How Secure Is RSA Algorithm?

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∟How Secure Is RSA Algorithm?

This section discusses the security of RSA public key encryption algorithm. RSA private key is not 100% secure. But if the private key uses larger value of n = p*q, it will take a very long time to crack the private key.

The security of RSA public key encryption algorithm is mainly based on the integer factorization problem, which can be described as:

Given integer n as the product of 2 distinct prime numbers p and q, 
find p and q.

If the above problem could be solved, the RSA encryption is not secure at all. This is because the public key {n,e} is known to the public. Any one can use the public key {n,e} to figure out the private key {n,d} using these steps:

Compute p and q by factorizing n.
Compute m = (p-1)*(q-1).
Compute d such that d*e mod m = 1 to obtain the private key {n,d}

If n is small, the integer factorization problem is easy to solve by testing all possible prime numbers in the range of (1, n).

For example, given 35 as n, we can list all prime numbers in the range of (1, 35): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31, and try all combinations of them to find 5 and 7 are factors of 35.

As the value of n gets larger, the integer factorization problem gets harder to solve. But it is still solvable with the use of computers. For example, the RSA-100 number with 100 decimal digits, or 330 bits, has been factored by Arjen K. Lenstra in 1991:

RSA-100 = 15226050279225333605356183781326374297180681149613
          80688657908494580122963258952897654000350692006139
 
    p*q = 37975227936943673922808872755445627854565536638199
        * 40094690950920881030683735292761468389214899724061

If you are using the above RSA-100 number as n, your private key is not private any more.

As of today, the highest value of n that has been factored is RSA-678 number with 232 decimal digits, or 768 bits, factored by Thorsten Kleinjung et al. in 2009:

RSA-768 = 12301866845301177551304949583849627207728535695953
          34792197322452151726400507263657518745202199786469
          38995647494277406384592519255732630345373154826850
          79170261221429134616704292143116022212404792747377
          94080665351419597459856902143413

    p*q = 33478071698956898786044169848212690817704794983713
          76856891243138898288379387800228761471165253174308
          7737814467999489
        × 36746043666799590428244633799627952632279158164343
          08764267603228381573966651127923337341714339681027
          0092798736308917