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simplifying expressions in GF(2)

asked 2015-02-16 03:45:32 +0200

freako89 gravatar image

updated 2015-02-16 03:46:12 +0200

Hi guys,

I know that for a variable $x$ in $GF(2)$, $x^2 = x$, and $2x=0$.

How do I simplify a polynomial expression in $GF(2)$ in the Sage interface?

For example, I should obtain


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answered 2015-02-16 14:35:21 +0200

vdelecroix gravatar image

The simplest is to work over boolean polynomials

sage: B.<a,b> = BooleanPolynomialRing(2)
sage: (a+b+1)^2
a + b + 1

But you can also use the more general quotient of polynomial rings (notice that in GF(2) a equals -a)

sage: R.<a,b> = PolynomialRing(GF(2), 'a,b')
sage: Rbar = R.quotient([a^2+a, b^2+b])
sage: abar,bbar = Rbar.gens()
sage: abar
sage: (abar+bbar+1)^2
abar + bbar + 1
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answered 2015-02-16 09:56:59 +0200

rws gravatar image

I'm not sure why you think that x^2=x and Sage gives

sage: R.<a,b> = PolynomialRing(GF(2), 'a,b')
sage: (a+b+1)^2
a^2 + b^2 + 1
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The question is how to manipulate expressions involving variables which are assumed to live in GF(2). In this context, x, a, b, etc. are thought of as elements in GF(2), not as indeterminates of polynomial rings over in GF(2). The question mentions polynomial expressions in such variables, but this does not mean formal polynomials.

slelievre gravatar imageslelievre ( 2015-02-16 14:44:21 +0200 )edit

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Asked: 2015-02-16 03:45:32 +0200

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Last updated: Feb 16 '15