nt.number theory – Calculate generators from $ Gamma_0 (N) $

I found the group in a newspaper $ Gamma_0 (18) $ can be generated by the following list of matrices:

  • $ displaystyle left ( begin {array} {rr} 7 & -1 \ 36 & 5 end {array} right) $

  • $ displaystyle left ( begin {array} {rr} 13 & -8 \ 18 & -11 end {array} right) $

  • $ displaystyle left ( begin {array} {rr} 71 & -15 \ 90 & -9 end {array} right) $

  • $ displaystyle left ( begin {array} {rr} 55 & -13 \ 72 & -17 end {array} right) $
  • $ displaystyle left ( begin {array} {rr} 7 & -2 \ 18 & -5 end {array} right) $
  • $ displaystyle left ( begin {array} {rr} 31 & -25 \ 36 & -29 end {array} right) $
  • $ displaystyle left ( begin {array} {rr} 1 & 1 \ 0 & 1 end {array} right) $
  • $ displaystyle left ( begin {array} {rr} -1 & 0 \ 0 & -1 end {array} right) $

E.g. $ 31 times (-29) – 36 times (-25) = -899 + 900 = $ 1 Actually, $ 36, 18, 19 equiv 0 pmod {18} $.

These generators look anything. There is a list of generators from $ Gamma_0 (N) $ to the $ N <100 $ ?

Is there a computer program or algorithm for finding generators of congruence groups, for example using a linear algebra package such as e.g. sage ?

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rational points – Two additional generators are required for an elliptical Z / 6 curve

We are looking for rank 8 elliptic curves with the torsion subgroup Z / 6 using newly discovered families that are similar to Kihara (Kihara family is described in https://arxiv.org/pdf/1503.03667.pdf).

Today we came across a curve

$ (0.8169768624655967629114128598.0, -451787550647310420612086468536366715869054405951830599.0) $

Both Magma Calculator (http://magma.maths.usyd.edu.au/calc/) and mwrank with $ -b12 $ Return 6 generators for this curve.
Magma V2.20-10 (STUDENT) no longer has enough memory to run the following code:

E := EllipticCurve((0,8169768624655967629114128598,0,-451787550647310420612086468536366715869054405951830599,0));

Sagemath returns $ 8 $ for the upper limit of the analytical rank, also for max_Delta =$ 3.3 $ (We are still testing for a higher max_Delta):

E = EllipticCurve((0,8169768624655967629114128598,0,-451787550647310420612086468536366715869054405951830599,0))  

Is there a way to find two more generators?

A similar question for that $ 6 $ <= Rank (E) <= $ 7 $ The situation has been successfully resolved by Jeremy Rouse (another elliptical Z / 6 curve generator required), but our software throttles when we try to follow his instructions.

We are ready to award a bounty from a $ 100 $ (How do I transfer my current call?) For both generators. Your name (together with ours) will also be published at the end of the page here: https://web.math.pmf.unizg.hr/~duje/tors/z6.html


BIP 32 HD Wallets – Why don't the keys generated in Bitcoin Core match those of online generators even though they use the same starting value?

I have the following problem in Bitcoin Core. With this command:

echo carpet begin bacon master draft fortune food cherry cage axis vault clown | 
bx mnemonic-to-seed | 
bx hd-new -v 70615956 | 
bx hd-to-ec -c bx-testnet.cfg | 
sed 's/$/01/' | 
bx base58check-encode -v 239

What I do is from the mnemonic carpet begin bacon master draft fortune food cherry cage axis vault clownI generate one hdseed for Bitcoin Core in regtest, Results:


Then I use the command:

bitcoin-cli -regtest sethdseed true cNQQqyR81GZXU3Pa1pK1gQQp6jgni5HyLYM99nL2JpzVRmdCyrFE

Where I change that hdseed for which I just got you and Bitcoin Core generates a key pool. When I put the same mnemonic on the BIP39 page, it gives me the same root key that gives me the command for this part:

echo carpet begin bacon master draft fortune food cherry cage axis vault clown | 
bx mnemonic-to-seed | 
bx hd-new -v 70615956

which is: tprv8ZgxMBicQKsPd5cY6ZZzsRzkcfDANp2y3jFtYNQZsp953sXiFNP9ZFod1fAhkgtCJZGvTcLJVdhFCj7VDNu68nuP4m2QGLcd4JpkteG8Ccc

However, the derivations of private keys generated by Bitcoin Core do not match those of BIP39 and the BIP32 root key of BIP39 does not match the master private key of Bitcoin Core hdseed if it matches what appears in the command. The command gets both the hdseed derivatives and the Bitcoin Core master key bitcoin-cli -regtest dumpwallet test_wallet

I get two big unknowns:

  1. Is the way in which I generate the hdseed from the mnemonic correct?
  2. Bitcoin Core generates its private master key hdseed? And if so, is this process reversible?

