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The dramatic surge in the utilisation of generative artificial intelligence (GenAI) underscores the need for a secure and efficient mechanism to responsibly manage, use and disseminate multi-dimensional data generated by artificial intelligence (AI). In this paper, we propose a blockchain-based copyright traceability framework called ring oscillator-singular value decomposition (RO-SVD), which introduces decomposition computing to approximate low-rank matrices generated from hardware entropy sources and establishes an AI-generated content (AIGC) copyright traceability mechanism at the device level. By leveraging the parallelism and reconfigurability of field-programmable gate arrays (FPGAs), our framework can be easily constructed on existing AI-accelerated devices and provide a low-cost solution to emerging copyright issues of AIGC. We developed a hardware-software (HW/SW) co-design prototype based on comprehensive analysis and on-board experiments with multiple AI-applicable FPGAs. Using AI-generated images as a case study, our framework demonstrated effectiveness and emphasised customisation, unpredictability, efficiency, management and reconfigurability. To the best of our knowledge, this is the first practical hardware study discussing and implementing copyright traceability specifically for AI-generated content.