Constructions Of Two Types Of Permutation Polynomials With Special Forms Over Finite Fields

Let q be a power of a prime.Let Fq be a finite field with q elements.A polyno-mial f(x)?Fq[x]is called a permutation polynomial(PP)if its associated polynomial mapping f:c?f(c)from Fq to itself is a bijection.PPs over finite fields have important applications in cryptography,coding theory and combinatorial design theo-ry.Since Hermite studied permutation polynomials over prime fields,the construction of permutation polynomials over finite fields has been a hot topic for more than one hundred years.In recant years,PPs over Fqm with the form(xq-x+?)s+cx and cx-xs+xqs attract many researchers' attention due to their simple algebraic representation and wide applications.In this thesis,we mainly investigated the relation between the general forms of two types of PPs,and obtain new PPs of the second form from the known ones of the first form by applying this relation.To this end,we firstly proposed four classes of PPs of the form cx-xs+xqs over Fq2,where(?).s=3q2+2q-1/4=3q+5/4(q-1)+1,q,c satisfy one of the following condition:(a).q?1(mod 8)and(-2/c)q+1/2=1,(b).q?5(mod 8)and(2/c)q+1/2=1;(?).s=(q+1)2/4=q+3/4(q-1)+1,q,c satisfy one of the following condition:(a).q?5(mod 8)and(-2/c)q+1/2=1,(b).q?1(mod 8)and(2/c)q+1/2=1;(?).s=q2+q+1/3=q+2/3(q-1)+1,q?1(mod 3);(?).(?),q is a even power of an odd prime.Apart from the above four PPs,we also listed all known PPs of the form cx-xs+xqs over Fq2.Based on the above relation,we obtain many new PPs of the form(xq-x+?)s+cx.As a contrast,the previous constructions always needed to choose special ?,but all new PPs we obtain exist for any ? ?Fq2.Moreover,we also checked the known PPS of the form(xq-x+?)s+cx,and deduced sporadic new PPs of the form cx-xs+Xqs.