Explicit cost analysis of Toom-4 multiplication for incomplete NTT in lattice-based cryptography
For cryptographers implementing lattice-based schemes, this offers a concrete cost model to optimize polynomial multiplication when modulus constraints prevent full NTT.
This work provides the first explicit cost analysis of Toom-4 multiplication within the incomplete NTT framework for lattice-based cryptography, deriving operation counts and identifying parameter ranges where Toom-4 outperforms Karatsuba.
Polynomial multiplication is fundamental in lattice-based cryptography. While the Number Theoretic Transform (NTT) enables fast multiplication, it imposes constraints on the modulus of the coefficient field. Hafiz et al. (2025) addressed this limitation by analyzing the incomplete NTT, which combines a truncated NTT with conventional multiplication methods In this work, we revisit Toom-4 multiplication in the context of incomplete NTT. Although Toom-4 is asymptotically faster than Karatsuba, its precise cost has not been expressed in a form compatible with the incomplete NTT framework. We present a concrete Toom-4 implementation and derive explicit operation counts that separate additions/subtractions and multiplications over the coefficient field. Our analysis based on addition chains yields a simple cost model for incomplete NTT. Using this model, we analyze hybrid strategies combining Toom-4, Karatsuba, and incomplete NTT. We identify parameter ranges where Toom-4 is advantageous and validate the predicted behavior experimentally.