Most public key crypto systems usefinitefield modulo arithmetic. This modulo arithmetic is applied on real numbers, binary values and polynomial functions. The computation cost is based on how it works with minimum use of scarce resources like processor and memory We have implemented the determination of the multiplicative inverse of a polynomial over GF(2p) with minimum computational cost. The "Extended Euclidean Algorithm" (EEA) has been demonstrated to work very well manually for integers and polynomials. However polynomial manipulation cannot be computerized directly. We have implemented the same by using simple bit wise shift and XOR operations. In small applications like smart cards, mobile devices and other small memory devices, this method works very well. To the best of our knowledge, the proposed algorithm seems to be the first, efficient and cost eective implementation of determining the multiplicative inverse of polynomials over GF(2p) using computers. As this is a pioneering work, the results could not be compared with that of any previous work.