| dbo:abstract |
In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. There are infinitely many different finite fields. Their number of elements is necessarily of the form pn where p is a prime number and n is a positive integer, and two finite fields of the same size are isomorphic. The prime p is called the characteristic of the field, and the positive integer n is called the dimension of the field over its prime field. Finite fields are used in a variety of applications, including in classical coding theory in linear block codes such as BCH codes and Reed–Solomon error correction, in cryptography algorithms such as the Rijndael (AES) encryption algorithm, in tournament scheduling, and in the design of experiments. (en) 在数学之中，有限域算术是一种在有限域之内的算术，因为域仅包括有限数量的元素，而有限域算术则相对于无限域算术，后者是包括无限数量的元素的算术（如在有理数之下的算术）。 由于并没有任何有限域是无限的，因此存在着无限多个不同的有限域。它们的势需要是能够在pn的形式下，这其中的p是一则素数，而n则是一则正整数，同时两个持有等量的有限域可以构成同构。素数p被称之为有限域的特征，而正整数n则被称之为有限域的向量空间的维数，凌驾于它的之上，最初域为最小的包括1F的子域。 有限域应用于各种领域，这其中包括在线性分组码之内的编码理论，譬如BCH码和里德-所罗门码，还有在密码学之中的演算法，比如Rijndael加密法之下的加密算法。 (zh) |
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http://www.samiam.org/galois.html%7Ctitle=AE's http://web.eecs.utk.edu/~plank/plank/papers/CS-07-593/%7Ctitle=Fast |
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在数学之中，有限域算术是一种在有限域之内的算术，因为域仅包括有限数量的元素，而有限域算术则相对于无限域算术，后者是包括无限数量的元素的算术（如在有理数之下的算术）。 由于并没有任何有限域是无限的，因此存在着无限多个不同的有限域。它们的势需要是能够在pn的形式下，这其中的p是一则素数，而n则是一则正整数，同时两个持有等量的有限域可以构成同构。素数p被称之为有限域的特征，而正整数n则被称之为有限域的向量空间的维数，凌驾于它的之上，最初域为最小的包括1F的子域。 有限域应用于各种领域，这其中包括在线性分组码之内的编码理论，譬如BCH码和里德-所罗门码，还有在密码学之中的演算法，比如Rijndael加密法之下的加密算法。 (zh) In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. There are infinitely many different finite fields. Their number of elements is necessarily of the form pn where p is a prime number and n is a positive integer, and two finite fields of the same size are isomorphic. The prime p is called the characteristic of the field, and the positive integer n is called the dimension of the field over its prime field. (en) |
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