Explicit cost analysis of Toom-4 multiplication for incomplete NTT in lattice-based cryptography

TL;DR

This paper provides an explicit cost analysis of Toom-4 multiplication within incomplete NTT frameworks in lattice-based cryptography, enabling optimized hybrid strategies.

Contribution

It introduces a concrete Toom-4 implementation with explicit operation counts and a simple cost model for incomplete NTT, facilitating performance optimization.

Findings

Toom-4 can be advantageous in specific parameter ranges.

Derived explicit operation counts for Toom-4 in incomplete NTT.

Validated cost model through experimental analysis.

Abstract

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.