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G.3. Finite Fields, Polynomials, and Vector Spaces using none toassign none on web,windows applicationbarcode c# tutorial Various algebraic none none objects, computational problems, and techniques play an important role in Complexity Theory. The most dominant such objects are nite elds as well as vector spaces and polynomials over such elds. Finite Fields.

We denote by GF(q) the nite eld of q elements and note that q may be either a prime or a prime power. In the rst case, GF(q) is viewed as consisting of the elements {0, . .

. , q 1} with addition and multiplication being de ned modulo q. Indeed, GF(2) is an important special case.

In the case that q = pe , where p is a prime and e > 1, the standard representation of GF( pe ) refers to an irreducible polynomial of degree e over GF( p). Speci cally, if f is an irreducible polynomial of degree e over GF( p), then GF( pe ) can be represented as the set of polynomials of degree at most e 1 over GF( p) with addition and multiplication de ned modulo the polynomial f . We mention that nding representations of large nite elds is a non-trivial computational problem, where in both cases we seek an ef cient algorithm that nds a representation (i.

e., either a large prime or an irreducible polynomial) in time that is polynomial in the length of the representation. In the case of a eld of prime cardinality, this calls for.

iPhone OS G.4. THE DETERMINANT AND THE PERMANENT generating a prime none none number of adequate size, which can be done ef ciently by a randomized algorithm (while a corresponding deterministic algorithm is not known). In the case of GF( pe ), where p is a prime and e > 1, we need to nd an irreducible polynomial of degree e over GF( p). Again, this task is ef ciently solvable by a randomized algorithm (see [24]), but a corresponding deterministic algorithm is not known for the general case (i.

e., for arbitrary prime p and e > 1). Fortunately, for e = 2 3e (with e being an integer), the polynomial x e + x e/2 + 1 is irreducible over GF(2), which means that nding a representation of GF(2e ) is easy in this case.

Thus, there exists a strongly explicit construction of an in nite family of nite elds (i.e., {GF(2e )}e L , where L = {2 3e : N}).

Polynomials and Vector Spaces. The set of degree d 1 polynomials over a nite eld F (of cardinality at least d) forms a d-dimensional vector space over F (e.g.

, consider the basis {1, x, . . .

, x d 1 }). Indeed, the standard representation of this vector space refers to d 1 the basis 1, x, . .

. , x d 1 , and (when referring to this basis) the polynomial i=0 ci x i is represented as the vector (c0 , c1 , . .

. , cd 1 ). An alternative basis is obtained by considering the evaluation at d distinct points 1 , .

. . , d F; that is, the degree d 1 polynomial p is represented by the sequence of values ( p( 1 ), .

. . , p( d )).

Needless to say, moving between such representations (i.e., representations with respect to different bases) amounts d 1 to applying an adequate linear transformation; that is, for p(x) = i=0 ci x i , we have d 1 1 1 1 p( 1 ) c0 p( 2 ) 1 2 d 1 c1 2 (G.

1) . = . .

. . .

. . .

. . .

. . .

d 1 cd 1 p( d ) 1 d d where the (full rank) matrix in Eq. (G.1) is called a Vandermonde matrix.

The foregoing transformation (or rather its inverse) is closely related to the task of polynomial interpolation (i.e., given the values of a degree d 1 polynomial at d points, nd the polynomial itself).

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