# Optimized Inversion Algorithm Finite-field inversion can be optimized when the extension degree is small and the field's algebraic structure is well-defined. Specifically, extension fields of degrees three and four enable inversion to be formulated in closed form, leveraging low-dimensional linear algebra and quadratic subextensions. For the quadratic extension ($$\mathbb{F}\_{p^2}$$), inversion admits a particularly simple closed form. Writing an element as $$x = x\_0 + x\_1 u$$, the inverse is obtained by first forming a base-field denominator $$x\_0^2 - \eta x\_1^2$$ and then multiplying its inverse by the conjugate $$x\_0 - x\_1 u$$. For the cubic extension ($$\mathbb{F}\_{p^3}$$), multiplication by a fixed element defines an $$\mathbb{F}\_p$$ -linear endomorphism of a three-dimensional vector space. Inversion can therefore be obtained by inverting this linear transformation, leading to explicit formulas involving determinants and adjugate matrices. For the quartic extension ($$\mathbb{F}*{p^4}$$), the field admits a natural representation as a quadratic extension over $$\mathbb{F}*{p^2}$$. This tower structure enables inversion via conjugation with respect to the intermediate field, reducing the problem to the inversion of a single element in $$\mathbb{F}\_{p^2}$$. ### 1. Inversion in $$\mathbb{F}\_{p^3}$$ #### 1.1 Field Definition Let $$ \mathbb{F}\_{p^3} = \mathbb{F}\_p\[u] / (u^3 - \xi), $$ where $$\xi \in \mathbb{F}*p$$ is fixed. Every element $$x \in \mathbb{F}*{p^3}$$ is written uniquely as $$ x = x\_0 + x\_1 u + x\_2 u^2, \qquad x\_0, x\_1, x\_2 \in \mathbb{F}\_p. $$ #### 1.2 Multiplication as a Linear Operator Multiplication by $$x$$ defines an $$\mathbb{F}*p$$-linear map $$L\_x : \mathbb{F}*{p^3} \to \mathbb{F}\_{p^3}$$ given by $$L\_x(y) = x y$$. With respect to the basis $${1, u, u^2}$$, this map is represented by $$ M\_x = \begin{pmatrix} x\_0 & \xi x\_2 & \xi x\_1 \\ x\_1 & x\_0 & \xi x\_2 \\ x\_2 & x\_1 & x\_0 \end{pmatrix}. $$ #### 1.3 Inversion via Linear Algebra Assume $$x \neq 0$$. Then $$L\_x$$ is invertible and the inverse element satisfies $$x^{-1} = L\_x^{-1}(1)$$. By [Cramer’s rule](https://en.wikipedia.org/wiki/Cramer%27s_rule), $$ M\_x^{-1} = \det(M\_x)^{-1} \operatorname{adj}(M\_x), $$ so the coordinates of $$x^{-1}$$ are given by the first column of $$\operatorname{adj}(M\_x)$$. #### 1.4 Explicit Inversion Formula 1. $$t\_0, t\_1, t\_2$$: The First Column of the Adjugate Matrix * $$t\_0 = x\_0^2 - \xi x\_1 x\_2$$ * This is the cofactor of the $$M\_x\[1, 1]$$, where $$M\_x\[r, c]$$ is the element at row $$r$$ and coloum $$c$$. * It is calculated as the determinant of the $$2 \times 2$$ matrix $$\begin{pmatrix} x\_0 & \xi x\_2 \ x\_1 & x\_0 \end{pmatrix}$$ remaining after crossing out the 1st row and 1st column. * $$t\_1 = \xi x\_2^2 - x\_0 x\_1$$ * This is the cofactor for the $$M\_x\[1, 2]$$. * The sign $$(-)$$ is already integrated into the simplified formula in the code. * $$t\_2 = x\_1^2 - x\_0 x\_2$$ * This is the cofactor for the $$M\_x\[1, 3]$$. These $$t\_0, t\_1, t\_2$$ constitute the **first column of the adjugate matrix** $$\text{adj}(M\_x)$$. In extension fields, the matrix has a **circulant** structure, meaning that knowing the first column is sufficient to determine the entire inverse.
