MAC-CDZ
Universal Families
editAn ϵ-almost ∆-universal (ϵ-A∆U)
editDefinition of An ϵ-almost ∆-universal (ϵ-A∆U)
editLet be an Abelian group. A family H of hash functions that maps from a set A to the set B is said to be ϵ-almost ∆-universal (ϵ-A∆U) w.r.t. , if for any distinct elements and for all :
H is ∆-universal (∆U) if .
An ϵ-almost universal family or (ϵ-AU)family
editAn ϵ-almost universal family or (ϵ-AU)family is one type of family in the universal hash function.
Definition of (ϵ-AU)family
editLet ϵ be any positive real number. An ϵ-almost universal (ϵ-AU) family H of hash functions mapping from a set A to a set B is a family of functions from A to B, such that for any distinct elements :
H is universal (U) if .
The definition above states that the probability of a collision is at most ϵ for any two distinct inputs. In the case is called universal, the smallest possible value for
An ϵ-almost strongly-universal family or (ϵ-ASU)family
editAn ϵ-almost strongly universal family or (ϵ-ASU)family is one type of family in the universal hash function.
Definition of (ϵ-ASU)family
editLet ϵ be any positive real number. An ϵ-almost strongly-universal (ϵ-ASU) family H of Hash functions maps from a set A to a set B is a family of functions from A to B, such that for any distinct elements and all :
and
H is strongly universal (SU) if .
The first condition states that the probability that a given input a is mapped to a given output b equals . The second condition implies that if a is mapped to b, then the conditional probability that is mapped to is upper bounded by ϵ.
ENH
editThis page is under construction
Theorem.1
Let be an ϵ-AΔU hash family from a set A to a set B. Consider a message . Then the family H consisting of the functions is ϵ-AU.
If , then this probability is at most ϵ, since is an ϵ-A∆U family. If but , then the probability is trivially 0. The proof for Theorem was described in [1]
ENH- family is a very fast universal hash family is the NH family used in UMAC:
Where ‘ ’ means ‘addition modulo ’, and . It is a -A∆U hash family.
Lemma.1
The following version of NH is -A∆U:
The proof for lemma.1 was described in[1]
Choosing w=32 and applying Theorem.1, One can obtain the -AU function family ENH, which will be the basic building block of MAC:
where all arguments are 32-bit and the output is 64-bit.
Welcome
editWelcome!
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We remove /* Proposed deletion of MAC (Message Authentication Code) – Wegman Carter */ by considering that the article is more to MMH and Badger which are the kinds of MAC Wegman-Carter. We also change the title to MMH-Badger MAC so that it will not overlap with the previous discussion about MAC.