Three classes of permutation quadrinomials in odd characteristic

In this paper, we construct three classes of permutation quadrinomials with Niho exponents of the form f (x) = alpha(0)x(r) + alpha(1)x(s1(pm-1)+r) + alpha(2)x(s2(pm-1)+r) + alpha(3)x(s3(pm-1)+r) is an element of F-pn [x], where p is an odd prime, n = 2m is a positive even integer, and (r, s(1), s(2), s(3)) = (1, -1/p(k)-2, 1, p(k)-1/p(k)-2), (1, p(k)+1/p(k)+2, 1, 1/p(k)+2) and (3, 1, 2, 3), respectively. The exponents of the first two classes are considered for the first time, and the third class covers all the permutation polynomials proposed by Gupta (Designs Codes and Cryptography 88, 1-17, 2020).