Intro I am somewhat new to cryptography. Repeating the basics of RSA from :  Textbooks say the one-way function is merely two primes (with some critical constraints) ...  may be a good base of this question, as well as . Also,  already asks the same thing as the question here, but in a more abstract way. The idea Why not calculating in advance, on many computers, a vast amount of multiplications of primes that you loop over systematically, in the constraints of RSA? And then you would save the outcome as keys in a Big Data system that you can quickly search through? You could use a Bloom filter, a key coordinator or other Big Data tricks to quickly find the computers that might have the multiplied value at hand, and then quickly find a value's underlying primes. You could have 10000 computers, with chosen different starting prime ranges. The first and second primes could be within a likely range that is needed to meet RSA standards of 4096 bit. You may also guess a bit, meaning to take a range that is rather around the square root of a chosen outcome, or take other tricks to guess a bit. Then it just depends on how many computers you use. Even if you hit just 0.001 % of the possible combinations you could already get the private key from the public key in 0.001 % of the cases. Or do I oversee something here. The question It cannot be that easy, can it? Can Big Data attack RSA by just calculating many prime multiplications in advance?
Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing prime keys up to $2^{2048}$?  I understand this would take an immense amount of time to do, but it would only be done once. Obviously, computing such an index would take a considerable amount of time and would have to be done with someone having the right amount of resource. Given the public key, you would be able to look up what the two primes are, and hence retrieve the private key instantly.  Breaking an 256 AES would then be easier, as AES keys are often generated using the RSA private key.  Would it be possible to build such an index? How long would that take in permutations and perhaps with the fastest computer in the world?
Is $\le$ defined by $\langle 𝑥_1, 𝑦_1 \rangle \le \langle 𝑥_2, 𝑦_2 \rangle$, if $𝑥_1 \le 𝑥_2 \land𝑦_1 \ge 𝑦_2$ a linear order? I need to tell if the relation $\le$ defined by relationship $\langle 𝑥_1, 𝑦_1 \rangle \le \langle 𝑥_2, 𝑦_2 \rangle$, if $𝑥_1 \le 𝑥_2$ $\land$ $𝑦_1 \ge 𝑦_2$ linear order?   I have already proven that the relation is an order, but I need to decide if it is a linear order and why. Thanks for any help! The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is , when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d&gt;1$?      — For instance if $p$ is prime (which is not coprime with $0$) then $$\mathbb{Z}[i]/(p) \cong \mathbb{F}_p[X]/(X^2+1) \cong \begin{cases} \mathbb{F}_{p^2} &amp;\text{if } p \equiv 3 \pmod 4\\ \mathbb{F}_{p} \times \mathbb{F}_{p} &amp;\text{if } p \equiv 1 \pmod 4 \end{cases}$$ (because $-1$ is a square mod $p$ iff $(-1)^{(p-1)/2}=1$).  — More generally, if $n=p_1^{r_1} \cdots p_m^{r_m} \in \Bbb N$, then  each pair of integers $p_j^{r_j}$ are coprime, so that by CRT we get $$\mathbb{Z}[i]/(n) \cong \mathbb{Z}[i]/(p_1^{r_1}) \times \cdots \times \mathbb{Z}[i]/(p_m^{r_m})$$  I was not sure how to find the structure of $\mathbb{Z}[i]/(p^{r}) \cong (\Bbb Z/p^r \Bbb Z)[X] \,/\, (X^2+1)$ when $p$ is prime and $r&gt;1$.  — Even more generally, in order to determine the structure of $\mathbb{Z}[i]/(a+bi)$ with $a+bi=d(x+iy)$ and $(x,y)=1$, we could try to use the CRT, provided that $d$ is coprime with $x+iy$ in $\Bbb Z[i]$. But this is not always true: for $d=13$ and $x+iy=2+3i$, we can't find Gauss integers $u$ and $v$ such that $du + (x+iy)v=1$, because this would mean that $(2+3i)[(2-3i)u+v]=1$, i.e. $2+3i$ is a unit in $\Bbb Z[i]$ which is not because its norm is $13 \neq ±1$.  — I was not able to go further. I recall that my general question is to known what $\mathbb{Z}[i]/(a+bi)$ is isomorphic to, when $a$ and $b$ are integers which are not coprime (for instance $a=p^r,b=0$ or $d=(a,b) = a^2+b^2&gt;1$).  Thank you for your help! Visiting Greece with back-to-back Schengen visas (One from Germany, other from Italy) I have a Schengen Visa issued by Germany, which expire on 11th Jan 2014, and I can remain in the zone till 11th Jan 23.59 PM. I have already visited Germany in the past with this visa, so I can travel whenever in Schengen zone directly from my country (Turkey).  I have a second Schengen Visa issued by Italy, which START on 12th Jan 2014, such that I can remain in Schengen zone from 12th Jan 00.00 AM onwards (I'm putting times to clarify which dates are inclusive).  I would like to travel to Greece with this visa: Entering the country on 6th January, and leaving the country on 19th January.  Is this allowed? Or do I have to leave the country first, and then re-enter through Italy? Getting an "Authentication token manipulation" error when trying to change my user password I am logging in to my Ubuntu Server using my username. Once I am logged in I am typing passwd command. Entering a new password but a second after getting following error messages:  passwd: Authentication token manipulation error passwd: password unchanged   What is wrong here? How can I change my password otherwise if I don't have access to that server physically, i.e. I am connecting remotely with ssh using terminal. How do I simulate in Photoshop what a piece will look like when printed on non-white paper? I'm designing a box in Photoshop that will be offset printed on unbleached paper. I haven't picked out a printer yet, so I don't know the exact kind of paper I'll be using and don't have access to it for proofs. For now I'm using a lightly-textured layer of tan color as a placeholder to mock it up.  