Generalized number-theoretic transformation for split rings

The number-theoretic transformation (NTT) is generally considered the most efficient method for computing polynomial multiplications with high dimensions and integer coefficients due to its quasilinear complexity.

What is the relationship between the NTT variants constructed by splitting the original polynomials into sets of lower degree subpolynomials such as K-NTT, H-NTT and G3-NTT? Can they be considered as special cases of a given algorithm under different parameterizations?

To solve the problems, a research team led by Yunlei Zhao published a new study in Limits of computer science.

The team proposed the first generalized splitting ring number theory transformation, called GSR-NTT. They then investigated the relationship between K-NTT, H-NTT, and G3-NTT.

In their research, they study generalized splitting ring polynomial multiplication based on the monic incremental polynomial variety and propose the first generalized splitting ring number theory transformation, called GSR-NTT. They show that K-NTT, H-NTT and G3-NTT can be considered special cases of GSR-NTT under different parameterizations.

They introduce a concise methodology for complexity analysis, based on which GSR-NTT can derive its optimal parameter settings. They provide GSR-NTT with further instantiations based on cyclic convolution polynomials and power-of-three cyclotomous polynomials.

They apply GSR-NTT to accelerate polynomial multiplication in the lattice-based scheme NTTRU and simple polynomial multiplication over power-of-three cyclotomous polynomial rings. The experimental results show that for NTTRU, GSR-NTT achieves speedups of 24.7%, 37.6%, and 28.9% for the key generation, encapsulation, and decapsulation algorithms, respectively, resulting in an overall speedup of 29.4%.

Future work may focus on implementing GSR-NTT on additional platforms.