption` should return a number.
|
|
|
|
```js
|
|
assert.strictEqual(typeof RSAEncryption(19, 37), 'number');
|
|
```
|
|
|
|
`RSAEncryption(19, 37)` should return `17766`.
|
|
|
|
```js
|
|
assert.strictEqual(RSAEncryption(19, 37), 17766);
|
|
```
|
|
|
|
`RSAEncryption(283, 409)` should return `466196580`.
|
|
|
|
```js
|
|
assert.strictEqual(RSAEncryption(283, 409), 466196580);
|
|
```
|
|
|
|
`RSAEncryption(1009, 3643)` should return `399788195976`.
|
|
|
|
```js
|
|
assert.strictEqual(RSAEncryption(19, 37), 17766);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function RSAEncryption(p, q) {
|
|
|
|
return true;
|
|
}
|
|
|
|
RSAEncryption(19, 37);
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
function gcd(a, b) {
|
|
if (b)
|
|
return gcd(b, a % b);
|
|
else
|
|
return a;
|
|
}
|
|
|
|
function RSAEncryption(p, q) {
|
|
let phi = (p - 1) * (q - 1);
|
|
|
|
let best = Number.MAX_SAFE_INTEGER;
|
|
let sum = 0;
|
|
|
|
for (let e = 0; e < phi; ++e) {
|
|
if (!(gcd(e, phi) == 1))
|
|
continue;
|
|
|
|
let msg = (gcd(p - 1, e - 1) + 1) * (gcd(q - 1, e - 1) + 1);
|
|
|
|
if (best == msg) {
|
|
sum += e;
|
|
} else if (best > msg) {
|
|
best = msg;
|
|
sum = e;
|
|
}
|
|
}
|
|
|
|
return sum;
|
|
}
|
|
```
|