feat: MDS mult. using FFT & special matrix#10
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Overall, looks good but a couple of comments:
- I think
mds_freq.rsfile is not longer a part of this PR for some reason. - Let's mention Hamish and Jacqueline in the code comments as well (in the
mds_freq.rs). - Let's add a sentence or two on performance improvements to the PR description.
feat: MDS mult. using FFT & special matrix
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This PR implements an optimization, discussed in the setting of the Poseidon hash function here, and relies on FFT-based fast matrix-vector multiplication techniques for circulant matrices. See here for a light exposition of the ideas.
By choosing an MDS matrix of a very special form, we are able to implement several optimizations on top of the FFT-based multiplication. More precisely, the MDS matrix we chose has components that are small powers of two in "frequency domain" and some of these powers are even equal to zero, provided we scale the (i)FFT appropriately. This translates to multiplications, in frequency domain, being substituted with shifts or even removed entirely. Moreover, the small powers of two entries permit very efficient scheduling of modular reductions. The special matrix we use was found using very optimized code that was kindly provided by the Polygon Zero team. The current implementation benefited, in addition, of the discussions and insights of Hamish Ivey-Law and Jacqueline Nabaglo.