Poster: Privacy in Distributed Mobile Networks

In order to achieve zero-knowledge proof (ZKP) in distributed mobile scenarios, we propose a two-stage multi-prover ZKP framework. Our method utilizes secure multi-party computation (MPC), which has advantages such as flexible adaptation, stable performance, and fewer restrictions compared to existing solutions. In addition, based on the properties of cyclic groups, we optimize secure multi-party summation, improving the balance between security and efficiency, as well as transferability of the algorithm.


INTRODUCTION
Privacy computing tasks, such as ZKP, typically require large computing resources.To solve the bottleneck of a single mobile device, tasks can be distributed across multiple devices.However, the ZKP has a limitation, which is that there can only be one prover.
Some works assume the servers are trustworthy [3,6,7], while other studies are only applicable to specific protocols [1,2,5], that is, low flexibility.OB22 [4] is applicable only to specific zk-SNARKs (like Groth16, Plonk), difficult to generalize, and requires complex modifications to the underlying ZKPs.Constrained by the MPC, it demands high bandwidth, and the performance tends to decrease exponentially as the number of parties increases.
Our contributions are as below: We propose a multi-prover ZKP that addresses the issue of multiple provers.We introduce the properties of cyclic groups into secure multi-party summation (SMPS) and secure multi-party multiplication (SMPM).

SCHEME DESIGN
We focus on the problem of multiple provers jointly calculating the sum of secret data   (SMPS) and providing proof to the verifier.We divide the proof generation process of ZKP into two stages, as shown in Fig. 1.

SMPS based on cyclic groups
To calculate the sum of  1 ,  2 , ...,   , we have each party add a random number   to their respective   .These random numbers are independently generated locally by each participant, kept confidential from each other, and satisfy  =1   = 0.The approach is as shown in equation (1).
The key challenge is how to ensure  =1   = 0 without a trusted third party.Our method is as Fig. 2, realized with the help of blind factor    .In order to prevent collusion attacks, we utilize the properties of cyclic groups to to generate different sequences of group elements.
Similar to SMPS, secure multi-party multiplication (SMPM) is also a very common and fundamental requirement in MPC.When it comes to multiplication, the calculations become more complex.Unlike addition, which has linear characteristics, multiplication involves cross computations that complicate the straightforward combination of encrypted values.This necessitates the use of more sophisticated encryption methods and intricate protocols.
So, the traditional methods for SMPS cannot be directly adapted to SMPM.However, our approach also offers an effective solution for SMPM with improved transferability.We simply need to adjust the condition from  =1   = 0 to  =1   = 1.The approach is as shown in equation (2).

Multi-prover ZKP protocol
In stage 1, we use SMPS to generate an aggregated witness, as shown in Fig. 3. Compared to using MPC throughout the entire proof generation process, our approach significantly reduces complexity and time overhead, and markedly decreases sensitivity to bandwidth.Fig. 4 shows the impact of different bandwidths.Slowdown refers to the ratio of the time overhead of a multi-prover proof system to the time overhead of a single-prover proof system.Network bandwidth has a significant impact on communication latency, throughput and scalability of interactive protocols.OB22 is very sensitive to changes in bandwidth, while our scheme is also applicable in environments with limited network bandwidth.
Table 1 presents a comprehensive comparison.Our solution is universally applicable to different types of ZKP protocols, therefore it has high flexibility.Bandwidth means whether the proof system can still maintain stability of performance in a bandwidthconstrained environment.Performance refers to the time cost under different conditions, and the smaller the time cost, the higher the performance.

CONCLUSION
We design a multi-prover zero-knowledge proof framework that integrates ZKP with MPC to address the challenge of secret data distributed among multiple provers.Our method stands out for its lightweight and flexible nature, without requiring substantial  modifications to many existing ZKP protocols, thereby significantly reducing the implementation workload.This method also has certain advantages in terms of applicable constraints and performance stability.Furthermore, we introduce the properties of cyclic groups into SMPS, effectively countering the collusion attacks that traditional methods face, thereby enhancing security.Our approach also supports a balance between performance and security.Additionally, this method is well-suited for secure multi-party multiplication.

Table 1 :
Comparison of Proof Systems