User talk:MAC-CDZ

= Universal Families =

Definition of An ϵ-almost ∆-universal (ϵ-A∆U)
Let $$(B,+)$$ 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. $$(B,+)$$, if for any distinct elements $$a, a^' \in A $$ and for all $$\delta \in B $$: $$Pr_{h \in H}[h(a)-h(a^')=\delta] \le \epsilon $$

H is ∆-universal (∆U) if $$ \epsilon = \frac {1}{\left\vert B \right\vert}$$.

An ϵ-almost universal family or (ϵ-AU)family
An ϵ-almost universal family or (ϵ-AU)family is one type of family in the universal hash function.

Definition of (ϵ-AU)family
Let 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 $$a, a^' \in A $$: $$Pr_{h \in H}[h(a)=h(a^')] \le \epsilon $$

H is universal (U) if $$\epsilon = \frac {1}{\left\vert B \right\vert}$$. The definition above states that the probability of a collision is at most ϵ for any two distinct inputs. In the case $$\epsilon = \frac {1}{\left\vert B \right\vert}$$ is called universal, the smallest possible value for $$\epsilon =\frac {a-b}{b(a-1)}$$

An ϵ-almost strongly-universal family or (ϵ-ASU)family
An ϵ-almost strongly universal family or (ϵ-ASU)family is one type of family in the universal hash function.

Definition of (ϵ-ASU)family
Let 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 $$ a, a^'\in A $$ and all $$ b, b^'\in B$$: $$Pr_{h \in H}[h(a)=b] = \frac {1}{\left\vert B \right\vert} $$

and $$Pr_{h \in H}[h(a)=b, h(a')=b'] = \frac {\epsilon}{\left\vert B \right\vert} $$

H is strongly universal (SU) if $$\epsilon = \frac {1}{\left\vert B \right\vert}$$. The first condition states that the probability that a given input a is mapped to a given output b equals $$\frac {1}{\left\vert B \right\vert}$$. The second condition implies that if a is mapped to b, then the conditional probability that $$a^' (a \ne a^')$$ is mapped to $$b^' (b \ne b^')$$ is upper bounded by ϵ.

=ENH= This page is under construction

Theorem.1

Let $$H^\triangle$$ be an ϵ-AΔU hash family from a set A to a set B. Consider a message $$(m, m_b) \in A \times B $$. Then the family H consisting of the functions $$h(m,m_b) = H^\triangle (m) + m_b $$ is ϵ-AU.

If $$ m \ne m^'$$, then this probability is at most ϵ, since $$H^\triangle$$ is an ϵ-A∆U family. If $$ m \ne m^'$$ but $$ m_b=m_b^'$$, 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:

$$NH_K (M)= \sum_{i=1}^ \frac{l}{2} (k_{(2i-1)} +_w m_{(2i-1)})\times (k_{2i} +_w m_{2i} ) \mod 2^{2w} $$

Where ‘$$+_w$$’ means ‘addition modulo $$2^w$$’, and $$m_i,k_i \in \big\{0,\cdots, 2^w-1\big\}$$. It is a $$2^{-w}$$-A∆U hash family.

Lemma.1

The following version of NH is $$2^{-w}$$-A∆U:

$$NH_K (M)=(k_1 +_w m_1 )\times(k_2 +_w m_2 ) \mod 2^{2w}$$

The proof for lemma.1 was described in[1]

Choosing w=32 and applying Theorem.1, One can obtain the $$2^{-32}$$-AU function family ENH, which will be the basic building block of MAC:

$$ENH_{k_1,k_2} (m_1,m_2,m_3,m_4 )=(m_1 +_{32} k_1)(m_2 +_{32} k_2) +_{64} m_3 +_{64} 2^{32} m_4$$

where all arguments are 32-bit and the output is 64-bit.

Welcome
Welcome!

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Removing Proposed deletion of MAC (Message Authentication Code) – Wegman Carter
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.

MAC-CDZ (talk) 04:47, 26 January 2011 (UTC)