The operation of computing$n$scalar multiplications in an elliptic curve group and then adding them together is called n-scalar multiplication.$n$-scalar multiplication is the essential operation for proof generation and verification in pairing-based trusted setup zero-knowledge succinct non-interactive argument of knowledge protocols, which enable the privacy-preserving features in blockchain applications. This paper proposed a method to compute$n$-scalar multiplication taking advantage of$3n$precomputed points. When instantiating over BLS12-381 curve, for$n=2^{c}\ (10\leq c\leq 22)$, which covers the majority of our purported applications, the proposed method showed 2.59% ∼ 12.26% theoretical speed improvement and demonstrated 1.63% ∼ 11.54% experimental improvement against Pippenger's bucket method.

Linear map commitments allow the prover to commit to a vector, with the ability to prove the image of a linear map acting on the vector. In this paper, we propose linear map commitments with updatable feature and perfectly hiding property. Updatable feature means that the prover can update the commitment more efficiently than recompute the commitment when some of the entries in the committed vector are changed. Perfectly hiding property ensures the commitment reveals no information about the committed vector before opening. Then we present the implementation of our updatable linear map commitment (ULMC) over the 256-bit BN curve recommended in the SM9 standard, which provides around 100-bit security. The implementation shows that our ULMC schemes are efficient enough to support the elementary database constructions that simultaneously permit batching membership test, linear combination test, updatable feature and authenticity. Finally, we show that the ULMC-powered elementary databases are capable of supporting various applications where privacy and trust are the first priority such as exam result management systems, Internet of Things (IoT) management systems and business operations between banks and enterprises.