On The Trace Of The Product Of K Elements In The Finite Field And Pseudorandom Binary Lattice Generated Base On The Quadratic Character In Finite Field

Let p be an odd prime,let q=pr with r?2 be an integer and consider the finite field Fq.we will consider the trace function Tr from IFq to FP which is of basic importance in finite fields.Let k? 2 be an integer,and let C1,C2,...,Ck are large subsets of F1.We consider the distribution of Tr(c1c2…ck)when c1 ? C1,c2 ? C2,…ck?Ck and we get that Tr(clc2…ck)is well distributed in IFp when C1,C2,…,Ck satisfies certain conditions.Moreover,we constructed large family of pseudorandom lattices by using the quadratic characteristics and multiplicative inverses in finite fields,and studied the cryptography properties:pseudorandom measure,collision and avalanche effect.In particular,Let ? be the quadratic character of IFq,f(x)?Fg[x].Define?(x)=?(x1,x2,…,xn)(?) We prove that if the polynomials xf(x)+1 has no zero in Fq then we have Qk(?)<<k(deg(f)+2)q1/2(1+logp)n.We further studied the collision and avalanche effect of large family of pseudo-random lattices.