Finite Field Arithmetic

Finite Field Arithmetic

Arithmetic in a finite field is different from standard integer arithmetic. There are a limited number of elements in the finite field; all operations performed in the finite field result in an element within that field.

While each finite field is itself not infinite, there are infinitely many different finite fields; their number of elements (which is also called cardinality) 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 and in cryptography algorithms such as the Rijndael encryption algorithm.

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—Blaise Pascal (1623–1662)

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