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**2022/957**

**Title:**

Caulk+: Table-independent lookup arguments

**Authors:**
Jim Posen and Assimakis A. Kattis

**Abstract:**

The recent work of Caulk introduces the security notion of position hiding linkability for vector commitment schemes, providing a zero-knowledge argument that a committed vector’s elements comprise a subset of some other committed vector. The protocol has very low cost to the prover in the case where the size m of the subset vector is much smaller than the size n of the one containing it. The asymptotic prover complexity is O(m^2 + m \log n), where the \log n dependence comes from a subprotocol showing that the roots of a blinded polynomial are all $n$th roots of unity. In this work, we show how to simplify this argument, replacing the subprotocol with a polynomial divisibility check and thereby reducing the asymptotic prover complexity to O(m^2), removing any dependence on n.

**ePrint:**
https://eprint.iacr.org/2022/957

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