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**2023/1090**

**Title:**

Bulletproofs With Stochastic Equation Sets

**Authors:**
Michael Brand, Benoit Poletti

**Abstract:**

Bulletproofs are general-purpose Zero Knowledge Proof protocols that allow a Prover to demonstrate to a Verifier knowledge of a solution to a set of equations, represented as a Rank 1 Constraint System.

We present a protocol extending the standard Bulletproof protocol, which introduces a second layer of interactivity to the protocol, by allowing the Verifier to choose the set of equations after the Prover has already committed to portions of the solution.

We show that such Verifier-chosen (or stochastically-chosen) equation sets can be used to design smaller equation sets with less variables that have the same proving-power as their larger, deterministic counterparts but are, in practice, orders of magnitude faster both in proof generation and in proof verification, and even reduce the size of the resulting proofs. We demonstrate this with an example from a real-world application.

Our method maintains all the classical benefits of the Bulletproof approach: efficient proof generation, efficient proof checking, extremely short proofs, and the ability to use Fiat-Shamir challenges in order to turn an interactive proof into a non-interactive proof.

**ePrint:**
https://eprint.iacr.org/2023/1090

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