When I am reading some algorithm related to error correction. To generate some polynomial it uses finite-field which is Galois field. I am not from mathematical background. Can anybody explain me in simple form to understand this ?
What is Galois Field
-
2@Jyrki EE/CS majors (at the graduate level) as well as math students (even those with weak backgrounds in algebra) can and do benefit from reading Blahut's book. But Sunny says elsewhere on Math.SE that "I am software programmer" and his other postings on Math.SE seem to indicate that he does not want or need such an extensive exposure to RS codes as Blahut's book provides. This is why I suggested other books that may well provide Sunny with what he needs. For the general readership of Math.SE, Blahut's book is indeed an excellent recommendation for algorithms relating to RS codes. – 2011-10-20
2 Answers
Galois field is the name that engineers (and especially those studying error correcting codes) use for what mathematicians call a finite field. In applications, the most commonly used Galois field is $\text{GF}(256)$, also called $\text{GF}(2^8)$. Its elements can be thought of as polynomials of degree $7$ or less with binary coefficients ($0$ or $1$). Addition of two field elements is addition of the two polynomials with coefficients being added modulo $2$. Multiplication is polynomial multiplication modulo a polynomial $m(x)$ of degree $8$, that is, multiply the two given polynomials (which may result in a polynomial of degree as much as $14$) and then divide by $m(x)$, throwing away the quotient and keeping only the remainder.
A Galois field is a finite field (from the Wikipedia article):
In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements.
-
3... and Rijndael is better known as AES these days, essentially the standard choice of block cipher everywhere. – 2011-10-20