implement given divisor as a specific hyperelliptic curve divisor

asked 2018-01-13 12:03:30 +0100

sherifasagewad
25 ●1 ●3 ●7

updated 2018-01-13 17:26:05 +0100

I get the values of a divisor for a hyperelliptic curve from a research paper to ensure the divisor is correct and i want to use this divisor on calculation.

so Idid the following:

A Secured Cloud System using Hyper Elliptic Curve and in sage:

p = 4112543547855339322343814790708185367671872426434747235319998473455582535888229747778325047393413053
K = GF(p)
R.<x> = K[]

f = x^5 + 7943193*x^4 + 6521255*x^3 + 1065528*x^2 + 3279922*x + 3728927
C = HyperellipticCurve( f )
J = C.jacobian()
X = J(K)

u, v = x^2 + 22457213658579645161*x + 62960708771725664757, 65279057408798633572*x + 32004384923913711271
D = X( [u,v] )

verbose 0 (3324: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation. verbose 0 (1083: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation. Error in lines 9-9 Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/jacobian_homset.py", line 145, in __call__ return JacobianMorphism_divisor_class_field(self, tuple(P)) File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/jacobian_morphism.py", line 388, in __init__ polys, C)) ValueError: Argument polys (= (x^2 + 22457213658579645161x + 62960708771725664757, 65279057408798633572x + 32004384923913711271)) must be divisor on curve Hyperelliptic Curve over Finite Field of size 4112543547855339322343814790708185367671872426434747235319998473455582535888229747778325047393413053 defined by y^2 = x^5 + 7943193x^4 + 6521255x^3 + 1065528x^2 + 3279922x + 3728927.

Does this mean the paper is wrong or I am at wrong. and if so, how can I modify my work?

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Please format your code, so that it is displayed properly. On this page, the markdown is - for instance - considering a star, *, to be something that starts italic text. For instance, *star* is shown as star . But we need of course a * in the above code with an other meaning.

In order to force a proper display, after applying the site markdown, one can proceed as follows:

copy and paste the python block (in a new line (after a new line)), then mark it, if not already marked, then press Control+K on it. (This will further indent the code in the typing frame.)
alternatively press that button with 101 and 010 in the "frame menu bar", where the question was posted.
or just indent once more (four spaces) before copy pasting...

dan_fulea ( 2018-01-13 13:42:36 +0100 )edit

If i woul have to trust either sage or a paper, then sage is my choice. In the given case, we may add asnegative "style arguments" for the paper:

the prime is "huge", but all coefficients in multiples of a "random divisor" are very "small". And which multiples are taken is also hidden in the darkness.
the test for $f-v^2= 0$ modulo $u$ fails for every "Mumfortd pair" published in the article.
we get a very accurate precision for the time taken by each computation, that has even more "relevant digits" than some "big coefficients in the Mumford representations", for instance: To create Divisor, it took 0.28114057028514255 Seconds - how can i give credit to such a paper?
references tend rather to mention elliptic curves, no "hype", and there is no page reference!

dan_fulea ( 2018-01-14 09:53:54 +0100 )edit

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answered 2018-01-14 15:22:52 +0100

dan_fulea
5537 ●5 ●43 ●95

The cited source above claims things in a two columns paper, that cannot be valid.

To support this, and to give an answer to the question "how can I modify my work?", please let us consider an other source using:

the same hyperelliptic curve
over the same prime
where there appears a divisor with Mumford representation $(u,v)$ with more "realistic" coefficients in $u,v$:

http://shodhganga.inflibnet.ac.in/bitstream/10603/102010/15/15_chapter%205.pdf

Let us take a look at Table 5.4. inside the above Chapter 5, and adapt the code to work with the new objects in this working reference:

p = 4112543547855339322343814790708185367671872426434747235319998473455582535888229747778325047393413053
K = GF(p)
R.<u> = K[]
f = u^5 + 7943193*u^4 + 6521255*u^3 + 1065528*u^2 + 3279922*u + 3728927

C = HyperellipticCurve( f )
J = C.jacobian()
X = J(K)

U = ( u^2
      + 1860389956661272000673008332624408105124473591357802495195699152127038244919646073105431091649161433*u
      + 3678638643325468767033409217184617082278646180398696096065435808423595790408365381053077562453250122 )

V = ( 1442288836874733903146967744806211994245652783209857240169344848296298432525255771657442734321755066*u
      + 921980454689397557939882885330855476778834577885555086747866424426336117619472682642574975583793355 )

D = X( [ U,V ] )

key = 11794224706464405453771880923682985303502368223256244790322242497664987902757947838816724010526943231
div = key * D

And sage confirms the printed result from this Chapter 5, page 107:

sage: print div
(x^2 + 1334417218673579196183082084171449472807365544624463016735162813781624145553691422830410095888029556*x + 3947000779241763670672412101498061083590545643436596730239170524313050659358606510365451650167089110, y + 3685255333461212784675197538912907368200372565862892461204210868447776269393357974512751786077530346*x + 4039903126531302725487748758957892552553012169705313271281907787620627040317525523860359172450156937)

Note: The computation also gives the warning:

verbose 0 (3324: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
verbose 0 (1083: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.

Note: This post is related to 40509 on this site.

Note: As mentioned in the comment, the coefficients of $(u,v)$ are expected to be (at least for $k\cdot D$) as "big" as the prime $p$...

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