In this paper, we present a novel approach to analyzing intrinsic properties of the Josephus function, J_k. The linear structure between extremal points of J_k is fully revealed, leading to the design of an efficient algorithm for evaluating J_k. We also derive algebraic expressions that describe how to recursively compute extremal points, including fixed points. The existence of consecutive extremal and also fixed points of J_k, for all k ≥ 2, is proven as a consequence, which generalizes Knuth’s result for k = 2. Moreover, we conduct an extensive comparative numerical experiment to illustrate the performance of the proposed algorithm for evaluating the Josephus function compared to established algorithms. The results show that the proposed scheme is highly effective in computing J_k(n) for large inputs.