Prime path coverage is a powerful structural testing criterion, but generating all prime paths in a directed graph remains computationally challenging due to the potentially exponential number of them. Existing approaches typically rely on enumerating large sets of candidate paths and filtering them, leading to high computational and memory overhead. In this paper, we present a new approach to prime path generation based on a structural characterization of prime paths in terms of strongly connected components. This characterization yields non-trivial necessary conditions for valid path endpoints and reduces the problem to constrained cycle enumeration in an augmented graph. As a result, we avoid explicitly enumerating all simple paths and instead generate only feasible candidates. Building on this insight, we design a streaming algorithm that outputs prime paths incrementally, using a Johnson-style traversal as a subroutine within a significantly reduced search space. The algorithm exploits SCC boundary crossings as natural pruning checkpoints - discarding partial paths the moment they are detected to be backward extendable, eliminating entire subtrees of the search space during traversal rather than filtering completed paths post hoc. We implement our method and evaluate it on a large dataset of real-world control-flow graphs extracted from open-source C++ and Python projects. The results demonstrate that our approach consistently outperforms existing methods, while maintaining stable inter-output delay in practice.