环上本原序列保熵压缩映射的研究
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摘要
设p是奇素数,整数e≥2,Z/(p~e)是整数模p~e的剩余类环。环Z/(p~e)上序列(?)有如下唯一的p-adic分解:(?)=(?)_0+(?)_1·p+…+(?)_(e-1)·p~(e-1),其中(?)是{0,1,…,p-1}上序列,称(?)是序列(?)的第i权位序列,(?)_(e-1)是(?)的最高权位序列,它们可自然视为素域GF(p)上序列。
设f(x)是Z/(p~e)上n次本原多项式,它是Z/(p~e)上周期为p~(e-1)·(p~n-1)的首一多项式,并且f(0)≠0(mod p)。设h(x)是{0,1,…,p-1)上多项式,degh(x) x~(p~(e-2)(p~n-1))-1=p~(e-1)·h(x)(mod f(x),p~e)。
记Z/(p~e)上所有由f(x)生成的线性递归序列之全体为G(f(x),p~e)。我们证明了,G(f(x),p~e)中本原序列最高权位在某些固定的位置上的0元素分布是唯一的。即,任给(?)=(a(t))_t≥0,(?)=(b(t))_t≥0∈G(f(x),p~e),(?)≠(?)(mod p),记(?)=h(x)(?)_0(mod p),若对使得α(t)≠0的非负整数t,都有a_(e-1)(t)=0当且仅当b_(e-1)(t)=0,则(?)=(?)。这一结论说明序列(?)_(e-1)在α(t)≠0的位置上的0元素分布情形包含序列(?)∈G(f(x),p~e)的所有信息。称这一特性为Z/(p~e)上本原序列的局部0保熵。它的意义主要表现为:一方面,它更为精确地描述了最高权位序列保熵的具体含义,进一步揭示了本原序列蕴涵信息的分布规律;另一方面,对于Z/(p~e)上本原序列一般保熵函数的研究,它可以提供了一个有力工具。同时,对于Z/(2~e)上本原序列,本文也得到了类似的结论。
基于Z/(p~e)上本原序列的局部0保熵的结果,本文证明了形如φ_(e-1)(x_0,x_1,…,x_(e-1))=x_(e-1)~k+η_(e-2)(x_0,x_1,…,x_(e-2)),2≤k≤p-1,的压缩函数是保熵的,其中η_(e-2)是素域GF(p)上任意一个e-1元多项式。即,任给(?),(?)∈G(f(x),p~e),若(?)≠(?)(mod p),则(?)=(?)当且仅当φ_(e-1)((?)_0,(?)_1,…,(?)_(e-1))=φ_(e-1)((?)_0,_(?)_1,…,(?)_(e-1))。进一步,若f(x)是Z/(p~e)上强本原多项式,上述形式的不同压缩函数和不同的本原序列对于导出序列的影响都是不同的。这一特性对于Z/(2~e)上本原序列的压缩函数是很难成立的。
FCSR序列中的极大周期序列(简称为l-序列)是一类性质优良的伪随机序列。由FCSR序列的代数表示,可知l-序列是Z/(p~e)上一次本原序列的mod 2导出序列,其中p是奇素数,2是mod p~e的一个本原元。设(?)是Z/(p~e)上n次本原序列,mod 2压缩序列
Let p be an odd prime, integer e ≥ 2, and Z/(p~e) be the integer residue ring modulo p~e. Any sequence -|a over Z/(p~e) has an unique /?-adic expansion: -|a = -|a0 + -|a1p +...+ -|a_e-p~(e-1), where each -|a_i is a sequence over {0, l,...,p-l}, i = 0, 1,..., e-1. The sequence -|a_i is called i-th level sequence of -|a, and -|a_(e-1) the highest-level sequence of -|a. They can be naturally considered as the sequences over the prime field GF(p).Let f{x) be a primitive polynomial of degree n over Z/(p~e), which is a monic polynomial of period p~(e-1) (p~n-1), such that f(0)≠0 (mod p). Let h(x) be a polynomial over {0, l,...,p-1}, degh(x) < n, such thatDenoted by G(f(x), p~e), the set of all linear recurring sequences generated by f(x) over Z/(p~e). We prove that, the distribution of zeros on some fixed positions of the highest-level sequence of any primitive sequence in G(f(x), p~e) is unique. That is, for -|a= (a(t)),t≥0 b = (b(t))t≥0∈G(f{x),p~e) such that -|a ≠0 (mod p), we set -|a = h(x)-|a0 (mod p). If a_(e-1) (f) = 0 iff b_(e-1) (t) = 0 for any integer t with a(t) ≠0, then -|a = -|b. This implies that the distribution of zeros of -|a_(e-1) on the positions t with a(t)≠ 0, contains all information of -|a∈G(f(x),p~e). This property is called zero-entropy-preservation of primitive sequences over Z/(p~e), which can disclose the distribution law of information contained in primitive sequences over Z/(pe). On the other hand, it can provide a powerful tool for the discussion of the injectivity of a general function acting on primitive sequences over Zl{pe). We also prove that the similar result over Z/(2~e) holds.
