Multiplicative Polynomial Inverse Over GF(73): Crisis of EEA and Its Solution

The multiplicative polynomial inverses of all elemental polynomials exist under each of all irreducible polynomials over the finite field GF(p(m)) where p is a prime integer and both p and m >= 2. For GF(2(8)), the Extended Euclidean Algorithm (EEA) successfully finds multiplicative inverses of all the 255 elemental polynomials under each of 30 irreducible polynomials. However, for GF(7(3)), the same algorithm cannot find multiplicative inverses of all the 342 elemental polynomials under each of its 112 monic irreducible polynomials. A simple algebraic method proposed in the paper finds all the 112 monic irreducible polynomials over GF(7(3)) along with the multiplicative inverses of all the 342 elemental polynomials under each of the 112 irreducible polynomials.