## 5 Proposed Approach and System
### Local Computation
Subsequently, the smart contract converts \(\bar{\mathbf{g}}^{(t)}\) from the finite field to the real domain as follows: \[\mathbf{g}^{(t)}=\frac{\phi^{-1}(\bar{\mathbf{g}}^{(t)})}{q^{2}}.\] (25)

## 6 Security Analysis

As outlined in Section 3.2, our threat model encompasses two types of security threats: privacy leakage and poisoning attacks. In this section, we concentrate solely on privacy protection, specifically ensuring the confidentiality of IoMT device gradients originating from edge nodes and blockchain systems. Indeed, the security of our PPBFL is based on our actively secure evaluation protocol, which not only ensures privacy protection in an information-theoretic sense, but also resists malicious attacks carried out by the participants (i.e., it provides active security). First, each IoMT device uploads only committed shares of its local gradients to edge nodes throughout the entire training process. Based on the security provided by verifiable Shamir's secret sharing, even if at most \(T-1\) nodes collude with one another, neither the edge nodes nor blockchain can infer any useful information about the devices from these committed shares or forge them. All operations conducted on edge nodes and within the blockchain are executed under a secure multiparty computation (MPC) framework, thus ensuring the confidentiality of device gradients. Next, we present a proof for our scheme using a hybrid approach [35]. Assuming there exists a set of malicious edge nodes, denoted as \(\mathbf{E}_{Att}\subset\{E_{1},\cdots,E_{M}\}\) holding \(|\mathbf{E}_{Att}|<T<M/3+1\), it is always possible to identify a simulator \(\mathbf{SIM}\) with adequate computational power such that, given parameters \(p\), \(T\), and \(M\), indistinguishability between simulator and real protocol can be assured in an information-theoretic sense--even when edge nodes belonging to \(\mathbf{E}_{Att}\) collaborate with one another. \[\mathbf{SIM}_{\mathbf{E}_{Att}}^{p,T,M}\equiv\mathbf{REAL}_{\mathbf{E}_{Att}}^ {p,T,M}. \tag{26}\]

Considering REAL's input of \(\mathbf{G}_{N}=(\mathbf{g}_{1},\cdots,\mathbf{g}_{n},\cdots,\mathbf{g}_{N})\) and SIM's input of \(\mathbf{G^{\prime}}_{N}=(\mathbf{g^{\prime}}_{1},\cdots,\mathbf{g^{\prime}}_{ n},\cdots,\mathbf{g^{\prime}}_{N})\) with identical distributions, we demonstrate the indistinguishability between \(\mathbf{REAL}_{\mathbf{E}_{Att}}^{p,T,M}\) and \(\mathbf{SIM}_{\mathbf{E}_{Att}}^{p,T,M}\) as well as the unforgeability against active edge nodes throughout the entire execution process of our scheme, as follows. * In this step, each edge node \(P_{m}\) receives the committed shares of device gradients. Each node \(P_{m}\) in the real protocol takes the shares denoted by \(\{[\![\![\![\! ## 7 Experimental Setup

We implemented our scheme on a private Ethereum blockchain setup. The smart contract (SC) layer was developed using the Solidity programming language and deployed on the private blockchain utilizing Truffle. Our experiments were conducted using PyTorch running on a workstation equipped with Ubuntu 20.04 OS, an AMD Ryzen Threadripper 3970X (32 cores, 64 threads, 3.7 GHz), an RTX 3090 Ti GPU, and 256 GB of RAM. Simultaneously, we simulated edge nodes using independent threads, with each thread implementing a real-world PyTorch classifier. Additionally, mobile phones (Huawei nova 7, 2.6 GHz, 8 cores, and 8 GB RAM) were employed as model Internet of Medical Things (IoMT) devices. Furthermore, our scheme operated within a finite field \(\mathbb{Z}_{p}\), with the secure framework based on a \((T,N)\)-threshold multi-party computation (MPC)-based protocol. The protocol utilized threshold multi-party computation (MPC), with the parameter \(p\) configured as \(2^{26}-5\) to avoid data overflow. Standard threshold parameters were predefined as \(T=4\) and \(N=10\) in this framework. ### Datasets and Settings

To evaluate the performance of our privacy-preserving blockchain federated learning (PP-BFL), we conducted experiments on two widely used datasets: PathMNIST [36] and OCTMNIST [37].