# An approximate quantity of multiplications in $\mathbb{F}_p$ amounting the same bit complexity as one inversion in $\mathbb{F}_p$

Consider a prime finite field $$\mathbb{F}_p$$ of quite large characteristics $$p$$, for example $$\log_2(p) \approx 256$$ bits. I would like to know an approximate quantity of multiplications in $$\mathbb{F}_p$$ amounting the same bit complexity as one inversion in $$\mathbb{F}_p$$. Of course, this depends on many factors, hence I propose to listen your ideas rather than exact results. May be, is there an article or book, which discusses this topic?

One algorithm to compute inverses in a finite field is to raise to a large power. In particular, $$x^{-1} = x^{p-2}$$ if you are working in $$\mathbb{F}_p$$ (by Fermat's little theorem). You can use standard algorithms for fast modular exponentiation to compute this with about $$2 \lg p$$ multiplications, or fewer (depending on the specific value of $$p$$ and the length of the shortest addition chain for $$p-2$$).
• I know these methods, I would like to understand their algebraic complexity, i.e., amount of necessary multiplications in $\mathbb{F}_p$ for one inversion in $\mathbb{F}_p$. However, in books I saw only bit complexity. – Dima Koshelev Feb 23 '19 at 10:37