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id	title	challengeType	forumTopicId	dashedName
5900f4231000cf542c50ff35	Problem 182: RSA encryption	5	301818	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^e 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^d mod n.

There exist values of e and m such that m^e mod n = m. We call messages m for which m^e 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^e 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.

assert(typeof RSAEncryption === 'function')

RSAEncryption should return a number.

assert.strictEqual(typeof RSAEncryption(19, 37), 'number');

RSAEncryption(19, 37) should return 17766.

assert.strictEqual(RSAEncryption(19, 37), 17766);

RSAEncryption(283, 409) should return 466196580.

assert.strictEqual(RSAEncryption(283, 409), 466196580);

RSAEncryption(1009, 3643) should return 399788195976.

assert.strictEqual(RSAEncryption(19, 37), 17766);

--seed--

--seed-contents--

function RSAEncryption(p, q) {

  return true;
}

RSAEncryption(19, 37);

--solutions--

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;
}

2.8 KiB Raw Blame History