Try using the following gp script:

multable( p, P, g ) =
{
P = Mod(1,p) *  P;
g = Mod( Mod(1,p) *  g, P );
d = poldegree( P );
n = p^d;

/*
    This version makes B[2] = Mod( 1, P ) which is wrong
    B = concat( [0], vector( n-1, i, g^(i-1) ) );

    The following line corrects the mistake
*/

B = concat( [0], vector( n-1, i, Mod( Mod(1,p), P ) * g^(i-1) ) );
R = vector( n );
for( i = 1, n,
    R[ 1 + subst( lift(lift( B[i] )), x, p )] = i-1;
);

for( a = 1, p^d,
     ae = B[a];
    for ( b = 1, p^d,
        be = B[b];
        re = ae * be;
        print1( R[ 1 + subst( lift(lift( re )), x, p )], "\t" );
    );
    print;
);

}
[download]

Save it in a file (say "ff.gp") and then invoke gp (it's an interactive environment for pari). A sample session might be

$ gp -q
? read ("/tmp/ff.gp")
? multable(2, x^3+x+1, x)
0       0       0       0       0       0       0       0
0       1       2       3       4       5       6       7
0       2       3       4       5       6       7       1
0       3       4       5       6       7       1       2
0       4       5       6       7       1       2       3
0       5       6       7       1       2       3       4
0       6       7       1       2       3       4       5
0       7       1       2       3       4       5       6
? 

The arguments to multable are the characteristic, the polynomial generating the field and a generator of the multiplicative group. To generate the addition table, substitute with re = ae + be on the due line.

Cheers

Antonio

The stupider the astronaut, the easier it is to win the trip to Vega - A. Tucket

Update: The mistake for g^0 + g^0 should be corrected now

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