Two classes of permutation polynomials with Niho exponents over finite fields with even characteristic

In this paper, by transforming the permutation problem into the root distribution problem in the unit circle of certain quadratic and cubic equations, we investigate the permutation behavior of the type $f(x)=x+x^{2^{3m}-2^{m}+1}+x^{2^{4m}-2^{3m}+2^{m}}$ over $mathbb{F}_{2^{4m}}$ and $f(x)=x+x^{2^{m}}+x^{2^{m+1}-1}+ax^{2^{2m}-2^{m}+1}$ over $mathbb{F}_{2^{2m}},$ respectively.

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