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.

翻译：有限半域是有限域上的可除代数，其中乘法运算不一定满足结合律。本文研究小规模半域及有限域扩张中乘法运算的复杂度。对于该运算，所需基域乘法运算的次数称为张量秩或乘法复杂度。其他基域运算包括加法与常数标量乘法，这些运算统称为加法复杂度。在递归应用时，张量秩决定了渐近复杂度界限的指数项，而其他运算则决定其常数项。对于小规模扩张，这两类度量具有同等重要性。本文确定了特征为2和3的若干半域及有限域的张量秩，并针对其加法复杂度提出了新的上下界。通过设计具有特定参数的线性码的短直线式程序，我们给出了在整体复杂度上优于现有技术的新算法。