UMAC
In cryptography, a message authentication code based on universal hashing, or UMAC, is a type of message authentication code (MAC) calculated choosing a hash function from a class of hash functions according to some secret (random) process and applying it to the message. The resulting digest or fingerprint is then encrypted to hide the identity of the hash function used. As with any MAC, it may be used to simultaneously verify both the data integrity and the authenticity of a message. UMAC is specified in RFC 4418, it has provable cryptographic strength and is usually a lot less computationally intensive than other MACs. UMAC's design is optimized for 32-bit architectures; a closely related variant of UMAC that is optimized for 64-bit architectures is given by VMAC.
Let's say the hash function is chosen from a class of hash functions H, which maps messages into D, the set of possible message digests. This class is called universal if, for any distinct pair of messages, there are at most |H|/|D| functions that map them to the same member of D.
This means that if an attacker wants to replace one message with another and, from his point of view the hash function was chosen completely randomly, the probability that the UMAC will not detect his modification is at most 1/|D|.
But this definition is not strong enough — if the possible messages are 0 and 1, D={0,1} and H consists of the identity operation and not, H is universal. But even if the digest is encrypted by modular addition, the attacker can change the message and the digest at the same time and the receiver wouldn't know the difference.
A class of hash functions H that is good to use will make it difficult for an attacker to guess the correct digest d of a fake message f after intercepting one message a with digest c. In other words,
needs to be very small, preferably 1/|D|.
It is easy to construct a class of hash functions when D is field. For example, if |D| is prime, all the operations are taken modulo |D|. The message a is then encoded as an n-dimensional vector over D (a1, a2, ..., an). H then has |D|n+1 members, each corresponding to an (n + 1)-dimensional vector over D (h0, h1, ..., hn). If we let
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