Engineering / Mathematics — Polynomial Algorithms
Polynomial Modular Inverses
Inverting a polynomial modulo another using the extended Euclidean algorithm, and the finite field application.
Executive summary
A polynomial is invertible modulo another exactly when they are coprime, and the inverse comes from the extended Euclidean algorithm.
This is the operation that makes finite field arithmetic possible, since field elements are polynomial residues and division requires inversion.
Learning objectives
- State the invertibility condition and the algorithm.
- Apply it to finite field arithmetic.
- Compare with the exponentiation alternative.
01The algorithm
Invertibility criterion
a is invertible modulo h in F[X] if and only if gcd(a, h) = 1.
Polynomial modular inverse
polynomials a, h over a fieldthe inverse of a modulo h, or a report of non-invertibility- Run extended Euclid on (a, h) to obtain d, s, t with as + ht = d.
- If deg d > 0, report that a is not invertible modulo h.
- Otherwise d is a non-zero constant; return s/d reduced modulo h.
O(deg(h)²) field operationsThe division by the constant d is the normalisation step. Extended Euclid returns a gcd that is a non-zero constant rather than exactly 1, and scaling is needed to make the identity read as ≡ 1.
02Finite field arithmetic
A finite field F_q = F_p[X]/(f) has elements represented as polynomial residues of degree below deg f. Division requires inverting such a residue, which is exactly this operation.
| Operation in F_{p^k} | Implementation | Cost |
|---|---|---|
| Addition | Coefficientwise in F_p | O(k) |
| Multiplication | Polynomial multiply then reduce mod f | O(k²) |
| Inversion, Euclid | Extended Euclid against f | O(k²), large constant |
| Inversion, exponentiation | a^{q−2} by repeated squaring | O(k² log q) |
| Frobenius, a ↦ a^p | Linear map; precomputable matrix | O(k²) or O(k) with a table |
Inversion is the expensive primitive, as in the integer case, and the same avoidance strategies apply. Projective coordinates in elliptic curve arithmetic exist precisely to defer inversions to a single one at the end.
03Euclid versus exponentiation
Two methods invert a field element, mirroring the integer situation exactly.
Extended Euclid
Asymptotically better and faster in practice, but branches on the operand values, so its running time depends on the input.
Fermat exponentiation
Compute a^{q−2} by a fixed exponentiation ladder. Slower but with data-independent control flow.
Batch inversion
Montgomery's trick inverts n elements with one inversion and about 3n multiplications, and applies unchanged here.
The batch trick is the most valuable of the three when many inversions are needed together, since it converts almost all of them into multiplications, which are both faster and easier to implement in constant time.
04Frequently asked questions
Why is the gcd a constant rather than 1?
Because the gcd in F[X] is defined only up to a unit, and units are the non-zero constants. Extended Euclid returns some constant multiple, and dividing by it normalises the identity.
Is inversion needed if the modulus is irreducible?
Yes — irreducibility guarantees every non-zero element has an inverse, but computing it still requires work. Irreducibility makes the operation always succeed rather than making it free.
How much slower is constant-time inversion?
The exponentiation route costs a factor of about log q more than Euclid. In elliptic curve implementations this is why inversions are batched or eliminated by projective coordinates rather than made constant time individually.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 405-406.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.