Finding the Greatest Common Divisor

Finding the Greatest Common Divisor

We can extend the analogy between polynomial arithmetic over a field and integer arithmetic by defining the greatest common divisor as follows. The polynomial c(x) is said to be the greatest common divisor of a(x) and b(x) if

1. c(x) divides both a(x) and b(x);
2. any divisor of a(x) and b(x) is a divisor of c(x).

An equivalent definition is the following: gcd[a(x), b(x)] is the polynomial of maximum degree that divides both a(x) and b(x).

We can adapt the Euclidean algorithm to compute the greatest common divisor of two polynomials. The equality in Equation (4.4) can be rewritten as the following theorem:

Equation 4-7

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[Page 118]

The Euclidean algorithm for polynomials can be stated as follows. The algorithm assumes that the degree of a(x) is greater than the degree of b(x). Then, to find gcd[a(x), b(x)],

EUCLID[a(x), b(x)]
1. A(x)  a(x); B(x)  b(x)
2. if B(x) = 0 return A(x) = gcd[a(x), b(x)]
3. R(x) = A(x) mod B(x)
4. A(x)  B(x)
5. B(x)  R(x)
6. goto 2

Find gcd[a(x), b(x)] for a(x) = x6 + x5 +x4 + x3 + x2 +x + 1 and b(x) = x4 + x2 + x + 1. A(x) = a(x); B(x) = b(x) Summary We began this section with a discussion of arithmetic with ordinary polynomials. In ordinary polynomial arithmetic, the variable is not evaluated; that is, we do not plug a value in for the variable of the polynomials. Instead, arithmetic operations are performed on polynomials (addition, subtraction, multiplication, division) using the ordinary rules of algebra. Polynomial division is not allowed unless the coefficients are elements of a field. [Page 119] Next, we discussed polynomial arithmetic in which the coefficients are elements of GF(p). In this case, polynomial addition, subtraction, multiplication, and division are allowed. However, division is not exact; that is, in general division results in a quotient and a remainder. Finally, we showed that the Euclidean algorithm can be extended to find the greatest common divisor of two polynomials whose coefficients are elements of a field. All of the material in this section provides a foundation for the following section, in which polynomials are used to define finite fields of order pn.