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A parallel transient electromagnetic simulator of linear complexity and linear speedup is developed to simulate very large-scale integrated circuits and packages from DC to very high frequencies. In this simulator, through suitable basis functions and linear algebraic techniques, we directly and rigorously decompose the original 3-D system matrix into multiple 1-D matrices with negligible computational overhead. Each one-dimensional matrix is made tridiagonal, and hence can be solved readily in linear complexity. We then achieve an almost embarrassingly parallel implementation of the linear-complexity electromagnetic solver with a low communication-to-computation ratio. Numerical experiments on very large-scale integrated circuits and packages have demonstrated superior performance and linear speedup of the proposed parallel transient simulator. It successfully simulates a large-scale combined die-package system from a real product, which involves more than 3.5 billion unknowns, in fast CPU run time.