2. $$t\_3$$: The Determinant $$ t\_3 = x\_0 t\_0 + \xi (x\_2 t\_1 + x\_1 t\_2) $$ * This is the [**Laplace expansion**](https://en.wikipedia.org/wiki/Laplace_expansion) along the first column of matrix $$M\_x$$. * It performs $$M\_x\[1, 1] \cdot t\_0 + M\[1, 2] \cdot t\_1 + M\[1,3] \cdot t\_3$$ 2. $$y = (y\_0, y\_1, y\_2)$$: The Final Inverse $$ (y\_0, y\_1, y\_2) = t\_3^{-1} \cdot (t\_0, t\_1, t\_2) $$ *** ### 2. Inversion in the Quartic Extension Field Direct inversion via a $$4 \times 4$$ matrix is inefficient. Instead, $$\mathbb{F}*{p^4}$$ is viewed as a quadratic extension over $$\mathbb{F}*{p^2}$$: $$ \mathbb{F}*p ;\xrightarrow{, v^2 = \xi ,}; \mathbb{F}*{p^2} ;\xrightarrow{, u^2 = v ,}; \mathbb{F}\_{p^4}. $$ #### 2.1 Tower Construction Let $$\mathbb{F}*{p^2} = \mathbb{F}*p\[v]/(v^2 - \xi)$$ and $$\mathbb{F}*{p^4} = \mathbb{F}*{p^2}\[u]/(u^2 - v)$$. Every element $$x \in \mathbb{F}*{p^4}$$ can be written as $$x = A + B u$$ with $$A, B \in \mathbb{F}*{p^2}$$. $$ \begin{align\*} x &= x\_0 + x\_1u + x\_2u^2 + x\_3u^3 \\ &= (x\_0 + x\_2u^2) + (x\_1 + x\_3u^2)u \\ &= (x\_0 + x\_2v) + (x\_1 + x\_3v)u \end{align\*} $$ #### 2.2 Quadratic Conjugation Define the conjugate of $$x$$ over $$\mathbb{F}*{p^2}$$ by $$\bar{x} = A - B u$$. This operation satisfies $$x \bar{x} \in \mathbb{F}*{p^2}$$. #### 2.3 First Level of Rationalization (Removing $$u$$) Our goal is to find $$x^{-1} = \frac{1}{A+Bu}$$. To remove $$u$$ from the denominator, we multiply the top and bottom by the conjugate $$\bar{x} = A - Bu$$: $$ x^{-1} = \frac{1}{A+Bu} \cdot \frac{A-Bu}{A-Bu} = \frac{A-Bu}{A^2 - B^2u^2} $$ Since $$u^2 = v$$, the denominator becomes $$D = A^2 - B^2v$$. Now, $$u$$ has been eliminated from the denominator, leaving only $$v$$. #### 2.4 Second Level of Rationalization (Removing $$v$$) Let's look at the new denominator $$D$$. Since $$A$$ and $$B$$ contain $$v$$, $$D$$ can be expanded into a linear form of $$v$$: $$D = D\_0 + D\_1v$$. To remove $$v$$, we multiply by its conjugate $$(D\_0 - D\_1v)$$: $$ x^{-1} = \frac{A - Bu}{D\_0 + D\_1v} \cdot \frac{D\_0 - D\_1v}{D\_0 - D\_1v} = \frac{(A - Bu) \cdot (D\_0 - D\_1v)}{D\_0^2 - D\_1^2v^2} $$ Since $$v^2 = \xi$$, the final denominator becomes $$N = D\_0^2 - D\_1^2\xi$$. This value $$N$$ (the Norm) is now a pure scalar (BaseField), free of both $$u$$ and $$v$$. #### 2.5 Inversion Formula Assume $$x \neq 0$$. Then $$D \neq 0$$, and the inverse of $$x$$ is $$ x^{-1} = D^{-1} (A - B u). $$ Thus inversion in $$\mathbb{F}*{p^4}$$ reduces to inversion in $$\mathbb{F}*{p^2}$$. #### 2.6 Inversion in the Quadratic Subfield Writing $$D = D\_0 + D\_1 v$$ with $$D\_0, D\_1 \in \mathbb{F}\_p$$, one has $$ D^{-1} = (D\_0^2 - \xi D\_1^2)^{-1} (D\_0 - D\_1 v). $$ This completes the inversion algorithm using only base-field arithmetic. ### 3. Performance Comparison with the [Itoh–Tsujii Algorithm](/intro/primitives/abstract-algebra/extension-field/inversion.md) This section compares the specialized inversion methods for low-degree extension fields with the Itoh–Tsujii algorithm from the perspective of arithmetic cost. The comparison is expressed in terms of operations over the prime field $$\mathbb{F}\_p$$. Throughout, it is assumed that: * Frobenius applications are free or negligible compared to multiplication, * squaring and multiplication in $$\mathbb{F}\_p$$ are counted separately *** #### 3.1 Arithmetic Cost Summary
FieldMethodSquareMultiplicationInversionStructural Cost
\mathbb{F}_{p^3}Matrix-based inversion391Linear algebra (determinant + adjugate)
\mathbb{F}_{p^3}Itoh–Tsujii3\approx 121Extension-field multiplications
\mathbb{F}_{p^4}Quadratic tower inversion6121Two successive quadratic reductions
