Solving X^2^2k+2^k+1+(X+1)^2^2k+2^k+1=b over 2^4k
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Let F(X)=X^2^2k+2^k+1 be the power function over the finite field 2^4k which is known as the Bracken-Leander function. In <cit.>, it was proved that the number of solutions in q^4 to the equation F(X)+F(X+1)=b is in {0,2,4} for any b∈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 <cit.>, determining differential spectrum of F(X)=X^2^2k+2^k+1 over 2^4k, is re-derived simply from our results.
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