# Efficient polynomial ring multiplication with coefficients in GF(2^s)

Is there something similar (or an other efficient approach) to the NTT for the multiplication of polynomials in polynomial rings modulo $$X^{2^n} + 1$$ with coefficients in $$\operatorname{GF}(p)$$ with $$p \equiv 1 \pmod{2^{n+1}}$$, but instead for polynomial rings with coefficients in $$\operatorname{GF}(2^s)$$?

• Not NTT, but you might be interested to see how BearSSL computes $\operatorname{GF}(2^{128})$ arithmetic for GHASH in software: crypto.stackexchange.com/a/66462 – Squeamish Ossifrage Feb 23 at 14:20
• On characteristic 2 additive FFTs serve this purpose. Chapter 3 of Mateer's thesis has a number of these, and several others have been discovered since then. – Samuel Neves Feb 23 at 15:24
• FFT-based GHASH with high bulk throughput (but also high setup overhead): eprint.iacr.org/2014/729 – Squeamish Ossifrage Feb 23 at 16:14
• Thank you very much. I will take a look at all this content – Pro7ech Feb 24 at 9:45