poseidon2: rounds, internal matrix for small state sizes #51
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In [0] some Grobner basis attacks were presented that apply either partially to the 384-bit and 512-bit security level and fully to the 1024-bit security level. Instead of updating the v1 round numbers for these higher security levels, we panic to alert the user that the choice may not be secure. For poseidon2, we implement the round selection including the adjustments for the improved grobner basis attacks in [0]. [0]: https://eprint.iacr.org/2023/537.pdf
We need to be able to compute the minimal polynomial in order to check that a randomly generated internal matrix is a secure choice, i.e. that it does not have arbitrarily long subspace trails. We need to be able to compute the minimal polynomial of a matrix and check it's irreducible. We can't yet do the former, but with Perron's criteron for monic polynomials we can do the latter with this commit.
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Closes #45: in this PR we implement the round calculation for Poseidon2, and add a panic to the impacted state sizes of Poseidon1's round calculation in light of the attacks presented in https://eprint.iacr.org/2023/537.
This PR also has the start of the internal matrix calculation for Poseidon2 for #40. The remaining work here is to implement functionality that computes the minimal polynomial of a proposed internal matrix. See comment here for more explanation.