Sorry for the leftovers I copied and pasted...
With the following I manage to create polynomials whose coefficients are in
Z_32 and they are modulo x^N-1.
N=100
q=32
PR.<xx> = Zmod(q)[]
Q.<x> = PR.quotient(xx^N - 1)
pp=Q.random_element()
The problem is that pp.inverse_mod(x^N-1) does not work
(NotImplementedError). Similarly, for pp^(-1) I get the error
NotImplementedError: The base ring (=Ring of integers modulo 32) is not a
field
I'm working on sage-6.3-x86_64-Linux
Justin proposed to use GF(), however, if I use the following (fixed the
garbages of the previous code):
q=32
FFQ.<t> = GF(q)
PR.<x> = PolynomialRing(FFQ)
Q.<xx> = PR.quotient(x^N - 1)
My coefficients belong to a finite field of size 32 I can invert etc, but
the coeffs are not in Z_32 as I would like to.
Thanks for your help
On Sunday, April 5, 2015 at 5:09:40 PM UTC+3, Dima Pasechnik wrote:
>
>
>
> On Sunday, 5 April 2015 13:20:49 UTC+1, absinthe wrote:
>>
>> Justin thanks for your reply. When I realised that I have posted to dev I
>> deleted the message and I posted to support. It looks like that you had
>> already answered there. Since it might help others which will look at
>> support and not dev, I copy and paste your dev reply here.
>> 2. Yes no problem :)
>> 1. The reason I resulted to Integers(p) was that I couldn't have the
>> proper cooefficients when using GF.
>> For the same "configuration" I tried the following
>> Trying with the following
>> p=32
>> N=100
>> FFQ.<t> = GF(q)#FiniteField(q)
>> PR.<x> = PolynomialRing(K)
>> Q.<xx> = PR.quotient(x^N - 1)
>> pp=Q.random_element()
>> while True:
>> try:
>> ppInv=pp.inverse_mod(xx^N-1)
>> break
>> except:
>> pp=Q.random_element()
>> print (pp*ppInv).mod(x^N-1)
>>
>> I get to a dead end as inverse_mod returns me a not implemented error
>> (sage-6.3-x86_64-Linux)
>>
>
> What is K there? And what is q?
> They are not defined in your code above.
>
> As well,
> it's not clear why you try to call inverse_mod(), as pp is already an
> element
> of the quotient ring modulo (x^N-1).
>
> But pp^{-1} need not exist, still, as Q is not a field for each N>1.
>
>
>
> On Sunday, April 5, 2015 at 3:59:42 AM UTC+3, Justin C. Walker wrote:
>>
>>
>> On Apr 4, 2015, at 17:25 , absinthe wrote:
>>
>> > Dear all,
>> >
>> > I'm trying to work with polynomials modulo x^N-1 whose coefficients
>> belong
>> > to Z_p (If it helps p is a power of a prime). I know that I'm doing
>> > something wrong, but I cannot figure out what so any help is welcome.
>>
>> I'm not sure how familiar you are with this stuff, so forgive me if this
>> is already clear to you.
>>
>> 1. When "p" is a prime power, Z/pZ is not a field (it's a ring, but not a
>> domain). If you want to deal with coefficients in a field, then you will
>> want to use "GF(p)", not "Integers(p)". And a minor syntactic wrinkle to
>> beware of is that when "p" (as above) is a prime power, and not a prime,
>> you need a second argument, to be used as the name of the "generator" of
>> F_p (as an extension of F_q, q being the prime in p).
>>
>> 2. Also, in computer algebra systems, you have to be careful about
>> parentheses, to get what you want. In particular, "X^N-1" and X^(N-1)" are
>> not the same.
>>
>> If this isn't helpful, we can look at this some more.
>>
>> HTH
>>
>> Justin
>>
>> --
>> Justin C. Walker
>> Curmudgeon at Large
>> Director
>> Institute for the Enhancement of the Director's Income
>> --
>> Build a man a fire and he'll be warm
>> for a night.
>> Set a man on fire and he'll be warm
>> for the rest of his life.
>>
>>
>>
>> On Sunday, April 5, 2015 at 4:00:47 AM UTC+3, Justin C. Walker wrote:
>>>
>>>
>>> On Apr 4, 2015, at 17:29 , absinthe wrote:
>>>
>>> > Dear all,
>>> >
>>> > I'm trying to work with polynomials modulo x^N-1 whose coefficients
>>> belong
>>> > to Z_p (If it helps p is a power of a prime). I know that I'm doing
>>> > something wrong, but I cannot figure out what so any help is welcome.
>>>
>>> Answered, possibly, on sage-devel...
>>>
>>> --
>>> Justin C. Walker, Curmudgeon at Large
>>> Institute for the Absorption of Federal Funds
>>> -----------
>>> My wife 'n kids 'n dogs are gone,
>>> I can't get Jesus on the phone,
>>> But Ol' Milwaukee's Best is my best friend.
>>> -----------
>>>
>>>
>>>
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