In this work, we propose new lattice-based protocols which are used to prove additive and multiplicative relations of committed integers. We introduce three new protocols. The first protocol proves additive relation of integers. In this framework, we introduce a new computational technique which splits the integers into chunks helping to achieve a significant improvement to the integer addition protocol proposed at CRYPTO’18 by reducing the computational costs significantly for commonly used integers of length \(L\in \{2^5,2^6,2^7\}\). Our second protocol presents a new way of proving multiplicative relations of polynomials and improves the performance of the existing polynomial multiplication protocol proposed at ESORICS’15 for small integers. Using these two developed protocols as building blocks, we present our third contribution to prove multiplicative relation of integers and achieve a notable reduction in computational complexity compared to the existing integer multiplication protocol presented at CRYPTO’18.

在这项工作中，我们提出了新的基于格的协议，这些协议用于证明承诺整数的加法和乘法关系。我们介绍了三个新协议。第一个协议证明了整数的加法关系。在此框架中，我们引入了一种新的计算技术，该技术通过将常用长度为\（L \ in \ {2 ^ 5,2 ^ 6,2 ^ 7 \} \）。我们的第二个协议提出了一种证明多项式乘法关系的新方法，并提高了ESORICS'15提出的针对小整数的现有多项式乘法协议的性能。使用这两个已开发的协议作为构建基块，与CRYPTO'18上提出的现有整数乘法协议相比，我们展示了我们的第三项贡献，即证明整数的乘法关系并显着降低了计算复杂性。