But in general I want to know why they give differently !!!

Thank you in advance.

ct.category theory – generators and colimit closures in higher categories

In ordinary categories $ mathcal {C} $there are nice conditions among those a generator is also one Coli with generator to the $ mathcal {C} $, In other words, under suitable conditions, if $ mathcal {C} $ contains a family of objects $ X _ { alpha} $ so for all $ C in mathcal {C} $ There is an effective epimorphism $ coprod X _ { alpha} to C $, then the smallest cocomplete subcategory of $ mathcal {C} $ with all $ X _ { alpha} $is $ mathcal {C} $ yourself. (See notes https://home.sandiego.edu/~shulman/papers/generators.pdf by Mike Shulman)

I wonder if something similar applies in higher categories where "higher category" could mean anything of a $ 2 $Category to one $ infty $-Category. Since we have an idea of ​​effective epimorphism in higher categories, we can also say what it means for a collection of objects to create the category. Under what conditions do these objects create the category under (iterated) colimits?

ac.commutative algebra – generators for ideals in the ring of multivariate Laurent polynomials

Consider the following problem:

Find an ideal $ I subset mathbb {Q} [x ^ { pm} _1, x ^ { pm} _2, x ^ { pm} _3] $ so that $ I_ {aff} subset mathbb {Q} [x_1, x_2, x_3] = I cap k [x_1, x_2, x_3] $ needs more generators than $ I $and homogenization $ I_ {proj} in k [x_0, x_1, x_2, x_3] $ needs more generators than $ I_ {aff} $,

It's pretty easy to give an example of an ideal $ I $ their homogenization $ I_ {proj} $ requires more generators – for example, take the ideal of twisted cubes $ I_ {t} $ = $ langle x_2 – x ^ 2_1, x_3 – x ^ 3_1 rangle $, However, I am not sure how exactly I should construct the ideal in the Laurent polynomial ring. One approach was to try to take the colon ideal

$ langle x_2 – x ^ 2_1, x_3 – x ^ 3_1 rangle $ : $ langle x_1, x_2, x_3 rangle $

This is equivalent to deleting the origin of the twisted cubes, but this simply leaves me with an ideal containing three generators. I have the feeling that I lack a substantial part of the understanding here. So if someone could help, I would be very grateful.



Computational Complexity – expressing a torsion point of an elliptic curve as a combination of generators

I face the following problem:
Suppose we have a finite field $ mathbb {F} _p $ and an elliptic curve $ E $ above defined. Suppose that for $ m in mathbb {Z} $ not a multiple of the characteristic of the basic field. So we have an isomorphism
$$ E[m] longleftrightarrow ( mathbb {Z} / m mathbb {Z}) ^ 2 $$
Suppose we know that
$$ E[m] subset E ( mathbb {F} _q) $$
from where $ q $ is a power of $ p $, Suppose we also had generators $ P, Q in E[m]$ and a third point $ R in E[m]$, I want to find $ a, b in [0,m-1]$ for which
$$ R = aP + bQ $$
What is the computational burden of this problem? The most efficient algorithm I think about is to solve a lot of ECDLP
$$ R-aP = bQ $$
from where $ a in [0,m-1]$, Of course, this has a computational burden $ O (m sqrt {m}) $ because there is computational effort for the single ECDLP algorithm $ O ( sqrt {m}) $,
Thanks for your time.

Pseudo-random generators – How Fibonacci LFSR works

In arithmetic, a linear feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state. Wiki quote.
Its output depends on the state and polynomial, not on two states.

The name (afair) of Fibonacci deals with the recursion relation, its sequence and the further interest in characteristic polynomials for generating sequences, but with the one that solves Fibonacci numbers in closed form and explores the characteristic equation (which may be present in the generator) minimum values) monically to make the longest time) was Lucas. I can not find any clue why Fibonacci is called LFSR, but it is the will of the inventors to call it.

The same applies to the LFG – delayed Fibonacci generator introduced by Marsaglia (in multiplicative form) to eliminate Additive 1 defects.

The real interest in randomizing was given by Galton (when it comes to the Romans throwing coins and stored bits), random numbers from Pearson, first PRNG is attributed by Neuman, first one works on Lehmer, and for Fibonacci his insights are nice , But he started his sequences to count rabbits that had no connection to PRNG.

Chained pseudo-random generators

I have two pseudo-random generators $ G_1 $ and $ G_2 $ from $ k $ a little too $ 2k $ little. I still have a PRG $ G_3 $ from $ k $ to $ 2k $ Bits. Now I'm building a new feature $ k $ to $ 4k $ as follows:

To let $ x $ His $ k $ Then apply bit $ G_3 $ on it $ x $, Then you go through the issue $ y $, Apply $ G_1 $ At first $ k $ Bits and $ G_2 $ on the last $ k $ and connect the resulting output. The resulting function is then a PRG. How do you prove that?