Exploring fields with shift registers
Radowicz, Jody L.
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The S-Boxes used in the AES algorithm are generated by field extensions of the Galois field over two elements, called GF(2). Therefore, understanding the field extensions provides a method of analysis, potentially efficient implementation, and efficient attacks. Different polynomials can be used to generate the fields, and we explore the set of polynomials x^ 2 + x + a^J over GF(2^n) where a is a primitive element of GF(2^n). The results of this work are the first steps towards a full understanding of the field that AES computation occurs in-GF(2^8). The charts created with the data we gathered detail which power of the current primitive root is equal to previous primitive roots for fields up through GF(2^16) created by polynomials of the form x^2 + x + a^i for a primitive element a. Currently, a C++ program will also provide all the primitive polynomials of the form x^2 + x+ a^i for a primitive element a over the fields through GF(2^32). This work also led to a deeper understanding of certain elements of a field and their equivalent shift register state. In addition, given an irreducible polynomial 2 f(x) = x^2 + a^i x + a^j over GF(2^n), the period (and therefore the primitivity) can be determined by a new theorem without running the shift register generated by f(x).
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