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A class of exponential sums and sequence families
Cryptography and Communications ( IF 1.2 ) Pub Date : 2019-05-16 , DOI: 10.1007/s12095-019-00368-4
Chengju Li , Qin Yue , Yongbo Xia , Wei Peng

Let m1 and m2 be two distinct positive integers with \(d=\gcd (m_{1}, m_{2})\). Let \(\mathbb {F}_{2^{m}}\) be the finite field with 2m elements, where m = m1m2/d. In this paper, we investigate the exponential sums$$ S(a,b)=\sum\limits_{x \in \mathbb{F}_{2^{m}}^{*}}(-1)^{\text{Tr}_{m_{1}}(ax^{\frac {2^{m}-1}{2^{m_{1}}-1}})+\text{Tr}_{m_{2}}(bx^{\frac {2^{m}-1}{2^{m_{2}}-1}})}, $$where \(a \in \mathbb {F}_{2^{m_{1}}}\), \(b \in \mathbb {F}_{2^{m_{2}}}\), and Trt denotes the trace function from \(\mathbb {F}_{2^{t}}\) to \(\mathbb {F}_{2}\). When d = 1, 2, 3, 4, we present the value distribution of the exponential sums S(a, b) explicitly. As an application, we construct three families of binary sequences with three-valued correlation.

中文翻译:

一类指数和和序列族

m 1m 2是两个具有\(d = \ gcd(m_ {1},m_ {2})\)的不同正整数。令\(\ mathbb {F} _ {2 ^ {m}} \)是具有2 m个元素的有限域,其中m = m 1 m 2 / d。在本文中,我们研究指数总和$$ S(a,b)= \ sum \ limits_ {x \ in \ mathbb {F} _ {{2 ^ {m}} ^ {*}}(-1)^ { \ text {Tr} _ {m_ {1}}(ax ^ {\ frac {2 ^ {m} -1} {2 ^ {m_ {1}}-1}})+ \ text {Tr} _ {m_ {2}}(bx ^ {\ frac {2 ^ {m} -1} {2 ^ {m_ {2}}-1}})}},$$其中\(a \ in \ mathbb {F} _ { 2 ^ {m_ {1}}} \)\(b \ in \ mathbb {F} _ {2 ^ {m_ {2}}} \)和Tr t表示从\(\ mathbb {F} _ {2 ^ {t}} \)\(\ mathbb {F} _ {2} \)的跟踪函数。当d = 1,2,3,4时,我们明确给出指数和Sab)的值分布。作为应用程序,我们构造了具有三值相关性的三个二进制序列族。
更新日期:2019-05-16
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