solutions of Differentially uniform mappings for cryptography
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Kaisa Nyberg provided a proof of number of zeros in inverse mapping in finite field ref. The proof is clear for me except the final step where she proved that the following equation has two solution when n is odd in $GF(2^n)$ and 4 solutions when n is even: $x(x^3 +alpha^3)= 0$ Q1) she applied $gcd(3,2^n-1)=1$ for her proof. could you please explain how to use gcd in finding zeros in $GF(2^n)$ ? Q2) I extended her approach to apply for $(x+alpha)^{-2} - x^{-2}=beta$ and obtained $x^{2}(x^6 +alpha^6)= 0$ in $GF(2^n)$ . I got stuck to prove the number of zeros when n is odd and even. i took this approach to understand the mathematical motivation taken for one of the Aria cipher sbox ( $x^{247}=x^{-8}$ ). could you please help me to continue proof number of solutions?
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