On Monday, 14 September 2015 09:44:48 UTC-7, eggartmumie wrote:
>
> Dear Vincent,
>
> thank You for Your advice.
> Probably I am extremely slow-witted, but I know PolynomialRing, polygen etc
> Using Your suggestion I get _polynomials_ like x0, x1, x2, x3, x4 or x0 +
> 2*x1
> which have a degree, a constant term, which can be partially
> differentiated etc.
> What I do _not_ get is a variable which assumes values in a certain finite
> field.
> Unfortunately SAGE let me specify var('x',domain = GF(3)) but ignores the
> specification of the domain:
> x is essentially a variable which assumes complex values.
>
You are right: Sage does not support "symbolic calculus over finite
fields".
On the other hand, every function over a finite field can be written as a
polynomial, more or less...
Thus, why would you ever need to do var('x',domain = GF(3)) ?
E.g.
sage: R.<x,y>=PolynomialRing(GF(3))
sage: p=x^2+y^2+1
sage: p
x^2 + y^2 + 1
sage: p(1,-1)
0
(so you see it's all consistent, as 1+1+1 = 0 mod 3...)
> I consider to write variable elements of F.<a> = GF(2^2)
> as c_1*a+c_o with complex variables c_o and c_1.
> But then I have to implement all arithmetic in F and its extension fields
> by hand.
>
what do you need to compute?
> This is not what I expect is the right way to do it in SAGE.
> What do You recommend? Best Eggart
>
>
>
> Am Montag, 14. September 2015 00:15:31 UTC+2 schrieb vdelecroix:
>>
>> Hello,
>>
>> You would better use polynomial variables
>>
>> sage: R = PolynomialRing(GF(3), 'x', 5)
>> sage: R.gens()
>> (x0, x1, x2, x3, x4)
>> sage: x0,x1,x2,x3,x4 = R.gens()
>> sage: x0 + 2*x1
>> x0 - x1
>>
>> But you can only do polynomial computations with them
>>
>> sage: x0 ** x1
>> Traceback (most recent call last):
>> ...
>> TypeError: non-integral exponents not supported
>>
>> Is that enough for your purpose?
>>
>> Vincent
>>
>> On 13/09/15 19:04, eggartmumie wrote:
>> > Hi,
>> >
>> > in the context of HFE pf post quantum cryprography
>> > I had to realize that symbolic variables are real or complex only.
>> > SAGE ignores a 'domain=F' in var('x',domain=F) for some finite field F!
>> > Is it possible to define symbolic variables as elements of a finite
>> field?
>> > I need such variables in order to determine the public set of
>> polynomials
>> > given the private conversion T\circ iso^{-1} \circ P\circ iso\circ S
>> > where S and T are matrix transformations of vectors over some field F,
>> > iso is the vector space isomorphism of F^n to the extension field E,
>> > and P is some polynomial on E.
>> > It is easy to determine the conversion of a given vector over F.
>> > But how can I compute the n multivariate polynomials if the argument
>> > is a symbolic vector over F?
>> > I am using SAGE 6.2.
>> > Any indication is very much appreciated, best Eggart
>> >
>>
>
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