基于模2pm的歐拉商的二元序列的線性復(fù)雜度
doi: 10.11999/JEIT190071
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西北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院 蘭州 730070
基金項(xiàng)目: 國(guó)家自然科學(xué)基金(61462077, 61562077, 61772022)，上海市自然科學(xué)基金(16ZR1411200)
Linear Complexity of Binary Sequences Derived from Euler Quotients Modulo 2pm
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College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
Funds: The National Natural Science Foundation of China (61462077, 61562077, 61772022), The Shanghai Municipal Natural Science Foundation (16ZR1411200)
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摘要: 基于歐拉商模奇素?cái)?shù)冪構(gòu)造的偽隨機(jī)序列均具有良好的密碼學(xué)性質(zhì)。該文根據(jù)剩余類環(huán)理論，利用模
$2{p^m}$ ($p$ 為奇素?cái)?shù)，整數(shù)$m \ge 1$ )的歐拉商構(gòu)造了一類周期為$2{p^{m + 1}}$ 的二元序列，并在${2^{p - 1}}\not \equiv 1 ({od}\,{p^2})$ 的條件下借助有限域${F_2}$ 上確定多項(xiàng)式根的方法，給出了序列的線性復(fù)雜度。結(jié)果表明，序列的線性復(fù)雜度取值為$2({p^{m + 1}} - p)$ 或$2({p^{m + 1}} - 1)$ 不小于其周期的1/2，能夠抵抗Berlekamp-Massey(B-M)算法的攻擊，是密碼學(xué)意義上性質(zhì)良好的偽隨機(jī)序列。-
關(guān)鍵詞:
- 有限域 /
- 二元序列 /
- 歐拉商 /
- 線性復(fù)雜度 /
- 極小多項(xiàng)式
Abstract: Families of pseudorandom sequences derived from Euler quotients modulo odd prime power possess sound cryptographic properties. In this paper, according to the theory of residue class ring, a new classes of binary sequences with period$2{p^{m + 1}}$ is constructed using Euler quotients modulo$2{p^m},$ where$p$ is an odd prime and integer$m \ge 1.$ Under the condition of${2^{p - 1}}\not \equiv 1 ({od}\,{p^2})$ , the linear complexity of the sequence is examined with the method of determining the roots of polynomial over finite field${F_2}$ . The results show that the linear complexity of the sequence takes the value$2({p^{m + 1}} - p)$ or$2({p^{m + 1}} - 1)$ , which is larger than half of its period and can resist the attack of Berlekamp-Massey (B-M) algorithm. It is a good sequence from the viewpoint of cryptography.-
Key words:
- Finite fields /
- Binary sequences /
- Euler quotients /
- Linear complexity /
- Minimal polynomial
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