\mathbb{F}_{p^4}Itoh–Tsujii\approx 8\approx 181Flat exponentiation in full extension
#### 3.2 Interpretation For both cubic and quartic extensions, the specialized inversion formulas reduce the number of base-field multiplications by avoiding full extension-field multiplication. * In the cubic case, inversion is obtained via explicit determinant and adjugate computation, keeping all arithmetic in $$\mathbb{F}\_p$$. * In the quartic case, inversion proceeds through a quadratic tower, so that all intermediate inversions occur in proper subfields. In contrast, the Itoh–Tsujii algorithm operates uniformly on the full extension field $$\mathbb{F}\_{p^k}$$ and requires additional extension-field multiplications, which expand to a larger number of base-field operations. #### 3.3 Asymptotic Perspective However, for small values of $$k$$, the constant factors dominate: * Specialized methods achieve lower arithmetic cost by exploiting explicit low-degree structure. * Itoh–Tsujii trades higher constant factors for generality and uniformity across extension degrees. As a result, for extension degrees $$k = 3$$ and $$k = 4$$, the specialized algorithms are typically preferred. ### A. Possibility of Inversion over a Quadratic Tower #### A.1 Setting Consider an extension field of degree $$2^n$$ over the prime field $$\mathbb{F}\_p$$. Assume that the field admits a recursive quadratic tower representation of the form $$ \mathbb{F}\_{p^{2^n}} ===================== \mathbb{F}*{p^{2^{n-1}}}\[u*{n-1}] / (u\_{n-1}^2 - \eta\_{n-1}), $$ where $$\eta\_{n-1} \in \mathbb{F}\_{p^{2^{n-1}}}$$ is a non-square. #### A.2 Quadratic Reduction Step At each level $$i$$ of the tower, an element of $$\mathbb{F}*{p^{2^{i+1}}}$$ can be written as $$x = x\_0 + x\_1 u\_i$$ with $$x\_0, x\_1 \in \mathbb{F}*{p^{2^i}}$$. The inverse of $$x$$ is computed by: * forming the denominator $$x\_0^2 - \eta\_i x\_1^2 \in \mathbb{F}\_{p^{2^i}}$$, * multiplying its inverse by the conjugate $$x\_0 - x\_1 u\_i$$. This reduces inversion in $$\mathbb{F}*{p^{2^{i+1}}}$$ to a single inversion in $$\mathbb{F}*{p^{2^i}}$$. #### A.3 Recursive Structure By recursively applying the quadratic reduction step, inversion in $$\mathbb{F}\_{p^{2^n}}$$ is reduced along the tower $$ \mathbb{F}*{p^{2^n}} \rightarrow \mathbb{F}*{p^{2^{n-1}}} \rightarrow \cdots \rightarrow \mathbb{F}\_{p^2} \rightarrow \mathbb{F}\_p. $$ As a result, exactly one inversion in the base field $$\mathbb{F}\_p$$ is required, together with a bounded number of multiplications and squarings in intermediate subfields. #### A.4 Applicability Conditions This approach is applicable if and only if: * the extension field is explicitly represented as a quadratic tower, * a suitable non-residue $$\eta\_i$$ is available at each level of the tower. If the field is instead given as a flat extension $$\mathbb{F}\_p\[x]/(f(x))$$ with $$\deg f = 2^n$$, a compatible quadratic tower representation may not be immediately available and may require nontrivial basis transformations. #### A.5 Limitations Although quadratic tower inversion is structurally simple and implementation-friendly, it is not universally optimal. For large values of $$n$$, the accumulated cost of intermediate multiplications may outweigh the benefits of avoiding extension-field exponentiation. Moreover, the existence of suitable non-residues is field-dependent and must be ensured by construction. > Written by [Soowon Jeong](mailto:undefined) of Fractalyze --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/extension-field/inversion/optimized-inversion-algorithm.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.