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The security of this system is based on the assumption that knowing the encrypting keys n and e does not allow one to compute the decrypting keys n and d. However, there are algorithms for doing exactly that!In fact, if p and q are so large that their binary rep- resentations require hundreds of digits, then the best known factoring algo-
rithms would require years before the identities of p and q could be revealed from n. In turn, the content of an encrypted message would remain secure long after its significance had deteriorated.
The security of this system is based on the assumption that knowing the encrypting keys n and e does not allow one to compute the decrypting keys n and d. However, there are algorithms for doing exactly that! One approach would be to factor the value n to discover the values p and q, and then to
determine d by finding a value k such that k(p - 1)(q - 1) + 1 is evenly divided by e (the quotient would then be d). On the other hand, the first step
in this process can be time-consuming—especially if the values of p and q were chosen to be large. In fact, if p and q are so large that their binary rep- resentations require hundreds of digits, then the best known factoring algo-
rithms would require years before the identities of p and q could be revealed from n. In turn, the content of an encrypted message would remain secure long after its significance had deteriorated.
To this date, no one has found an efficient way of decrypting messages based on RSA cryptography without knowing the decrypting keys, and thus, public-key encryption based on the RSA algorithm is widely used to obtain privacy when communicating over the Internet.
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