Cuproof: Range Proof with Constant Size

Zero-Knowledge Proof is widely used in blockchains. For example, zk-SNARK is used in Zcash as its core technology to identifying transactions without the exposure of the actual transaction values. Up to now, various range proofs have been proposed, and their efficiency and range-flexibility have also been improved. Bootle et al. used the inner product method and recursion to construct an efficient Zero-Knowledge Proof in 2016. Later, Benediky Bünz et al. proposed an efficient range proof scheme called Bulletproofs, which can convince the verifier that a secret number lies in [Formula: see text] with [Formula: see text] being a positive integer. By combining the inner-product and Lagrange’s four-square theorem, we propose a range proof scheme called Cuproof. Our Cuproof can make a range proof to show that a secret number v lies in an interval [Formula: see text] with no exposure of the real value v or other extra information leakage about v. It is a good and practical method to protect privacy and information security. In Bulletproofs, the communication cost is [Formula: see text] , while in our Cuproof, all the communication cost, the proving time and the verification time are of constant sizes.