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

2026-05-17

AI Summary

This work addresses the limitations of traditional Number Theoretic Transform (NTT), which imposes strict constraints on the modulus, and the lack of precise cost modeling for Toom-4 multiplication within existing incomplete NTT frameworks. For the first time, the paper constructs an explicit computational cost model for Toom-4 that is compatible with incomplete NTT, deriving exact counts of additions, subtractions, and multiplications over the coefficient field through concrete implementation. Leveraging addition chain theory, it establishes a concise analytical framework for cost evaluation. Using this model, the study systematically assesses hybrid strategies combining Toom-4, Karatsuba, and incomplete NTT, experimentally validates the model’s accuracy, and demonstrates that Toom-4 outperforms alternative methods within specific parameter regimes.

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.

Problem

• Toom-4 multiplication
• incomplete NTT
• lattice-based cryptography
• polynomial multiplication
• cost analysis

Innovation

• Toom-4
• incomplete NTT
• cost analysis
• addition chains
• lattice-based cryptography