Based on the result of zero-entropy-preservation of the primitive sequence over Z/(pe); we prove that the compressing function of form(pe-i(x0;xu-..;xe-i) = 4_i + rje-2(xo;xh...;xe_2); 2 < k
设f(x)是Z/(p~e)上n次本原多项式,它是Z/(p~e)上周期为p~(e-1)·(p~n-1)的首一多项式,并且f(0)≠0(mod p)。设h(x)是{0,1,…,p-1)上多项式,degh(x)
记Z/(p~e)上所有由f(x)生成的线性递归序列之全体为G(f(x),p~e)。我们证明了,G(f(x),p~e)中本原序列最高权位在某些固定的位置上的0元素分布是唯一的。即,任给(?)=(a(t))_t≥0,(?)=(b(t))_t≥0∈G(f(x),p~e),(?)≠(?)(mod p),记(?)=h(x)(?)_0(mod p),若对使得α(t)≠0的非负整数t,都有a_(e-1)(t)=0当且仅当b_(e-1)(t)=0,则(?)=(?)。这一结论说明序列(?)_(e-1)在α(t)≠0的位置上的0元素分布情形包含序列(?)∈G(f(x),p~e)的所有信息。称这一特性为Z/(p~e)上本原序列的局部0保熵。它的意义主要表现为:一方面,它更为精确地描述了最高权位序列保熵的具体含义,进一步揭示了本原序列蕴涵信息的分布规律;另一方面,对于Z/(p~e)上本原序列一般保熵函数的研究,它可以提供了一个有力工具。同时,对于Z/(2~e)上本原序列,本文也得到了类似的结论。
基于Z/(p~e)上本原序列的局部0保熵的结果,本文证明了形如φ_(e-1)(x_0,x_1,…,x_(e-1))=x_(e-1)~k+η_(e-2)(x_0,x_1,…,x_(e-2)),2≤k≤p-1,的压缩函数是保熵的,其中η_(e-2)是素域GF(p)上任意一个e-1元多项式。即,任给(?),(?)∈G(f(x),p~e),若(?)≠(?)(mod p),则(?)=(?)当且仅当φ_(e-1)((?)_0,(?)_1,…,(?)_(e-1))=φ_(e-1)((?)_0,_(?)_1,…,(?)_(e-1))。进一步,若f(x)是Z/(p~e)上强本原多项式,上述形式的不同压缩函数和不同的本原序列对于导出序列的影响都是不同的。这一特性对于Z/(2~e)上本原序列的压缩函数是很难成立的。
FCSR序列中的极大周期序列(简称为l-序列)是一类性质优良的伪随机序列。由FCSR序列的代数表示,可知l-序列是Z/(p~e)上一次本原序列的mod 2导出序列,其中p是奇素数,2是mod p~e的一个本原元。设(?)是Z/(p~e)上n次本原序列,mod 2压缩序列
Let p be an odd prime, integer e ≥ 2, and Z/(p~e) be the integer residue ring modulo p~e. Any sequence -|a over Z/(p~e) has an unique /?-adic expansion: -|a = -|a0 + -|a1p +...+ -|a_e-p~(e-1), where each -|a_i is a sequence over {0, l,...,p-l}, i = 0, 1,..., e-1. The sequence -|a_i is called i-th level sequence of -|a, and -|a_(e-1) the highest-level sequence of -|a. They can be naturally considered as the sequences over the prime field GF(p).Let f{x) be a primitive polynomial of degree n over Z/(p~e), which is a monic polynomial of period p~(e-1) (p~n-1), such that f(0)≠0 (mod p). Let h(x) be a polynomial over {0, l,...,p-1}, degh(x) < n, such thatDenoted by G(f(x), p~e), the set of all linear recurring sequences generated by f(x) over Z/(p~e). We prove that, the distribution of zeros on some fixed positions of the highest-level sequence of any primitive sequence in G(f(x), p~e) is unique. That is, for -|a= (a(t)),t≥0 b = (b(t))t≥0∈G(f{x),p~e) such that -|a ≠0 (mod p), we set -|a = h(x)-|a0 (mod p). If a_(e-1) (f) = 0 iff b_(e-1) (t) = 0 for any integer t with a(t) ≠0, then -|a = -|b. This implies that the distribution of zeros of -|a_(e-1) on the positions t with a(t)≠ 0, contains all information of -|a∈G(f(x),p~e). This property is called zero-entropy-preservation of primitive sequences over Z/(p~e), which can disclose the distribution law of information contained in primitive sequences over Z/(pe). On the other hand, it can provide a powerful tool for the discussion of the injectivity of a general function acting on primitive sequences over Zl{pe). We also prove that the similar result over Z/(2~e) holds.
Based on the result of zero-entropy-preservation of the primitive sequence over Z/(pe); we prove that the compressing function of form(pe-i(x0;xu-..;xe-i) = 4_i + rje-2(xo;xh...;xe_2); 2 < k
引文
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