Thanks a lot for the code and the detailed explanations, John! I got the code running, and it computes the y-s which I need.
About (1) and (2): > (1) the expression for y output is a polynomial in the xi with > coefficients pollys in z, while I expect you hoped that would be the > other way round; Yes. Indeed I need the polynomial coefficients to be expressions of xi- s. Maybe I can try to fix this with some symbolic manipulations. As to (2): > (2) The expression x*y does not simplify to 1, but it does satisfy > (x*y)^2==x*y. > I wonder if this means that the expression I get for y is not in fact the multiplicative inverse of x mod the polynomial? Thank you very much for your time and help! I On May 29, 5:44 pm, "John Cremona" <[EMAIL PROTECTED]> wrote: > Here is what I was talking about in my first reply. It does not do > exactly what you want, in two ways: > (1) the expression for y output is a polynomial in the xi with > coefficients pollys in z, while I expect you hoped that would be the > other way round; > (2) The expression x*y does not simplify to 1, but it does satisfy > (x*y)^2==x*y. > > If someone else can suggest how to deal with (2) this might make a > nice worked examples for the documentation. > > John > > # Define the field with 2 elements and the polynomial ring over it > # with variable Z > > F2=GF(2) > R.<Z>=PolynomialRing(F2) > > # Define the polynomial f and check that it is irreducible: > > f=Z^8+Z^4+Z^3+Z+1 > assert f.is_irreducible() > > # Define the field GF(2^8) with generator z satisfying f as its > # minimal polynomial: > > F256.<z>=GF(256,'z',f) > > # Define a polynomial ring in 8 variables over F2 > > S=PolynomialRing(F256,8,'x') > > # Quotient out by the ideal containing xi^2-xi for all 8 generators. > # The variables in the quotient are called x0,x1,...,x7 > > I=S.ideal([xi^2-xi for xi in S.gens()]) > Sbar.<x0,x1,x2,x3,x4,x5,x6,x7>=S.quo(I) > > # Define the "generic" element x: > > x = x0+x1*z^1+x2*z^2+x3*z^3+x4*z^4+x5*z^5+x6*z^6+x7*z^7 > > # If all is well this should be true: > > assert x^256 == x > > # > > y = x^254 > > # ideally, we should have x*y = x^255 = 1, but unfortunately not: > > assert not x*y==1 > assert not x^255==1 > > # However > > one = x*y > assert one^2==one > > 2008/5/28 vpv <[EMAIL PROTECTED]>: > > > > > Hello, > > > I am trying to solve the following equation for y in SAGE: > > > x*y = 1 (mod z^8+z^4+z^3+z+1) > > > where > > > x = x0+x1*z^1+x2*z^2+x3*z^3+x4*z^4+x5*z^5+x6*z^6+x7*z^7 > > y = ? > > > x0,...,x7 are elements of GF(2). I do not know their values. I am > > searching for y in parametric form i.e. as a polynomial of z of degree > > 7 with coefficients - some functions of x0,...,x7. > > > I define in SAGE: > > > P.<x0,x1,x2,x3,x4,x5,x6,x7> = BooleanPolynomialRing(8, order='lex') > > Z.<z> = PolynomialRing(P) > > > I try to do the following in SAGE > > > y = inverse_mod(x0+x1*z^1+x2*z^2+x3*z^3+x4*z^4+x5*z^5+x6*z^6+x7*z^7, > > z^8+z^4+z^3+z+1) > > > but it does not work. > > > Alternatively, I try to define a ring of univariate polynomials of z > > with coeffients in P, every element of which is reduced (mod > > z^8+z^4+z^3+z+1), but I am not able to get the right syntax to do this > > in SAGE. > > > Any help is appreciated. > > > Thanks! --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
