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Does applying a strong hash function like SHA-256 to the ECB-encryption of a message (using some secret key $K$) produce a secure mac? For example, given a message $m$, would a simple mac construction $H(E_K(m))$ be considered a secure mac if we used a strong hash $H$ like SHA-256?

Compared to standard HMAC, this construction seems simpler and might even execute a little faster too. Also, it doesn't seem like this mac scheme is vulnerable to length extension attacks either since without knowledge of $K$, it doesn't seem like the attacker can "extend" the input to the hash function $H$ since the output of $E_K(m)$ never gets "exposed" to the attacker but only consumed as just some intermediate computation step inside of $H(E_K(m))$.

Of course, the standard $\text{HMAC}(K,m)$ construction is likely more secure against the usage of "weak hash functions", so I'm purposefully requiring $H$ in my construction to be a "strong" hash function (e.g., SHA-256) that should be collision-resistant and (of course) preimage-resistant as well.

Likewise, this key $K$ would only be used for generating mac's only, and not "shared" for other encryption purposes elsewhere. This is because if some "other part of the application" reuses $K$ for general encryption elsewhere, an attacker might take advantage of that to determine $c=E_K(p)$ for some known or chosen (or even "derived") plaintext $p$, and thus trivially forge some message $m = p$ along with its valid mac $H(c)$.

**Edit: this is essentially the reverse of this scheme...

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    $\begingroup$ Note that HMAC is still a single pass of the underlying H over the data - even if the ECB + hash scheme works for larger amounts of data (and I wouldn't be sure about that either because of key reuse), it would be a two pass scheme, rather than a single pass scheme. $\endgroup$
    – Maarten Bodewes
    Sep 22 at 12:02
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No, the proposed construction is not secure, unless the block size $b$ of the block cipher is unusually large, or much wider than the MAC. Assuming $p$ known distinct message/MAC pairs $(m_i,h_i)$ with $b$-bit message $m_i$ and $h_i$ at least $b$-bit, there's a simple attack of expected cost dominated by $2^b/p$ hashes and searches among the $h_i$.

The attack simply hashes arbitrary distinct $b$-bit values $c$ until $H(c)$ is one of the $h_i$. With good probability, the encryption of the corresponding $m_i$ is $c$. This allows to trivially compute the MAC for $m_i\mathbin\|m_i$ as $H(c\mathbin\|c)$, which counts as forgery.

The attack is quite feasible when the block cipher is $64$-bit, e.g. 3-keys 3DES or IDEA. With AES-256 and $p=2^{40}$, it costs a borderline plausible $2^{88}$ hashes and searches among $p$ values.

Note: the construction as stated in the question works only for messages of size a multiple of $b$. Common block encryption padding (like appending a one bit then $0$ to $b-1$ zero bits as necessary to reach the end of a block) can fix this, but the attack is easily adapted.

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