I have studied the design of block ciphers like DES and AES.
I know the actual algorithms, but I don't intuitively "grok" the most important property of them:
The most important property, is that if you have the key and the ciphertext, you can calculate easily the plaintext, but if you have the plaintext and ciphertext, you cannot easily calculate the key.
That is, the block cipher is a function $$f(P,K)= C$$ and the essential property is that the inverse w.r.t. P is very easily caclulated, but the inverse w.r.t. K is not.
I don't intuitively understand why this assymmetry is true, even though I know how to actually do the decryption algorithm. Is there an intuitive explanation of this asymmetry?