# Construction of Isomorphism between Galois Fields

I am trying to create an isomorphism between GF($2^8$) and an extension field in GF($(2^4)^2$). The finite field GF($2^8$) uses the Rijndael(AES) irreducible polynomial.In this field I have found 128 generators(G1).

I have found 3 irreducible polynomials(IR1) in GF($2^4$). For each irreducible polynomial, I have found 120 irreducible polynomials(IR2) in the extension field GF($(2^4)^2$). For each of the 360 pairs of (IR1,IR2) I have found the 128 generators(G2).

So, now I am trying to create an isomorphism using (G1,G2) by mapping $G1^i \to G2^i$ ( by doing multiplications in their respective fields). What I find is that many pairs of (G1,G2) are not forming isomorphisms . This was verified by checking if map(1) +map(2) =map(3).

Is this supposed to happen? Why/Why not? Please explain in detail.

Note:My knowledge in isomorphisms is not great and was unable to find good online sources for the same.

• – user991 Aug 20 '17 at 4:02
• I'm voting to close this question as off-topic because it is about general mathematics. – fkraiem Aug 20 '17 at 5:35
• You are trying to do an isomorphism to simplify the math required to calculate the SBox? – b degnan Aug 20 '17 at 14:43
• @fkraiem: you often seem to see a clear border between mathematics and cryptography, which I have difficulties to see. – user27950 Aug 20 '17 at 21:11
• Yes, I am trying to implement the S-box on an FPGA. I had read the Canright paper and want to down to the GF((2^4)^2) field to make the implementation faster. – Aditya Pradeep Aug 22 '17 at 1:35