Why isn't ElGamal a symmetric-key scheme? I was asked about this question by a friend.
He considers that Alice and Bob share a secret key s=g^(xy).
So why isn't it considered a symmentric-key scheme?
According to Wikipedia:


  Symmetric-key algorithms[1] are algorithms for cryptography that use the same cryptographic keys for both encryption of plaintext and decryption of ciphertext. The keys may be identical or there may be a simple transformation to go between the two keys.


I observe that the shared s is quite easy to compute by Alice and Bob.

I know there's misunderstanding, but I can't convince myself now.
g^x is the public key of Alice and x her private key, while g^y is the public key of Bob and y his private key.

s=g^(xy) is the shared secret between Alice and bob, that can only be computed by them.

Thus, ElGamal is an asymmetric algorithm computing a shared secret that can be used as a symmetric key.