Length of key: symmetric vs asymmetric algorithms I am studying the fundamental concept of cryptography. I found the following statements:


You can compare symmetric algorithms based on the key (e.g., CAST-128 bit “weaker” than AES-256)
You cannot compare asymmetric algorithms based on key length.
More importantly, never compare asymmetric vs. symmetric key lengths!


The third point seems obvious. Can you explain me the other two?
Point 1: If two symmetric algorithms with no attacks better than brute force but different key lengths are to be compared, then simply comparing the key lengths tells you which is stronger. A larger key is stronger. Note that keys larger than 256 bits are typically slower and provide no practical increase in security (as in, even with quantum computers they'll take centuries to brute-force). Note that the "no attacks better than brute force" is the key here, FEAL-NX with a 128-bit key is thoroughly broken and far less secure than AES with a 128-bit key.

Point 2: The security of asymmetric algorithms is less related to key length because there are attacks better than brute force for all existing asymmetric algorithms. For a trivial example RSA uses large prime numbers, which must be odd, and thus one can skip the even numbers when performing trial division, thus dividing the brute force search time in half. The actual improvements on brute force search are more complicated and vary significantly from algorithm to algorithm.

Point 3: As symmetric and asymmetric ciphers are totally different things with different purposes it's like comparing the cargo capacity of a freight train with that of a container ship. The same units (mass or volume) are used, but they do very different things and can't simply be exchanged. Trains don't float.