New explicit injective compressing mappings on primitive sequences over Zpe

Linear feedback shift registers over residue rings play a vital role in communication theory and cryptography. Let p be an odd prime and e >= 2 an integer. For any integer N >= 2, let Z(N) denote the residue ring modulo N. Let sigma(x) be a primitive polynomial over Z(pe), and G'(sigma(x), p(e)) the set of primitive linear recurring sequences generated by sigma(x). Given a mapping phi : Z(pe) -> Z(N), its induced mapping (phi) over cap transforms a sequence (..., u(i-1), u(i), u(i+1),...) to (..., phi(u(i-1)), phi(u(i)), phi(u(i+1)),...). Then phi is called an injective compressing mapping (w.r.t. s-uniformity) if for any two distinct sequences (u) under bar, (v) under bar is an element of G' (phi(x), p(e)), at least one element of Z(N) (s is an element of Z(N)) is distributed in (phi) over cap((u) under bar) differently from in (phi) over cap((u) under bar). It has been desirable to construct explicit injective compressing mappings (w.r.t. s-uniformity). Let the i-th coordinate a(i) of a is an element of Z(pe) be given by a = Sigma(e-1)(i=0) a(i)p(i), a(i) is an element of Z(p). In this correspondence, it is proved that any permutation polynomial in the (e -1)-th coordinate is an injective compressing mapping w.r.t. s-uniformity for all (but one) s is an element of Z(p), and the efficiently implemented bitwise right-shift operator is an injective compressing mapping. Furthermore, two families of new injective compressing mappings are given in the form of coordinate polynomials.