# A sketch for a STARK-based accumulator

We can define a version of MIMC that works as follows: SimplifiedMiMCHash(x, d) = f^{512}(x), where f(x) = x^3 + d; that is, we apply the permutation x \rightarrow x^3 + d 512 times.

Security claim: partial collision resistance - if y = SimplifiedMimChash(....SimplifiedMiMCHash(SimplifiedMiMCHash(x, d1), d2)...dn) it is infeasible to find (d1', d2' ... dn') \ne (d1, d2 .. dn) such that y = SimplifiedMimChash(....SimplifiedMiMCHash(SimplifiedMiMCHash(x, d1'), d2')...dn')

[NOTE: I think there are better ways to do this that more directly lean on traditional collision resistance properties of these arithmetically cheap hash functionsâ€¦]

We now define the accumulator as follows. The accumulator A starts at 0, and then every time a value v is added we set A := SimplifiedMiMCHash(A, v).

For proofs of inclusion or exclusion, we set up a STARK with three tapes: the accumulator state A, the witness W consisting of a sequence of 512-value repeats of values that get added to the accumulator, a loop progress counter M which starts at 1 and a product trace P which starts at 1. Let \omega be a 512th root of unity, and x be the value you want to prove inclusion or exclusion of. We add the following constraints:

• M[i] = 1 or W[i] = W[i-1] (ie. W is only allowed to change at multiples of 512)
• M[i] = M[i-1] * \omega (incrementing M; note that it loops around to 1 every 512 steps)
• A[i] = A[i-1]^3 + W[i]
• P[i] = P[i-1] * (x - W[i-1])

We check the boundary conditions (i) A[0] is the starting accumulator, (ii) A[n] is the ending accumulator, (iii) P[0] = 1. The goal is that P will stay nonzero as long as x is never used in the witness, and will permanently become zero if x is used in the witness even once.

The STARK construction is very simple, with only 4 state objects to worry about; it should not be difficult to convert the existing MIMC-STARK code to implement this construction. Note that it should be fairly straightforward to replace MIMC in this construction with Jarvis. This means that we can use STARKs for proving history inside of Plasma, and even potentially to prove contract non-double-resurrection for sharding, more quickly than a fully complete STARK system that supports more complicated operations.

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OK letâ€™s break it.

A collision in a binary field:

MiMCHash(1,1)=MiMCHash(1,0)=1.

A collision in a prime field F.

MiMCHash(1,0) = MiMCHash(MiMCHash(1,0),0)=1.

Finding a collision on inputs of the same length is less trivial but I am pretty sure is doable because of repetitive structure of the function.

Can this be fixed by adding a simple boundary condition and/or field definition which excludes 0 and 1

Many things can be fixed, but cryptographic design produces best outputs in a different way. If one needs a secure hash function, it is better to start with requirements.

Is this a fixed set of values such as the first 512 prime numbers? Does this mean the accumulator can only accommodate 512 entries?

The goal here is to have a secure hash function of the form h(x, y) = z, which can be calculated arithmetically ideally with one state object, though 2 state objects or slightly more would also work too.

The nice thing about such a primitive is that it is also useful for Merkle branches.

I absolutely agree that coming up with something for which we try to establish traditional collision resistance properties is the right approach here.

A hash function of the form h(x,y) = z can be any hash function H(u) = z by setting u=x||y. Here H can be Jarvis, MiMC, InversionHash, or any STARK-friendly design, so there are many of them.

Is there any other specific property you want to have here?

Also if you want an iterative hash function that takes arbitrary long inputs, then there are basically two ways to do that:

1. Take a STARK-friendly secure permutation and use it in the sponge mode. It must be sufficiently wide to provide 128-bit collision and preimage resistance, but we know how to design such permutations.

2. Take a STARK-friendly secure compression function and use it in the Merkle-Damgard or Haifa mode. The compression function itself can be based on a blockcipher (like Jarvis, SHA-2) or built from scratch (like Groestl or MD6).

can be any hash function H(u) = z by setting u=x||y .

How do you do || in a field?

Depends on the size of x and y and the field size and the structure of the function. The simplest case is when both x and y are field elements and the hash function operates in the same field, then I do not see a problem.

Letâ€™s suppose x and y are field elements, and we want to output a field element hash_input = x || y. How do you do that?

Also note that if your answer is â€śMerkle-Damgard constructionâ€ť, then note that requires as a building block a function f(x, y) \rightarrow z which can then be applied many times to get a function f(x_1, x_2, x_3 ... x_n) \rightarrow z so it still requires solving the original problem.

No I mean if x and y are field elements and the internal state of H is a tuple of field elements (say a pair, but can be a quadruple or else) then you can do that trivially.

How to build H over a tuple of field elements? Like AES, for example, or Groestl. In the Inversion Hash we use an MDS matrix to mix the field elements with each other and an inversion S-box for degree saturation. Then you get a permutation, but it can be also used for a block cipher, then with MD/MP/MMO you get a compression function then a full hash function. But the latter is more complicated I think.