A secure hash (also called cryptographically secure hash, or cryptographic hash) is a hash function
- Cannot be reversed - you can't find an item that matches a particular hash value
- Cannot be forced - you can't find two items with the same hash value
- Does not show any relation between two similar items - you cannot recompute the hash for an item you do not fully know based on another item you do not fully know, but where you know the relation between them.
... in a computionally feasible amount of time, of course.
Secure hashes are used to make message digests for digital signatures, and in most implementation of Message Authentication Codes, called HMACs (for Hash Message Authentication Code).
Examples of secure hashes are SHA-1 and MD5.
The reason for the high length of secure hash functions (usually 128 or 160 bits) is birthday attacks.