High-Radix/Mixed-Radix NTT Multiplication Algorithm/Architecture Co-Design Over Fermat Modulus

Polynomial multiplication using Number Theoretic Transform (NTT) is crucial in lattice-based post-quantum cryptography (PQC) and fully homomorphic encryption (FHE), with modulus <inline-formula><tex-math notation="LaTeX">q</tex-math> <mml:math><mml:mi>q</mm...

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Published in:IEEE transactions on computers Vol. 74; no. 10; pp. 3519 - 3533
Main Authors: Xing, Yile, Li, Guangyan, Ye, Zewen, Luk, Ryan W. L., Chen, Donglong, Yan, Hong, Cheung, Ray C. C.
Format: Journal Article
Language:English
Published: IEEE 01.10.2025
Subjects:
conflict-free memory access
Convolution
Cryptography
Discrete Fourier transforms
Fermat number transform
Hardware
high radix
mixed radix
Polynomial multiplication
Polynomials
Training
Transforms
Urban areas
Vectors
Writing
ISSN:0018-9340, 1557-9956
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Abstract Polynomial multiplication using Number Theoretic Transform (NTT) is crucial in lattice-based post-quantum cryptography (PQC) and fully homomorphic encryption (FHE), with modulus <inline-formula><tex-math notation="LaTeX">q</tex-math> <mml:math><mml:mi>q</mml:mi></mml:math><inline-graphic xlink:href="chen-ieq1-3590972.gif"/> </inline-formula> significantly affecting performance. Fermat moduli of the form <inline-formula><tex-math notation="LaTeX">2^{2^{n}}+1</tex-math> <mml:math><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math><inline-graphic xlink:href="chen-ieq2-3590972.gif"/> </inline-formula>, such as 65537, offer efficiency gains due to simplified modular reduction and powers-of-2 twiddle factors in NTT. While Fermat moduli have been directly applied or explored for incorporation into existing schemes, Fermat NTT-based polynomial multiplication designs remain underexplored in fully exploiting the benefits of Fermat moduli. This work presents a high-radix/mixed-radix NTT architecture tailored for Fermat moduli, which improves the utilization of the powers-of-2 twiddle factors in large transform sizes. In most cases, our design achieves a 30%-85% reduction in DSP area-time product (ATP) and a 70%-100% reduction in BRAM ATP compared to state-of-the-art designs with smaller or equivalent modulus, while maintaining competitive LUT and FF ATP, underscoring the potential of Fermat NTT-based polynomial multipliers in lattice-based cryptography.
AbstractList Polynomial multiplication using Number Theoretic Transform (NTT) is crucial in lattice-based post-quantum cryptography (PQC) and fully homomorphic encryption (FHE), with modulus <inline-formula><tex-math notation="LaTeX">q</tex-math> <mml:math><mml:mi>q</mml:mi></mml:math><inline-graphic xlink:href="chen-ieq1-3590972.gif"/> </inline-formula> significantly affecting performance. Fermat moduli of the form <inline-formula><tex-math notation="LaTeX">2^{2^{n}}+1</tex-math> <mml:math><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math><inline-graphic xlink:href="chen-ieq2-3590972.gif"/> </inline-formula>, such as 65537, offer efficiency gains due to simplified modular reduction and powers-of-2 twiddle factors in NTT. While Fermat moduli have been directly applied or explored for incorporation into existing schemes, Fermat NTT-based polynomial multiplication designs remain underexplored in fully exploiting the benefits of Fermat moduli. This work presents a high-radix/mixed-radix NTT architecture tailored for Fermat moduli, which improves the utilization of the powers-of-2 twiddle factors in large transform sizes. In most cases, our design achieves a 30%-85% reduction in DSP area-time product (ATP) and a 70%-100% reduction in BRAM ATP compared to state-of-the-art designs with smaller or equivalent modulus, while maintaining competitive LUT and FF ATP, underscoring the potential of Fermat NTT-based polynomial multipliers in lattice-based cryptography.
Author Cheung, Ray C. C.
Xing, Yile
Li, Guangyan
Ye, Zewen
Chen, Donglong
Luk, Ryan W. L.
Yan, Hong
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Snippet Polynomial multiplication using Number Theoretic Transform (NTT) is crucial in lattice-based post-quantum cryptography (PQC) and fully homomorphic encryption...
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SubjectTerms conflict-free memory access
Convolution
Cryptography
Discrete Fourier transforms
Fermat number transform
Hardware
high radix
mixed radix
Polynomial multiplication
Polynomials
Training
Transforms
Urban areas
Vectors
Writing
Title High-Radix/Mixed-Radix NTT Multiplication Algorithm/Architecture Co-Design Over Fermat Modulus
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