freako89's profile - activity

Hi Guys,

By writing this:

B.<a,b> = BooleanPolynomialRing()

Not only a Boolean Polynomial Ring in 'a' and 'b' is defined, but 'a' and 'b' are also treated as boolean variables.

However, if we write in this manner:

B = BooleanPolynomialRing(names = ['a','b'])

We'll obtain a Boolean Polynomial Ring in 'a' and 'b', but we don't even get 'a' and 'b' as variables.

Is there any way to resolve the issue in the second method, especially, if we have a boolean polynomial ring of >1000 variables? Thanks in advance!

Hi guys, let's we want to get the polynomial expression for a S-Box.

sage: S = mq.SBox(7,6,0,4,2,5,1,3) sage: P = S.ring()

We can get the indirect representation...

sage: S.polynomials()
[x0*x2 + x1 + y1 + 1,
 x0*x1 + x1 + x2 + y0 + y1 + y2 + 1,
 x0*y1 + x0 + x2 + y0 + y2,
 x0*y0 + x0*y2 + x1 + x2 + y0 + y1 + y2 + 1,
 x1*x2 + x0 + x1 + x2 + y2 + 1,
 x0*y0 + x1*y0 + x0 + x2 + y1 + y2,
 x0*y0 + x1*y1 + x1 + y1 + 1,
 x1*y2 + x1 + x2 + y0 + y1 + y2 + 1,
 x0*y0 + x2*y0 + x1 + x2 + y1 + 1,
 x2*y1 + x0 + y1 + y2,
 x2*y2 + x1 + y1 + 1,
 y0*y1 + x0 + x2 + y0 + y1 + y2,
 y0*y2 + x1 + x2 + y0 + y1 + 1,
 y1*y2 + x2 + y0]

or the direct representation

sage: P.<y0,y1,y2,x0,x1,x2> = PolynomialRing(GF(2),6,order='lex')
sage: S.polynomials([x0,x1,x2],[y0,y1,y2], groebner=True)
[y0 + x0*x1 + x0*x2 + x0 + x1*x2 + x1 + 1,
 y1 + x0*x2 + x1 + 1,
 y2 + x0 + x1*x2 + x1 + x2 + 1]

My question is, given the direct representation of the S-Box, can we retrieve the indirect representation for the S-Box?

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

$$(a+b+1)^2=a^2+b^2+1+2a+2b+2ab=a+b+1$$