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Let's $\ a,b\ $ be two (pseudo)random independent values, and $m$ be a message, where $a,b,m \in \mathtt{F}_p$, where $p$ is a large prime number.


Question 1: Is $\ am+b\ $ an information theoretic MAC?

if yes

Question 2: Is there any paper/textbook uses it?

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    $\begingroup$ You need $a \neq 0 \pmod p$. $\endgroup$ – CodesInChaos Feb 7 '17 at 14:35
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    $\begingroup$ This is the single block special case of polynomial MACs. See Poly1305 or GHash. $\endgroup$ – CodesInChaos Feb 7 '17 at 14:37
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    $\begingroup$ If $a,b$ are pseudorandom and not truly random, then it cannot be information theoretic. $\endgroup$ – Yehuda Lindell Feb 7 '17 at 14:59
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    $\begingroup$ Unless a pair $a, b$ is used for only one message, it can't be informational theoretic (or even practically secure). And, changing the key for each message isn't a standard property of MACs. $\endgroup$ – poncho Feb 7 '17 at 22:01
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Just collecting some of the comments, if you make sure that $a$ and $b$ are chosen uniformly at random with $a \neq 0 \mod p$, then the given function is a information-theoretic MAC, basically because it satisfies the following property:

If $h_k(m) = am + b$ with $k= (a,b)$ and $\tau$ is the codomain of $h_k$, then for all $m\neq m'$ and $t,t'\in\tau$ we have $$\operatorname{Pr}[h_k(m) = t \ \wedge\ h_k(m') = t'] = \frac{1}{|\tau|^2}$$ (probability taken over random choice of $k$)

When a function satisfy this property we say it is a strongly universal function, and the main result is that such a function is a $1/|\tau| -$secure MAC, that is, the advantage of an attacker is at most $1/|\tau|$ (notice we can not ask for this advantage to be zero because any adversary can always try to guess a valid tag).

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  • $\begingroup$ If I have two messages $m_0, m_1$ and get the MAC values for those two messages $h_k(m_0)$ and $h_k(m_1)$, I can compute the MAC $h_k(m_2)$ for any message $m_2$. How is this information-theoretic? $\endgroup$ – poncho Feb 7 '17 at 23:08
  • $\begingroup$ @poncho This is one-time information-theoretic. I don't know about any other information-theoretic definition (this definition comes from Katz-Lindell book) $\endgroup$ – Daniel Feb 7 '17 at 23:08
  • $\begingroup$ IMHO, if it doesn't meet the standard definition of a Message Authentication Code, it probably should be called something else; possibly a Universal Hash function... $\endgroup$ – poncho Feb 7 '17 at 23:16
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    $\begingroup$ @poncho I partially agree with you, but consider for example OTP: it is information-theoretic secure using a one-time game, it makes certain sense to involve an analogous scenario for the MAC context (but I am aware that the MAC security definition involves oracle access, which is not the case for EAV-secrecy). However, I'm a beginning on this, so I really do not know. $\endgroup$ – Daniel Feb 7 '17 at 23:25

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