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Packet classification has been studied for decades; it classifies packets into specific flows based on a given rule set. As software-defined network was proposed, a recent trend of packet classification is to scale the five-tuple model to multi-tuple. In general, packet classification on multiple fields is a complex problem. Although most existing software-based algorithms have been proved extraordinary in practice, they are only suitable for the classic five-tuple model and difficult to be scaled up. Meanwhile, hardware-specific solutions are inflexible and expensive, and some of them are power consuming. In this paper, we propose a universal multi-dimensional packet classification approach for multi-core systems. In our approach, novel data structures and four decomposition-based algorithms are designed to optimize the classification and updating of rules. For multi-field rules, a rule set is cut into several parts according to the number of fields. Each part works independently. In this way, the fields are searched in parallel and all the partial results are merged together at last. To demonstrate the feasibility of our approach, we implement a prototype and evaluate its throughput and latency. Experimental results show that our approach achieves a 40％ higher throughput than that of other decomposed-based algorithms and a 43％ lower latency of rule incremental update than that of the other algorithms on average. Furthermore, our approach saves 39％ memory consumption on average and has a good scalability.