Hash function that produces short hashes?

You can use any commonly available hash algorithm (eg. SHA-1), which will give you a slightly longer result than what you need. Simply truncate the result to the desired length, which may be good enough.

For example, in Python:

>>> import hashlib
>>> hash = hashlib.sha1("my message".encode("UTF-8")).hexdigest()
>>> hash
'104ab42f1193c336aa2cf08a2c946d5c6fd0fcdb'
>>> hash[:10]
'104ab42f11'