Let $F(X)=X^{ 2^{ 2k}+2^k+1}$ be the power function over the finite field $\GF{ 2^{ 4k}}$ which is known as the Bracken-Leander function. In \cite{ BCC10,BL10,CV20,Fu22,XY17}, it was proved that the number of solutions in $\GF{ q^4}$ to the equation $F(X)+F(X+1)=b$ is in $\{ 0,2,4\}$ for any $b\in \GF{ q^4}$ and the number of the $b$ giving $i$ solutions have been determined for every $i$. However, no paper provided a direct and complete method to solve such an equation, and this problem remained open. This article presents a direct technique to derive an explicit solution to that equation. The main result in \cite{ BCC10,BL10,Fu22,XY17}, determining differential spectrum of $F(X)=X^{ 2^{ 2k}+2^k+1}$ over $\GF{ 2^{ 4k}}$, is re-derived simply from our results.
Categories: cs.IT math.IT math.NT