Methods of finding multiplicative inverses in GF(28)

Finding the multiplicative inverse of an element in Galois Field(p), GF(p) for small values of p such as 5 or 7 is no problem. One can find the multiplicative inverse by constructing multiplication tables and establish the desired value directly. The look-up table procedure, when implemented through software, is fast and handy, and is employed in the design of S-boxes of the Rijndael encryption algorithm used in advanced encryption standard (AES). However, a second choice, if execution time is not the consideration, is the use of extended Euclid’s algorithm. The two methods of finding multiplicative inverse in GF(28) presented in many books are discussed in this paper. An effort is made to present the working of these methods in a detailed manner which is not found in any text. A comparison of these methods vis-à-vis time, transistor count and complexity is also made. Section 1 contains a brief introduction to the topic, Section 2 reviews multiplication in polynomial arithmetic. Multiplicative inverses in GF(28) are taken up in Sections 3, and 4 concludes the paper.