Low latency data-flow graphs for simultaneous modular inversion of many inputs

Abstract

Montgomery's trick accelerates simultaneous modular inversion of $N$ inputs by amortizing a single shared inversion, but auxiliary multiplications for complement products are typically scheduled in a linear, serial form. We construct a maximally parallelizable data-flow graph (DFG) that computes all $\overline{x}$ complement~products by scheduling auxiliary multiplications into idle multiplier slots during accumulation of the product of all inputs, and that of the shared inversion. This scheduling ensures the post-inversion phase adds exactly one multiplication layer of latency regardless of $N$, yielding a critical path latency of $\lceil \log_2 N \rceil$ multiply layers, one inversion, and one final parallel multiply layer.