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 -~----------~----~----~----~------~----~------~--~---
