Computational Explorations on Semifields

A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the number of required base field multiplications is the tensor rank, or the multiplicative complexity. The other base field operations are additions and scalings by constants, which together we refer to as the additive complexity. When used recursively, the tensor rank determines the exponent while the other operations determine the constant of the associated asymptotic complexity bounds. For small extensions, both measures are of similar importance. In this paper, we establish the tensor rank of some semifields and finite fields of characteristics 2 and 3. We also propose new upper and lower bounds on their additive complexity, and give new associated algorithms improving on the state-of-the-art in terms of overall complexity. We achieve this by considering short straight line programs for encoding linear codes with given parameters.

💡 Research Summary

The paper investigates the computational complexity of multiplication in small semifields and finite‑field extensions of characteristics 2 and 3. A semifield is a finite division algebra over a finite field whose multiplication need not be associative; when an identity element exists it is called a semifield, otherwise a pre‑semifield. The authors focus on two complementary measures of complexity: the tensor rank (also called multiplicative complexity), which counts the number of base‑field multiplications required, and the additive complexity, which counts additions and scalar multiplications by constants. While the tensor rank determines the exponent in recursive algorithms, the additive complexity determines the constant factor, and both are crucial for small extensions where table‑lookup approaches are infeasible due to side‑channel resistance requirements.

Theoretical framework.
Multiplication is modelled as a 3‑tensor (T\in (F^n)^\ast\otimes (F^n)^\ast\otimes F^n). A decomposition (T=\sum_{i=1}^r L_i\otimes R_i\otimes P_i) yields three matrices (L,R,P) (the lrp representation). The minimal possible (r) is the tensor rank. By a theorem of Brocket and Dobkin, any such decomposition forces the three matrices to be generator matrices of an (