Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over $\BBZ /(p^{e})$

Zheng, Qun-Xiong; Qi, Wei

doi:10.1109/tit.2009.2034782

Cited by 16 publications

(18 citation statements)

References 14 publications

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“…First, since and , it follows from Condition 2) of Theorem 19 and Lemma 23 that there exist two positive integers and such that (17) and ( (19) Note that is also a typical primitive polynomial of degree over , and so by (2) (22) and (23) together with the fact that is an odd integer give a contradiction to the assumption that . If then (22) and (23) …”

Section: Lemma 23mentioning

confidence: 97%

“…Then is a sequence over and is called a compressing sequence derived from . When is a primitive sequence over , many cryptographical properties of such compressing sequence have been studied during the last 20 years [2]- [17], in particular the distinctness of compressing sequences, that is, if and only if , where and are two primitive sequences generated by a primitive polynomial over . Obviously, for a given primitive polynomial over , if the compressing sequences of all primitive sequences generated by are pairwise distinct, then there is a one-to-one correspondence between primitive sequences and their compressing sequences, which implies that every compressing sequence preserves all the information of its original primitive sequence.…”

Section: Introductionmentioning

confidence: 99%

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On the Distinctness of Binary Sequences Derived From Primitive Sequences Modulo Square-Free Odd Integers

Zheng

Tian

2013

IEEE Trans. Inform. Theory

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Let be a square-free odd integer and the integer residue ring modulo . This paper studies the distinctness of primitive sequences over modulo 2. Recently, for the case of , a product of two distinct prime numbers and , the problem has been almost completely solved. As for the case that is a product of more prime numbers, the problem has been quite resistant to proof. In this paper, a partial proof is given by showing that a class of primitive sequences of order over is distinct modulo 2, where is a positive integer. Besides as an independent interest, this paper also involves two distribution properties of primitive sequences over , which are related closely to our main results.

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Section: Lemma 23mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

On the Distinctness of Binary Sequences Derived From Primitive Sequences Modulo Square-Free Odd Integers

Zheng

Tian

2013

IEEE Trans. Inform. Theory

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“…Zhu and Qi [33,34] proved that a = b iff ϕ(a) and ϕ(b) are 0-uniform, i.e., under the compressing map x → x e−1 the distribution of 0 in the compressed sequence implies all information of the original sequence. Zheng and Qi [27,31] proved that for s ∈ F p and k ∈ F * p , under the compressing map x → x e−1 + η(x 0 , . .…”

Section: Introductionmentioning

confidence: 99%

Injectivity on distribution of elements in the compressed sequences derived from primitive sequences over ℤ p e $\mathbb {Z}_{p^{e}}$

Wang¹,

2018

Cryptogr. Commun.

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Let p ≥ 3 be a prime and e ≥ 2 an integer. Let σ(x) be a primitive polynomial of degree n over Z/p e Z and G ′ (σ(x), p e ) the set of primitive linear recurring sequences generated by σ(x). A compressing map ϕ on Z/p e Z naturally induces a map ϕ on G ′ (σ(x), p e ). For a subset D of the image of ϕ, ϕ is called to be injective w.r.t. D-uniformity if the distribution of elements of D in the compressed sequence implies all information of the original primitive sequence. In this correspondence, for at least 1 − 2(p − 1)/(p n − 1) of primitive polynomials of degree n, a clear criterion on ϕ is obtained to decide whether ϕ is injective w.r.t. D-uniformity, and the majority of maps on Z/p e Z induce injective maps on G ′ (σ(x), p e ). Furthermore, a sufficient condition on ϕ is given to ensure injectivity of ϕ w.r.t. D-uniformity. It follows from the sufficient condition that if σ(x) is strongly primitive and the compressing map ϕ(x) = f (xe−1), where f (xe−1) is a permutation polynomial over Fp, then ϕ is injective w.r.t. D-uniformity for ∅ = D ⊂ Fp. Moreover, we give three specific families of compressing maps which induce injective maps on G ′ (σ(x), p e ).

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“…Последовательность u n−1 изучалась многими авторами (см., например, работы [3]- [18]). Результаты этих исследований показали, что последователь-ность u n−1 , полученная из ЛРП u максимального периода над кольцом Z p n , пригодна для использования в качестве псевдослучайной последовательности.…”

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“…, a n−1 определены равенством (1). Такие отображения Ψ достаточ-но широко исследуются на пригодность в качестве функции усложнения (см., например, работы [6], [18] и цитированную в них литературу).…”

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Частотные Характеристики Разрядных Последовательностей Линейных Рекуррент Над Кольцами Галуа

Камловский¹

2013

Известия Российской академии наук. Серия математическая

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Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over $\BBZ /(p^{e})$

Cited by 16 publications

References 14 publications

On the Distinctness of Binary Sequences Derived From Primitive Sequences Modulo Square-Free Odd Integers

On the Distinctness of Binary Sequences Derived From Primitive Sequences Modulo Square-Free Odd Integers

Injectivity on distribution of elements in the compressed sequences derived from primitive sequences over ℤ p e $\mathbb {Z}_{p^{e}}$

Частотные Характеристики Разрядных Последовательностей Линейных Рекуррент Над Кольцами Галуа

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