Find number coprime to modulus

Question

I've been doing some research on the RSA encryption system, it's pretty simple to code one using small primes, as there aren't too many possibilities and performance is not a real issue.

Here's how RSA Works:

• First, you generate a n value, product from 2 primes picked up randomly:

  n = p * q
  

  • Then you need your public key, called e. It can be any number coprime to φ(n):

    mcd(φ(n), e) = 1

  • According to the Euler function, φ(n) = (p-1)(q-1)

  • You still need your private key, known as d, which is the inverse of e on modulo **φ(n).

  • So d * e = 1 (mod φ(n))

  • n and e are your public values, when d is your private key. Nobody should know p or q, as with those values it would be possible to get φ(n), and use it to calculate d.
• To encrypt a message, we use M = m^e (mod n), where m is the original message, and M is the encrypted one. So, everybody can send you an encrypted message you can read, by using your public values.
• To decrypt a message, you use d, as it's the inverse of e, so M^d = (m^e)^d = m (mod n). The message can be decrypted, but only using d.

Knowing this, to encrypt a message we just need to get two random primes, calculate n, e, d, and make our encryption and decryption methods.

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This seems pretty easy in theory, now let's turn this into java code:

I first choose two random primes, the bigger the number, the stronger the encryption. So I'll choose p = 1644623 and q = 1644751. According to this page, they are primes.

BigInteger p = new BigInteger("1644623");
BigInteger q = new BigInteger("1644751");

Then, on my init method, I initialize n, e and d:

BigInteger p = new BigInteger("1644623");
BigInteger q = new BigInteger("1644751");

BigInteger n;
BigInteger e;
BigInteger d;

void init() {
    n = p.multiply(q);

    BigInteger pn = p.subtract(BigInteger.ONE).multiply(q.subtract(BigInteger.ONE));

    e = n.subtract(pn);
    while (!mcd(pn, e).equals(BigInteger.ONE)) {
        e = e.add(BigInteger.ONE);
        System.out.println(e);
    }

    d = e.modPow(new BigInteger("-1"), pn);
}

I used BigInteger instead of long, as the values used are extremely big.

In theory, everything works. Practically, pn = 2704992034500, so checking if mcd(pn, e) = 1, and if not adding 1 to e and trying again it's not an option. This program is running since 30 minutes, at an average of 150.000 numbers / second. e = 548151505 and still not found mcd(pn, e) = 1.

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Is there any way to find e in a reasonable time?

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Answer 1

Any prime number will be co-prime to the modulus, by definition, so you don't need to search for one, just pick a prime number. Since the encryption key is public, most RSA implementations use something that makes it easy to compute the modular exponentiation: 3 and 65537 = 2 ** 16 + 1 are both prime, and both are commonly used for the encryption key.