Abstract We present , a zkSNARK solution for large-scale matrix multiplication. Classical zkSNARK protocols typically underperform in data analytic contexts, hampered by the large size of datasets and the superlinear nature of matrix multiplication. excels in its scalability. The prover time of scales linearly with respect to the number of non-zero elements in the input matrices. For $$n \times n$$ n × n matrix multiplication with N non-zero elements across three input matrices, employs a structured reference string (SRS) of size O ( n ), and achieves RAM usage of $$O(N+n)$$ O ( N + n ) , transcript size of $$O(\log n)$$ O ( log n ) , prover time of $$O(N+n)$$ O ( N + n ) , and verifier time of $$O(\log n)$$ O ( log n ) . The prover time, notably at $$O(N+n)$$ O ( N + n ) and surpassing all existing protocols, includes $$O(N+n)$$ O ( N + n ) field multiplications and O ( n ) exponentiations and pairings within bilinear groups. These efficiencies make effective for linear algebra on large matrices common in real-world applications. We evaluated with $$2^{15} \times 2^{15}$$ 2 15 × 2 15 input matrices each containing 1 G non-zero integers, which necessitate 32 T integer multiplications in naive matrix multiplication. recorded prover and verifier times of 150.84s and 0.56s, respectively. When applied to $$1M \times 1M$$ 1 M × 1 M sparse matrices each containing 1 G non-zero integers, it demonstrated prover and verifier times of 1, 384.45s and 0.67s. Our approach outperforms current zkSNARK solutions by successfully handling the large matrix multiplication task in experiment. We extend matrix operations from field matrices to group matrices, formalizing group matrix algebra. This mathematical advancement brings notable symmetries beneficial for high-dimensional elliptic curve cryptography. By leveraging the bilinear properties of our group matrix algebra in the context of the two-tier commitment scheme, achieves efficiency gains over previous matrix multiplication arguments. To accomplish this, we extend and enhance Bulletproofs to construct an inner product argument featuring a transparent setup and logarithmic verifier time.

Matrix multiplication is a common operation in applications like machine learning and data analytics. To demonstrate the correctness of such an operation in a privacy-preserving manner, we propose zkMatrix, a zero-knowledge proof for the multiplication of committed matrices. Among the succinct non-interactive zero-knowledge protocols that have an O(log n) transcript size and O(log n) verifier time, zkMatrix stands out as the first to achieve O(n2) prover time and O(n2) RAM usage for multiplying two n X n matrices. Significantly, zkMatrix distinguishes itself as the first zk-SNARK protocol specifically designed for matrix multiplication. By batching multiple proofs together, each additional matrix multiplication only necessitates O(n) group operations in prover time.

In federated learning (FL), data owners "share" their local data in a privacy preserving manner in order to build a federated model, which in turn, can be used to generate revenues for the participants. However, in FL involving business participants, they might incur significant costs if several competitors join the same federation. Furthermore, the training and commercialization of the models will take time, resulting in delays before the federation accumulates enough budget to pay back the participants. The issues of costs and temporary mismatch between contributions and rewards have not been addressed by existing payoff-sharing schemes. In this paper, we propose the Federated Learning Incentivizer (FLI) payoff-sharing scheme. The scheme dynamically divides a given budget in a context-aware manner among data owners in a federation by jointly maximizing the collective utility while minimizing the inequality among the data owners, in terms of the payoff gained by them and the waiting time for receiving payoff. Extensive experimental comparisons with five state-of-the-art payoff-sharing schemes show that FLI is the most attractive to high quality data owners and achieves the highest expected revenue for a data federation.