Diminution of Extended Euclidean Algorithm for Finding Multiplicative Inverse in Galois Field

This manuscript deals with the theorem on diminution of the Extended Euclidean Algorithm for finding the multiplicative inverse of non-zero elemental polynomials of Galois field  with respect to a monic irreducible polynomial  over , where  is a prime and is any positive integer. This method became successful in finding the multiplicative inverse of all those non-zero polynomials for which the Extended Euclidean Algorithm fails. We find the inverse of all 342 non-zero elemental polynomials of  using an irreducible polynomial  We also used Cayley Hamilton’s theorem for finding the multiplicative inverse of the non-zero elemental polynomials of a finite field.