Singular

I want to generate a random element in a finite field, (may not be a prime field), do I have to write dynamic module? And I can't find a file to reference with the sourcecode written in the structure of dynamic module. Besides, how can I use NTL in singular? Should I write dynamic module too? Thank you

Hi,
you can use the function random in a prime field or sparsepoly in a non prime field (Please check the manual). NTL is used by default for univariate gcd and factorization. If you need NTL for anything besides this let me know.

but sparsepoly can not generate a random element of the field with equal posibility...

What do you mean by



Use the command random for random integers within a specified range

http://www.singular.uni-kl.de/Manual/la ... htm#SEC340

Say, GF(3^3), I want randomly generate a element in this field, and each one with equal possibility to be outputted.

Hi
Try something like
LIB "random.lib";
ring r= 3,x,dp;
poly f= sparsepoly (0,2,50,4);

ring s=(3,a),(x,y),dp;
minpoly= a3+2a+1;
map m= r,a;
m (f);
The sparsepoly (0,2,50,4) creates a random polynomial in x with coefficients in the range [1,4) of degree between 0 and 2. And each monomial is zero with probability 0,5.
The last 4 lines maps this polynomial to a ring s with coefficients from Z/3Z[a]/ (a3+2a+1).

Each coefficient of the monomial in f is zero with probability 1/3. So f must be defined as
poly f = sparsepoly (0,2,0,4);

Don't you think so?

If you want a uniformly random element of GF (3^3) you're right.

Thanks. But when I define a ring:
ring s=(9,a),(x,y),dp;
s;
// # ground field : 9
// primitive element : a
// minpoly : 1*a^2+2*a^1+2*a^0
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C

then i can't reduce the elememts like a^6 or a7x2+a9y;
what i mean is to present the coefficients by the basis { 1, a },
in fact a^6 is (-a-1);

In your case GF (3^2) is represented by a primitive element. See http://www.singular.uni-kl.de/Manual/latest/sing_29.htm#SEC39 for an example of this.