103 lines
2.8 KiB
Markdown
103 lines
2.8 KiB
Markdown
---
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id: 5900f4231000cf542c50ff35
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title: 'Problem 182: RSA encryption'
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challengeType: 5
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forumTopicId: 301818
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dashedName: problem-182-rsa-encryption
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---
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# --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=p*q` and `φ=(p-1)(q-1)`. 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]`. 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.
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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=c<sup>d</sup> mod n.
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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.
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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`. 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.
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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.
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# --hints--
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`RSAEncryption` should be a function.
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```js
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assert(typeof RSAEncryption === 'function')
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```
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`RSAEncryption` should return a number.
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```js
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assert.strictEqual(typeof RSAEncryption(19, 37), 'number');
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```
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`RSAEncryption(19, 37)` should return `17766`.
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```js
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assert.strictEqual(RSAEncryption(19, 37), 17766);
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```
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`RSAEncryption(283, 409)` should return `466196580`.
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```js
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assert.strictEqual(RSAEncryption(283, 409), 466196580);
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```
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`RSAEncryption(1009, 3643)` should return `399788195976`.
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```js
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assert.strictEqual(RSAEncryption(19, 37), 17766);
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```
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# --seed--
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## --seed-contents--
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```js
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function RSAEncryption(p, q) {
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return true;
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}
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RSAEncryption(19, 37);
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```
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# --solutions--
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```js
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function gcd(a, b) {
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if (b)
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return gcd(b, a % b);
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else
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return a;
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}
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function RSAEncryption(p, q) {
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let phi = (p - 1) * (q - 1);
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let best = Number.MAX_SAFE_INTEGER;
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let sum = 0;
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for (let e = 0; e < phi; ++e) {
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if (!(gcd(e, phi) == 1))
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continue;
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let msg = (gcd(p - 1, e - 1) + 1) * (gcd(q - 1, e - 1) + 1);
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if (best == msg) {
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sum += e;
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} else if (best > msg) {
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best = msg;
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sum = e;
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}
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}
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return sum;
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}
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```
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