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I am faced with a problem of selecting an appropriate message authentication code (MAC) for a particular application. These are the constraints:

I have two 64-bit integers $r_a$ and $r_s$ as input, where:

  1. $r_a < 2^{52}$ and $r_s < 2^{52}$
  2. $r_a$ is a multiple of 4, i.e. $r_a = 0 \pmod{4}$
  3. $r_s$ is a multiple of 16, i.e. $r_s = 0 \pmod{16}$

This makes for an effective 98 bits of input.

My goal is to get a message authentication code for those 98 bits of input. The message authentication code $M$ must be between 2 and 10 bits in length, i.e. $2 \le |M| \le 10$. However, more is better, so the ideal goal is a 10-bit MAC.

The security goal to be achieved is the following:

The general flow of the protocol goes like this: Alice submits $(r_a, r_s)$ to Steve. Steve returns the concatenation of $r_a$ and the message authentication code $M = F(K, r_a || r_s)$ for a key $K$ only known to Steve, i.e. $r_{auth} = r_a || M$. At a later point, Alice submits $r_{auth}$ to Steve again. Alice can have as many $(r_a, r_s)$ tuples as the limits of the respective input integers allow. If $M$ is correct, Alice is permitted to use $r_a$ as a resource.

I want to prevent that an attacker can forge a value of $r_{auth}$ for any pair of $(r_a, r_s)$. If $M$ is incorrect, Steve changes the key used to generate $M$ and reissues all $r_{auth}$ in existence.

My goal is to minimize the likelihood that a valid $r_{auth}$ is discovered by an attacker. Of course, an attacker should also not learn $K$ as that would imply the ability to forge any $r_{auth}$.

The performance constraints are the following:

The generation of $M$ must not take more than 50 cycles on the target platform. The target platform is a RISC CPU:

  1. with 64-bit integers
  2. no rotate instructions
  3. no constant-time multiplication
  4. no native 64x64->128 multiplication, only 64x64->64
  5. no hardware support for AES, GHASH or Keccak

I have looked at SipHash in particular. SipHash-2-4 takes about 64 cycles (at 8 cycles per round with 2 rounds for $r_a$ and another 2 rounds for $r_s$ and 4 rounds of finalization). The SipHash paper notes that SipHash-2-4 is expected to provide the maximum PRF security possible. There is a distinguisher for four rounds. However, considering the very limited output size of at most 10 bits and the ability to change $K$ on a detected forgery, I wonder if lower round counts would be reasonable. If so, however, it is unclear how to best choose them.

Poly1305 is both held back by either having to use 32-bit limbs (or really awkward 64-bit multiplication implementation) and the lack of an obvious way to generate the one-time keys.

What would be the optimal (existing) MAC function given these constraints? There seems to be a lot of choice for long-input MACs, but short-input MACs seem to be a rare issue.

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  • $\begingroup$ So 1024 refreshes of the key later the adversary has a 0.5 chance of getting authorized to the resource? Sure it is better than 512 tries without refresh but... $\endgroup$
    – Maarten Bodewes
    Dec 4, 2023 at 13:02

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