Changes from September 20 meeting #35
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daira
commented
Sep 20, 2024
daira
commented
- Define shortcut
$[i, j]$ notation and use it consistently. - Clarifications to the definition of a correctness-preserving translation.
- Make notation in the definition of
$\mathcal{R}_{\mathsf{concrete}}$ more consistent. - Move explanations of the FIND_ROW_MAPPING algorithm to its definition.
- Complete the (informal) proof for FIND_ROW_MAPPING.
Signed-off-by: Daira-Emma Hopwood <daira@jacaranda.org>
…ion. Signed-off-by: Daira-Emma Hopwood <daira@jacaranda.org>
…more consistent. Co-authored-by: Mary Maller <mary.maller@ethereum.org> Co-authored-by: Jack Grigg <thestr4d@gmail.com> Signed-off-by: Daira-Emma Hopwood <daira@jacaranda.org>
Co-authored-by: Mary Maller <mary.maller@ethereum.org> Co-authored-by: Jack Grigg <thestr4d@gmail.com> Signed-off-by: Daira-Emma Hopwood <daira@jacaranda.org>
daira
commented
Sep 20, 2024
| * Completeness is preserved: given a satisfying instance $x$ and witness $w$ for the abstract circuit, $w' = \mathcal{F}(w)$ is a satisfying witness for the concrete circuit with instance $\mathcal{I}(x)$. | ||
| * Knowledge soundness is preserved: given a satisfying instance $x'$ and witness $w'$ for the concrete circuit, we can efficiently compute some satisfying witness $w$ for the abstract circuit with instance $\mathcal{I}^{-1}(x')$. | ||
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| We also claim that a correctness-preserving translation in this sense, when used with a concrete proof system that is zero-knowledge, necessarily yields an overall proof system for the abstract relation that is zero-knowledge. That is, informally, no additional information about the abstract witness is revealed beyond the fact that the prover knows such a witness. | ||
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| > Aside: we could have required there to be an efficient reverse witness translation function $\mathcal{F}' \mathrel{⦂} \mathbb{F}^{m' \times n'} \rightarrow \mathbb{F}^{m \times n}$ from concrete witnesses to abstract witnesses, and then used $w = \mathcal{F}'(w')$ in the definition of knowledge soundness preservation. We do not take that approach because strictly speaking it would be an overspecification: we do not need the satisfying abstract witness to be *deterministically* and efficiently computable from the concrete witness; we only need it to be efficiently computable. Also, in general $w$ could also depend on the instance $x'$, not just $w'$. In practice, specifying such a function $\mathcal{F}'$ is likely to be the easiest way to prove knowledge soundness preservation. |
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I'm open to just requiring this to be a deterministic function
Co-authored-by: Mary Maller <mary.maller@ethereum.org> Co-authored-by: Jack Grigg <thestr4d@gmail.com> Signed-off-by: Daira-Emma Hopwood <daira@jacaranda.org>
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