**Bulletproofs+: Shorter Proofs for Privacy-Enhanced Distributed Ledger**

*Heewon Chung and Kyoohyung Han and Chanyang Ju and Myungsun Kim and Jae Hong Seo*

**Abstract: **We present a new short zero-knowledge argument for the range proof and the arithmetic circuits without a trusted setup. In particular, the proof size of our protocol is the shortest of the category of proof systems with a trustless setup. More concretely, when proving a committed value is a positive integer less than 64 bits, except for negligible error in the $128$-bit security parameter, the proof size is $576$ byte long, which is of $85.7\%$ size of the previous shortest one due to B\"unz et al.~(Bulletproofs, IEEE Security and Privacy 2018), while computational overheads in both proof generation and verification are comparable with those of Bulletproofs, respectively.
Bulletproofs is established as one of important privacy enhancing technologies for distributed ledger, due to its trustless feature and short proof size. In particular, it has been implemented and optimized in various programming languages for practical usages by independent entities since it proposed. The essence of Bulletproofs is based on the logarithmic inner product argument with no zero-knowledge. In this paper, we revisit Bulletproofs from the viewpoint of the first sublinear zero-knowledge argument for linear algebra due to Groth~(CRYPTO 2009) and then propose Bulletproofs+, an improved variety of Bulletproofs. The main ingredient of our proposal is the zero-knowledge weighted inner product argument (zk-WIP) to which we reduce both the range proof and the arithmetic circuit proof. The benefit of reducing to the zk-WIP is a minimal transmission cost during the reduction process. Note the zk-WIP has all nice features of the inner product argument such as an aggregating range proof and batch verification.

**Category / Keywords: **cryptographic protocols / Zero Knowledge Argument, Range Proof, Inner Product

**Date: **received 17 Jun 2020

**Contact author: **jhsbhs at gmail com,heewonchung@hanyang ac kr

**Available format(s): **PDF | BibTeX Citation

**Version: **20200618:154806 (All versions of this report)

**Short URL: **ia.cr/2020/735

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