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Generating random polynomials in R/I

asked 2014-11-19 14:31:31 -0500

amaloz gravatar image

updated 2015-01-14 04:56:22 -0500

FrédéricC gravatar image

I'm trying to generate random polynomials in ring R/I. Here's an example:

P.<x> = PolynomialRing(ZZ)
R = P.quotient(x**256 + 1)
I = R.ideal(g) # for some appropriate polynomial g
S = QuotientRing(R, I)
S.random_element()

Instead of this last command outputting a polynomial, it outputs a small integer. Looking through the sage code, the QuotientRing_nc class doesn't appear to be overloading random_element, so I guess the default random_element method from the Ring class is being called, which just outputs random integers.

Is there any way to generate random polynomials in S, or is that functionality not implemented currently?

Thanks.

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answered 2014-11-19 19:02:50 -0500

tmonteil gravatar image

The problem with the .random_element() methods in Sage is that the distribution is usually far from being uniform. In some cases, it is very hard to sample according to a particular distribution (even theoretically).

In this case the distribution is indeed concentrated to the integers, as it is specified in the documentation, see:

sage: S.random_element?

Note however that the elements are not integers but belong to S:

sage: S.random_element().parent()
Quotient of Univariate Quotient Polynomial Ring in xbar over Integer Ring with modulus x^256 + 1 by the ideal (xbar^4 - 3)
sage: S.random_element().parent() == S
True

Other distributions are not implemented for such quotient rings, so there is room for interesting development, which is more than welcome !

In the meantime, you can still do something like:

sage: S(ZZ[x].random_element(degree=256))
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Asked: 2014-11-19 14:31:31 -0500

Seen: 279 times

Last updated: Nov 19 '14