What I'm trying to figure out is, how do I figure out how my design will look printed on natural paper rather than bleached white paper? I know that the brightness will be diminished and the hue will be altered, but I need to know how much so I can compensate for it as much as possible.  Is there a way to get at least a rough but reliably-simulated idea of how my prints will come out? Does it have to do with blend modes, or color separations? I'm at a bit of a loss here. How precise is the frequency of the AC electricity network? If I make a digital clock which first detects the AC frequency roughly (whether it is 50, 60, 100Hz, etc), then uses it as its clock pulse source, how precise would it be? Would it work world wide precisely?  For instance, in USA, what is the mean and standard deviation of the 60Hz AC frequency? Can we have a perfect field of characteristic $p$ with a non-archimedean valuation? Fspy add-on not installing Can I connect to a sandbox using JWT Flow and Prod connect app? I'm trying to connect to a sandbox using the sfdx cli and the JWT flow through force:auth:jwt:grant. According to , we can use the same connected app information from Production to connect to our Scratch Org.  Is it something that should work for sandbox too ? I'm getting a "Failed: Not approved" error.  Connected app in sandbox comes from the refresh but has a different consumer key. The Permission set is set correctly on my user.  Is there a way to connect to a recently refreshed sandbox using the JWT flow (without manually changing something like the consumer key) ? Openlayers v4.0.1 support Google Maps Javascript API? I want to know if the latest version of Openlayers (v4.0.1) supports Google Maps as a Tile Layer. I can't find any document ion about this. If Openlayers doesn't support Google Maps, could someone  tell me if there is any way to do this? Why do I get the error "failed unmounting /var" during shutdown? I have a fresh install of Debian 9.0.0. It has a separate 1 GB partition for /var formatted at ext4. When I shutdown from the terminal, using shutdown -h now, there is a bright-red error message stating "failed unmounting /var".  What is the fix for this error? How to Connect & Share your Internet Connection (Wired & Wireless) I have Ubuntu 12.04 on a HP 430 notebook and this has a single wired internet connection and I would like to share this with a LG Optimus One with Android 2.3.3. I followed  instructions and created a hotspot as shown below, but UbuntuHost is not visible on the LG Optimus One when I enable the Wifi on it.  To summarize, I want my Laptop with an internet connection to act as a WiFi router and access the internet from other devices (LG Optimus One with Android in this case).  What additional steps have to be done? Removing files with strange names Possible Duplicate:          Somehow I ended up with a file named "-r".  How do I remove it?  rm -r doesn't work.  I tried 'rm -i `ls -a`' to step through the file names, but it didn't prompt me to delete this one.  Edit A very hacky approach was to use python's os.unlink function.  That worked, but I'm curious to hear other ways. Drop "not constructive", combine "noisy", reword "rude" and "other" comment flags What Happened to The Latex Symbols In the Libreoffice-Texmaths Equation Editor? Is there any way to increase the drop rate of items? Merge two custom post types into one admin page? I am building a site with 4 custom post types. To make things easier for our client/admin, I'm wondering if there is a way to merge those custom post types into one list in the backend? Like if you look at normal backend admin page with the list and columns of posts, can a page like that pull in 3 other post types for easy sorting/searching?  I can't combine the CPTs into one, they must remain seperate outside of one 'viewing' page in the admin area if possible.  Any thoughts on how to do this?? Or even where to start looking/thinkging of how to develop this? I am currently studying  answer on the Moore-Penrose pseudoinverse and the Euclidean norm by the user &quot;Etienne dM&quot;. The first point of their answer proceeds as follows:  Let $x$ be $A^+y$.  Let me begin by the second point. For all $z$, we have: \begin{align} \lVert Az-b \rVert_2^2 &amp;= \lVert Ax-b \rVert_2^2 + \lVert A(z-x) \rVert_2^2 + 2 (z-x)^TA^T(Ax-b)\\ &amp; \geq  \lVert Ax-b \rVert_2^2 + 2 (z-x)^TA^T(Ax-b) \end{align} Moreover, because $(AA^+)^T = AA^+$, $$ A^T(Ax-b) = ((AA^+)A)^Tb - A^Tb = 0$$ Thus, we prove that for all $z$, $\rVert Az-b \lVert_2^2 \geq\rVert Ax-b \lVert_2^2$, that is to say $A^+b$ is as close as possible to $y$ in term of the Euclidian norm $\lVert Ax-b\rVert_2$.   I realise that $x = A^+ y$, but I don't understand how any of this implies that &quot;$A^+ b$ is as close as possible to $y$ in terms of the Euclidean norm $\lVert Ax-b\rVert_2$&quot;. I would greatly appreciate it if people would please take the time to explain this to me. I am trying to run an executable file from a network drive on Windows 7 Professional (64-bit) in order to install a program, but, each time I try, it doesn't run and the following message appears:  {path-name} The specified path does not exist. Check the path, and then try again.   I am definitely connected to the network drive as I can see all of the directories and files, and can even open non-executable file types. Also, I am fairly confident that this is neither a firewall nor a permissions issue. Here comes Bob.  Bob is coming here.  Would anyone please explain what is  the difference between these?