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The worst-case execution time of the hybrid AG generator scales quadratically with input assets, and linearly with input exploits and graph size (number of nodes). Using a serial algorithm to generate a large AG is a tremendous challenge in terms of execution time. To address this issue, we employ parallel programming to accelerate the algorithm and realize it on high performance computing (HPC) hardware. In Section 4, we provide design details and performance evaluation of an efficient AG generator based on this strategy.

Our AGs are generated in a BFS manner, as shown by the serial algorithm in Figure 2. An AG is expanded from a root node $ s_0$ defined by the input sets $ Q_0$ and $ T_0$. A frontier queue $ \Omega$ stores the nodes to be expanded. It is propagated by a while-loop from the root node to its descendant nodes until the entire graph is generated. A hash table $ H_d$ stores the hash values of discovered nodes to avoid duplicate storage and expansion. For a specific node $ s_c$ dequeued from $ \Omega$ for expansion, a combinatorial process binds each exploit (vulnerability) with every asset (the for-loop in line 5) or asset pair (the for-loop in line 8) in the network. If the preconditions of a binding are satisfied by the current node, then it is successful and stored as an applicable binding in a queue $ B_s$. After all the applicable bindings are discovered, the for-loop in line 11 derives neighbors and outgoing edges for the current node $ s_c$ from these bindings. For each binding, the postconditions of the binding's exploit are applied to the current node, which makes a change to its quality set and/or topology set, and derives a neighbor node $ s_n$. If searching for $ s_n$ in the hash table $ H_d$ is a miss, then $ s_n$ is a new node. It will be hashed to $ H_d$, stored in the node set $ F$ and enqueued to $ \Omega$ for expansion. The edge $ s_c \rightarrow s_n$ will be stored in the edge set $ E_d$. If the search results in a hit, then only the edge will be stored.

The serial AG generation is not precisely equivalent to generic BFS for graph traversal, where target graphs are known before the algorithm runs. The nodes and edges in generic BFS algorithms are typically given in compressed sparse row (CSR) or coordination list (COO) formats. In AG generation, except the root, all the nodes and edges are unknown at the beginning. The task on each node comprises discovering all its neighbors, storing the corresponding edges and enqueuing newly discovered neighbors for future expansion. The neighbors of any node are discovered by the three inner for-loops on the fly rather than being read from some arrays. The example in Figure 3 illustrates the discovery and storage of a neighbor for a given node $ n$ defined by its quality set $ Q_n$ and topology set $ T_n$. The discovery starts by binding exploit k with host 2 in the target network. After binding, the precondition (in red font) is found satisfied, because it exists in the quality set $ Q_n$. Therefore, the binding is successful and a neighbor $ n1$ is derived by applying the postcondition (in blue font) on node $ n$.

During the expansion of node $ n$, all the possible bindings between the asset set $ A$ and the exploit set $ E_x$ are examined. An exploit is local if its preconditions refer to a single asset and remote if its preconditions refer to two connected assets. The number of bindings to examine for each node is $ |A||E_x|$, if all the exploits are local, and $ |A|^2|E_x|$, if all the exploits are remote. In the worst case where all the exploits are remote, the computational complexity of the serial AG generator is $ O(|A|^2|E_x||F|)$.

Here, the serial AG generator is parallelized on shared-memory systems. Distributed-memory systems are out of the discussion scope and will be the focus of our future research. We build our parallel AG generator upon the parallelism strategy adopted in References [20, 21], but with significant improvement.

In our GPU implementation, the serial phase runs on the CPU as a host program. When the CPU frontier $ \Omega ^{(m)}$ is filled with enough nodes for every warp, the $ parallel\_generator()$ is invoked as the GPU kernel by the host. During kernel execution, every warp fetches the same number of nodes from $ \Omega ^{(m)}$ and expands in its local frontier. To avoid conflicts when multiple warps access a shared memory unit simultaneously, we use atomic primitives from CUDA to implement the lock mechanism.

Figure 7 shows the speedup values of the parallel AG generators over the serial version. Both the OpenMP and CUDA AG generators are able to accelerate the serial algorithm with more than 10$ \times$ speedup. For target AGs tree1 and chain1, the OpenMP AG generator runs faster than the CUDA version due to the small problem size. For target AGs tree2 and chain2, their relatively large problem size unleashes the computational power of the GPU. As a result, the CUDA version outperforms the OpenMP version.

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