关于Dickson多项式的不动点问题

ON THE FIXED POINTS OF DICKSON POLYNOMIAL D_n(x,1)

设,p>3是素数,证明了,当p(?)±1(mod5)或p(?)±1(mod7),且p(?)±1(mod8)或p≡11(mod30),等等,均存在有限域F_p上的d次置换多项式g_d(x,1),使其恰有5个不动点0,±1,±2,并由此提出一个猜想.此结果在运用置换多项式g_d(x,1)构造RSA公开密钥码体制的研究中,有重要意义.

If p is a prime, let F1 denotes the finite field of order p. For a E Fris called a Dickson polynomial of degree over Fr , where [] denotes the greatest integer function. Fora a= 0, Dickson polynomial D.(x,a) is a permutation polynomial of Ff if and only if (n, p2 - 1) = 1. Lets(n,p) = {b|Dn(b,1) =b,bEfr, (,pz - 1) =1}- i.e. the s(,p) denotes the set of fixed points of Dickson polynomial D,(x, 1) of degree over F, , where D,(x, 1) is a permutation polynomial of F,. We prove the followingTheorem. If p satisfies one in following conditions:then there is a Dickscn polynomial D.(x, 1) of degree n over Ft such that We have the followingConjecture 1 Let p > 3 , then there is a Dickson polynomial D.(x,1) of degree n over F, such that |s(,p) |=5.In 1985, Nobauer proved that |s(n,p)| = 1/2((n+ 1 ,p + 1) + (n + 1 ,p - 1) + (n - 1,p+1)+ ( n- 1,p - 1)) - 2. So, the conjecture 1 is equivalent toConjecture 2 Let p > 3 , then there is a positive integer n such that (n,p2- 1) = 1 and(n+ 1,p+ 1) + (n+ 1,p- 1) + (n- 1,p+ 1) + (n- 1,p- 1) = 14. Let p > 3 , it is dear, {0, + 1,+ 2} , s(n,p), and |s(n,p) | = 5. So, the conjecture 2 is equivalent toConjecture 3 Let p > 3 , then there is a Dickson polynomial D,(x, 1) of degree n over Ft such that s(n,p) = {0, + 1, + 2).