Research On The Local Entropy-Preservation Of Compressing Sequences Derived From Primitive Sequences Over Z/(p~e)

Let p be a prime number, e≥2 be an integer, and Z/(pe) be the integer residue ring modulo pe. Any sequence a over Z/(pe) has a unique p-adic expansion a = a0 + a1?p +…+ ae-1?pe-1, where ai is a sequence over {0, 1,…, p ? 1} which can be regarded as a sequence over the Galois field GF(p).The compressing sequences derived from primitive sequences over Z/(pe) was a class of nonlinear sequences independently introduced by Chinese and Russian scholars in the middle of 1980s. For the past two decades, plentiful results have been obtained on such nonlinear sequences. In this paper, the local entropy-preservation of the compressing sequences derived from e-variable functions over Z/(p) of the form xe-1 +η(x0, x1,…, xe-2) and primitive sequences over Z/(pe) is studied. The main results are as follows.1. Assume p is an odd prime number. It is proved that if the coefficient of xe p? ? 21…x1 p-1 x0p-1 in the expression ofη(x0, x1,…, xe-2) is not equal to (p + 1)/2, then the compressing sequences derived by applying functions xe-1 +η(x0, x1,…, xe-2) on primitive sequences over Z/(pe) are local entropy-preservation. In detail, for a given strongly primitive polynomial f(x) over Z/(pe), and two primitive sequences a and b generated by f(x) over Z/(pe), if there exists an element s∈Z/(p) such that the distribution of s of the sequence ae-1 +η(a0, a1,…, ae-2) is the same as that of the sequence be-1 +η(b0, b1,…, be-2) at positions t withα(t)≠0, i.e. ae-1(t) +η(a0(t), a1(t),…, ae-2(t)) = s if and only if be-1(t) +η(b0(t), b1(t),…, be-2(t)) = s for all t withα(t)≠0, then a = b, where the sequenceαis an m-sequence over Z/(p) uniquely determined by f(x) and a0.2. Assume p = 2, an even prime number. It is proved that the sequences derived by applying Boolean functions xe-1 +η(x0, x1,…, xe-2) on primitive sequences over Z/(2e) are local entropy-preservation. In detail, for a given strongly primitive polynomial f(x) over Z/(2e), and two primitive sequences a and b generated by f(x) over Z/(2e), if there exists an s∈Z/(2), such that the distribution of s of the sequence ae-1 +η(a0, a1,…, ae-2) is the same as that of the sequence be-1 +η(b0, b1,…, be-2) at positions t withα(t)≠0, then a = b, whereαis an m-sequence over Z/(2) uniquely determined by f(x) and a0.