Functions　with　low　differential　uniformity　have　wide　applications　in　cryptography.　In　this　paper,　by　using　the　quadratic　character　of　Fpn∗,　we　further　investigate　the　(-1)-differential　uniformity　of　these　functions　in　odd　characteristic:　(1)　f1(x)=xd,　where　d=-pn-12+pk+1,　n　and　k　are　two　positive　integers　satisfying　ngcd(n,k)　is　odd;　(2)　f2(x)=(xpm-x)pn-12+1+x+xpm,　where　n=3m;　(3)　f3(x)=(x3m-x)3n-12+1+(x3m-x)3n-12+3m+x,　where　n=3m.　The　results　show　that　the　upper　bounds　on　the　(-1)-differential　uniformity　of　the　power　function　f1(x)　are　derived.　Furthermore,　we　determine　the　(-1)-differential　uniformity　of　two　classes　of　permutation　polynomials　f2(x)　and　f3(x)　over　Fpn　and　F3n,　respectively.　Specifically,　a　class　of　permutation　polynomial　f3(x)　that　is　of　P-1N　or　AP-1N　function　over　F3n　is　obtained.　©　The　Author(s),　under　exclusive　licence　to　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature　2023.