Assume $F$ is a pseudo-random permutation (PRP) then the tweakable block-cipher based on the Merkle-Damgård construction (take this as the way I understand, here is the equation):

$F_k[t](m) := F_{F_k(t)}(m)$

is a secure tweakable block cipher. This is marked with two stars in my notes.

First question: is it right or wrong?

Second question (if it is right): will a standard argument - that distinguishes the real world from the ideal world - work?

  • 1
    $\begingroup$ Minematsu (Section 5) showed this is secure up to $2^{n/2}$ queries for an $n$-bit tweak. $\endgroup$ Dec 27 '18 at 17:12
  • $\begingroup$ @SamuelNeves care to turn that into an answer? $\endgroup$
    – Maarten Bodewes
    Dec 28 '18 at 2:03

Minematsu studied this construction, and some generalizations thereof, in Section 5 of Beyond-Birthday-Bound Security Based on Tweakable Block Cipher. Minematsu, however, uses a PRF to derive keys instead of the block cipher.

The idea is simple: the tweakable SPRP security of this construction, for $q$ queries distributed arbitrarily over $t \le q$ different tweaks, is the security of $t$ instances of $F$, plus the distance from the key derivation to uniformly distributed keys.

The security of $t$ instances of $F$ is otherwise known as multi-key or multi-user security, and we can bound it as $t \cdot \mathbf{Adv}_{F}^{\text{sprp}}(\mathcal{D})$, i.e., at most $t$ times the security of any single instance.

So we have $$ \mathbf{Adv}^{\widetilde{\text{sprp}}}_{F}(\mathcal{D}) \le \mathbf{Adv}^{\text{prf}}_F(\mathcal{D'}) + t \cdot \mathbf{Adv}^{\text{sprp}}_{F}(\mathcal{D''}) \,, $$ for a distinguisher $\mathcal{D'}$ making $t$ queries and a distinguisher $\mathcal{D''}$ making at most $q$ queries. The first distance can be refined by the PRP-PRF switch, so we end up with $$ \mathbf{Adv}^{\widetilde{\text{sprp}}}_{F}(\mathcal{D}) \le \frac{\binom{q}{2}}{2^n} + (t + 1) \cdot \mathbf{Adv}^{\text{sprp}}_{F}(\mathcal{D''}) \,. $$

So this construction is secure until approximately $2^{n/2}$ blocks are queried and, assuming a perfect block cipher with $\mathbf{Adv}(\mathcal{D}) \le t/2^k$, around $2^k/t$ evaluations.

Imagine for example a concrete instance of AES-128 where an attacker queries $2^{64}$ different tweaks $x_i = \text{AES}_{\text{AES}_k(t_i)}(0)$ then tries $2^{64}$ keys $k_i$ until $\text{AES}_{k_j}(0) = x_i$. Each attempt has success probability $2^{64-128}$, since there are $2^{64}$ distinct keys. Then, $\text{AES}_{\text{AES}_k(t_i)}(1)$ must also be equal to $\text{AES}_{k_j}(1)$.


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