Standard – composite field conversion

Have just read this and this is all I understand about the paper:

  1. There are various way to represent the element of GF(2^k) depending on the choice of the basis or the particular construction method. If k is the product of two integers as k=m.n, then it is possible to derive a different representation method by defining GF(2^k) over the ground field GF(2^n). An extension field defined over a subfield of GF(2^k) other than the prime field GF(2) is know as the composite field. We will use GF((2^n)^m) to denote the composite field. Since there is only one field with 2^k elements, both the binary and the composite fields refer to this same field. However, their representation methods are different, and it is possible to obtain one representation from the other.
  2. For GF(2^8), A=(a_7\alpha^7, a_6\alpha^6, ..., a1\alpha,a_0) where the composite field representation in GF((2^n)^m) is {\bar{A}={\bar{a}_{00},\bar{a}_{01},\bar{a}_{10},\bar{a}_{11},\bar{a}_{20},\bar{a}_{21},\bar{a}_{30},\bar{a}_{31}}}
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