Paper 2023/1284

Improving logarithmic derivative lookups using GKR

Shahar Papini, StarkWare
Ulrich Haböck, Polygon Labs

In this informal note, we instantiate the Goldwasser-Kalai-Rothblum (GKR) protocol to prove fractional sumchecks as present in lookup arguments based on logarithmic derivatives, with the following impact on the prover cost of logUp (IACR eprint 2022/1530): When looking up $M\geq 1$ columns in a (for the sake of simplicity) single column table, the prover has to commit only to a single extra column, i.e. the multiplicities of the table entries. In order to carry over the GKR fractional sumcheck to the univariate setting, we furthermore introduce a simple, yet (as far as we know) novel transformation for turning a univariate polynomial commitment scheme into a multilinear one. The transformation complements existing approaches and might be of independent interest for its elegant way to prove arbitrary powers of the lexicographic shift over the Boolean hypercube.

Note: Further typos corrected.

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Cryptographic protocols
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lookup argumentsGKRmultilinear commitment scheme
Contact author(s)
spapini @ starkware co
uhaboeck @ polygon technology
2023-09-18: last of 2 revisions
2023-08-27: received
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      author = {Shahar Papini and Ulrich Haböck},
      title = {Improving logarithmic derivative lookups using GKR},
      howpublished = {Cryptology ePrint Archive, Paper 2023/1284},
      year = {2023},
      note = {\url{}},
      url = {}
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