On Two Pseudorandom Sequences

The work in this thesis contains two parts: Let Z/(p^e)be the integer residue ring modulop^ewith p an odd prime and integer e≥3.We firstly discuss on the injective property of compression maps acting on primitive sequences over Z/(p^e),and large quantities of injective functions over Z/(p),which are of form are derived.Besides,the influence of the injective function defined as above acting on the primitive sequences is partly given.Secondly,we focus on the linear complexity of nonlinear filtered sequences.Let(?)be a binary m-sequence of degree n,andαa root of(?)'s minimal polynomial,then the following results are gained:(1)Let(?)be the nonlinear filtered sequence derived by applying the Boolean function with a unique maximum order term x_i1x_i2...x_ikon(?).Ifα^i₁,α^i₂,...,α^i_kare linear independent over F₂ and F₂(α^i₁,α^i₂,...,α^i_k)= F_2^n*.with n^*≠n,then the linear complexity of b denoted by LC((?)) satisfies(2)We point out the problems in reference 36,and get the following conclusions:Let k be a positive integer with 2≤k≤n,gcd(d,n)=1,andα^δa normal element in F_2ⁿover F2,where 1≤δ<2ⁿ-1.Let(?)b be the nonlinear filtered sequence of(?),whose nonlinear filtered function is a Boolean function with a unique maximum order term x_δx_δ2^dx_δ2^2d...x_δ2^(k-1)d,then LC((?))is lower bounded by(?).Denote the number of normal elements in F_2ⁿover F₂ by v(n), if2≤k≤n-2,then the number of different x_δx_δ2^dx_δ2^2d...x_δ2^(k-1)dis(?)(n)·v(n)/2.