This paper presents a new short zero-knowledge argument for the range proof and arithmetic circuits without a trusted setup. In particular, it can achieve the shortest proof size of the proof system categories without a trusted setup. More specifically, when proving that 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 bytes long, which is 85.7% the size of the previous shortest proof due to Bünz et al. (Bulletproofs, IEEE Security and Privacy 2018). Similarly, circuit satisfiability can be proven with less communication overhead. Nevertheless, computational overheads in both proof generation and verification are comparable with those of Bulletproofs. Bulletproofs is established as one of the important privacy-enhancing technologies for a 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 usage by independent entities since it was proposed. The essence of Bulletproofs is based on the logarithmic inner product argument with no zeroknowledge. This paper revisits 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 component is the zero-knowledge weighted inner product argument (zk-WIP) which enables to reduce both the range proof and the arithmetic circuit proof. It already has zeroknowledge properties, there is no additional information when reducing zk-WIP, and it incurs a minimal transmission cost during the reduction process. Note that zk-WIP has all characteristics of the inner product argument, such as an aggregating range proof and batch verification; thus, Bulletproofs+ is superior to Bulletproofs in all aspects.INDEX TERMS Zero-knowledge proofs, range proofs, arithmetic circuit.

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