[agwa]

That's not how the math works.

First, 2048 bits is not 512 bits doubled twice, but rather doubled 1536 times (512 doubled twice would be 514). If this were a symmetric cipher, you could stop here and conclude that a 2048 bit key was 2^1536 times stronger than a 512 bit key.

However, RSA has diminishing returns on security as you increase the key length. The strength is determined by the complexity of the GNFS, the fastest known way of breaking RSA[1]. That tells us that breaking 256-bit RSA takes ~2^46 operations, 512-bit RSA takes ~2^63, 2048-bit RSA takes ~2^116, and 4096-bit RSA takes ~2^156. 2^116 is a lot of operations - they say the amount of energy required to break that would be nearly enough to boil all the water on earth.

[1] http://crypto.stackexchange.com/questions/8687/security-stre...

[mikekchar]

> they say the amount of energy required to break that would be nearly enough to boil all the water on earth.

Not wanting to be alarmist, but what you're saying is that someone breaking my ssh key (which is 2048 bits) is the end of the world...

[therealmarv]

great answer!