以下为句子列表：
英文: Hardware implementation for the all-erasure RS decoder is proposed, including the finite field arithmetic operation units.
中文: 本文并提出纯粹抹除式里德所罗门解码器的硬体实现，包括有限体中的算数运算模组。
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英文: Polynomials over the finite field include cyclotomic polynomials and irreducible polynomials constructed on different theories. In this paper, their properties and ways for their construction are explored and discussed.
中文: 摘要有限域上的多项式有分圆多项式和不可约多项式，其中各由数条原理构成。文中对其性质和构造方法进行了具体研究与论述。
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英文: The primary advantage that elliptic curve systems have over systems based on the multiplicative group of a finite field (and also over systems based on the intractability of integer factorization) is the absence of a subexponential-time algorithm (such as
中文: 与基于有限域的乘法群系统(及基于整数分解的难解性之上的系统)相比,椭圆曲线系统的优势在于,迄今为止还没找到这类群上离散对数的次指数时间算法(如微积分类).
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