Vector Commitments with Proofs of Smallness: Short Range Proofs and More

Vector commitment schemes are compressing commitments to vectors that make it possible to succinctly open a commitment for individual vector positions without revealing anything about other positions. We describe vector commitments enabling constant-size proofs that the committed vector is small (i.e., binary, ternary, or of small norm). As a special case, we obtain range proofs featuring the shortest proof length in the literature with only 3 group elements per proof. As another application, we obtain short pairing-based NIZK arguments for lattice-related statements. In particular, we obtain short proofs (comprised of 3 group elements) showing the validity of ring LWE ciphertexts and public keys. Our constructions are proven simulation-extractable in the algebraic group model and the random oracle model.