Thank you very much for your reply! The output for y which you sent below is exactly what I am looking for. Can you please paste also the SAGE definitions which you use to construct the field GF(2^8) (mod z^8+z^4+z^3+z+1), the polynomial ring S, and finally the quotient of S by the 8 relations (x0^2=x0, x1^2=x1, ..., x7^2=x7)?
Here is how I try to define the above constructions, but as my SAGE programming is not very good I don't seem to get it right: Z = IntegerModRing(2^8) R = PolynomialRing(F256,'z'); S = R.quotient(z^8+z^4+z^3+z+1, 'x'); P.<x0,x1,x2,x3,x4,x5,x6,x7> = PolynomialRing(S) When I try to construct an element of P like this: b = (x0+x1*z^1+x2*z^2+x3*z^3+x4*z^4+x5*z^5+x6*z^6+x7*z^7) I get the following error in SAGE: "TypeError: unsupported operand parent(s) for '*': 'Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7 over Univariate Quotient Polynomial Ring in x over Integer Ring with modulus z^8 + z^4 + z^3 + z + 1' and 'Univariate Polynomial Ring in z over Boolean PolynomialRing in x0, x1, x2, x3, x4, x5, x6, x7'" Thanks for your help! On May 28, 11:58 pm, "John Cremona" <[EMAIL PROTECTED]> wrote: > You could define GF(2^8) using your polynomial as modulus, then define > the polynomial ring S in 8 variables x0,...,x7 over that, write x as > an element in that ring. The inverse of x is also x^254, but you > want to evaluate this with the side conditions xi^2=xi. So take the > quotient of S by those 8 relations first. This might be better > handled by the PlyBori package, but I am not familiar with that. > > I tried this and it worked. The output (your y) starts like this, > where xibar is your xi and a is your z. > > (a^6 + a^4 + a)*x0bar*x1bar*x2bar*x3bar*x4bar*x5bar*x6bar + (a^7 + a^5 > + a^2)*x0bar*x1bar*x2bar*x3bar*x4bar*x5bar*x7bar + (a^6 + a^4 + a + > 1)*x0bar*x1bar*x2bar*x3bar*x4bar*x6bar*x7bar + (a^7 + a^5 + a^2 + > a)*x0bar*x1bar*x2bar*x3bar*x5bar*x6bar*x7bar + (a^2 + > 1)*x0bar*x1bar*x2bar*x4bar*x5bar*x6bar*x7bar + (a^6 + a^4 + > a^3)*x0bar*x1bar*x3bar*x4bar*x5bar*x6bar*x7bar + (a^7 + a^5 + > a^4)*x0bar*x2bar*x3bar*x4bar*x5bar*x6bar*x7bar + (a^5 + a^3 + > 1)*x1bar*x2bar*x3bar*x4bar*x5bar*x6bar*x7bar + (a^7 + a^6 + > a^2)*x0bar*x1bar*x2bar*x3bar*x4bar*x5bar + (a^7 + a^5 + a^2 + > 1)*x0bar*x1bar*x2bar*x3bar*x4bar*x6bar + (a^7 + > a^4)*x0bar*x1bar*x2bar*x3bar*x5bar*x6bar + (a^6 + a^4 + a^2 + > a)*x0bar*x1bar*x2bar*x4bar*x5bar*x6bar + (a^7 + a^5 + > a)*x0bar*x1bar*x3bar*x4bar*x5bar*x6bar + (a^7 + a^6 + a^4 + > a)*x0bar*x2bar*x3bar*x4bar*x5bar*x6bar + (a^6 + a^5 + > a)*x1bar*x2bar*x3bar*x4bar*x5bar*x6bar + (a^7 + a^4 + a + > 1)*x0bar*x1bar*x2bar*x3bar*x4bar*x7bar + (a^6 + a^4 + > 1)*x0bar*x1bar*x2bar*x3bar*x5bar*x7bar + (a^6 + a^4 + a^3 + a + > 1)*x0bar*x1bar*x2bar*x4bar*x5bar*x7bar + (a^7 + a^6 + a^3 + > a^2)*x0bar*x1bar*x3bar*x4bar*x5bar*x7bar + (a^6 + a^4 + a^3 + a^2 + a > + 1)*x0bar*x2bar*x3bar*x4bar*x5bar*x7bar + (a^7 + a^6 + a^4 + a^3 + > a^2 + a + 1)*x1bar*x2bar*x3bar*x4bar*x5bar*x7bar + (a^5 + a^4 + a^3 + > a^2 + 1)*x0bar*x1bar*x2bar*x3bar*x6bar*x7bar + (a^6 + a^4 + > a^2)*x0bar*x1bar*x2bar*x4bar*x6bar*x7bar + > (a^6)*x0bar*x1bar*x3bar*x4bar*x6bar*x7bar + (a^7 + a + > 1)*x0bar*x2bar*x3bar*x4bar*x6bar*x7bar + (a^6 + > a^3)*x1bar*x2bar*x3bar*x4bar*x6bar*x7bar + > > John Cremona > > 2008/5/28 David Joyner <[EMAIL PROTECTED]>: > > > > > It seems you should be able to represent multiplication by y as a > > matrix equation, > > which you might have luck inverting. > > > On Wed, May 28, 2008 at 5:27 PM, vpv <[EMAIL PROTECTED]> wrote: > > >> 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 -~----------~----~----~----~------~----~------~--~---
