A link between two classes of permutation polynomials Abstract   The study of permutation polynomials over finite fields has attracted many scholarsu0027 attentions due to their wide applications and there are several different forms of permutations over finite fields. However, there is little literature on the relation between different forms of permutations.  In this paper, we find an equivalent relation between permutation polynomials of the form    f  (  x  )    x  h   (    x      2    n    xe2x88x92  1    )     over      F      2    2  n        and permutations of the form    F  (  x  )        H    1    (  xcex94  )      H    1    (  xcex94  )      H    2    (  xcex94  )      x    on      F      2    n       , where    xcex94      1      x    2      x    xcexb4     ,    xcexb4  xe2x88x88    F      2    n        with      Tr      2    n      (  xcexb4  )    1    and      H    1    ,    H    2      are the xcoordinate and 1coordinate of h (see  Definition 2.4 ), respectively. A special case with      H    1        H    2      1   , which means    F  (  x  )      H    1     (    1      x    2      x    xcexb4    )     x   , has been already studied  8 ,  16 ,  21 ,  22 . We summarize the known permutations and then construct 8 new classes of permutations of such form, from which one can derive some permutation polynomials of the form    x  h   (    x      2    n    xe2x88x92  1    )     over      F      2    2  n        directly. Finally, we give an asymptotic result about      (    1      x    2      x    xcexb4    )      2    k        x    to be a permutation over      F      2    n        using the HasseWeil bound.

More permutation polynomials with Niho exponents which permute Fq2 Abstract   Constructions of permutation polynomials over finite fields have attracted much interests in recent years, especially those with few terms, such as trinomials, due to their simple form and additional properties. In this paper, we construct several classes of permutation trinomials over      F      p    2  k        with Niho exponents of the form    f  (  x  )    x      xcexbb    1      x    s  (    p    k    xe2x88x92  1  )    1        xcexbb    2      x    t  (    p    k    xe2x88x92  1  )    1     ; some necessary and sufficient conditions for the polynomial    f  (  x  )    to permute      F      p    2  k        are provided. Specifically, for    p    5   , new permutation trinomials are presented. We also give recursive constructions of permutation polynomials using selfreciprocal polynomials.