I have known that the vector is sampled from Gaussian distribution in lattice-based cryptography because the distribution of the vector $\mod{\mathcal{P}(\mathbf{B})}$ approximates to uniform distribution.  It is proved by below lemma.     For any $s &gt; 0$, $\mathbf{c} \in \mathbb{R}^n$, and lattice $\Lambda(\mathbf{B})$, the statistical distance between $\mathcal{D}_{s,\mathbf{c}} \mod{\mathcal{P}(\mathbf{B})}$ and the uniform distribution over $\mathcal{P}(\mathbf{B})$ is at most $\frac{1}{2} \rho_{1/s}(\Lambda(\mathbf{B})^*\backslash\{\mathbf{0}\})$. In particular, for any $\epsilon &gt;0$ and any $s \geq \eta_{\epsilon}(\mathbf{B})$, the statistical distance is at most $\Delta(\mathcal{D}_{s,\mathbf{c}} \mod{\mathcal{P}(\mathbf{B})},\mathcal{U}(\mathcal{P}(\mathbf{B}))) \leq \epsilon/2$.   But if the vector is sampled from uniform distribution at first, the distribution of the vector $\mod{\mathcal{P}(\mathbf{B})}$ is uniform. So, I thought that Gaussian sampling is not needed for above reason, because Gaussian sampling requires large costs.  Another reason I think is that the smaller vector is preferred because almost lattice problem is about finding short vector and Gaussian variable has higher probability as closer to the center. Is it right?  If my thought is not correct, let me know the reason using Gaussian distribution in lattice-based cryptography.
The Wikipedia article on  mentions two methods of sampling "small" polynomials namely uniform sampling and discrete Gaussian sampling. Uniform sampling is clearly the simplest, involving simply uniformly selecting the coefficients from the set of small coefficients, thereby guaranteeing they will be small. However, the articles mention that security proofs rely on discrete Gaussian sampling.  However, I have a bit of difficulty wrapping my head around    Why discrete Gaussian sampling would be better than uniform, security-wise?   I mean, usually uniform is preferable - it maximizes the entropy/information in each sample, making the outcome more unpredictable. I could understand it if the Gaussian sampling technique would somehow serve to avoid (at least with high probability) "weak" parts of the space of outcomes but I have found no information/justification that this is the case.  Another reason I can see why discrete Gaussian might be better is that there is, after all, a chance (albeit a small one) one would select a non-small coefficient, so the space of outcomes is actually larger. However, given that this probability of hitting this extra space is (by design) quite small but also that one would likely immediately reject such a sample (because the algorithm calls for a small polynomial), this seems to be irrelevant.  It is not because I am against discrete Gaussian sampling I just want to understand why it is more secure. Also, I want to understand if it really IS more secure or it is simply easier to prove security in that model. I would actually consider that very interesting if it is the case because it raises the question of why is it easier to prove the security under conditions that are intuitively worse (i.e. discrete Gaussian vs the intuitively better uniform case)? If it is so intuitively clear that uniform sampling is better, surely it must be possible to transform that intuition into a security proof of RWE also under uniform sampling. Otherwise, I would fear the intuition must with high likelihood be wrong or at least dubious.
Shortcut for "null if object is null, or object.member if object is not null" Why don't we have an isomorphism between $R[x]$ and $ R[[x]]$? Interior product (contraction) symbol See the above excerpt from John Lee's Introduction to Smooth Manifolds. In differential topology and linear algebra, there's an operator called the interior product or contraction operator, denoted $\iota_X(\omega)$ or alternatively with some other "half box" like operator $X_|\omega$. Does anyone know how to markup that operator in LaTeX?   Google has turned up nothing except for this thread:   That thread did receive an accepted answer, which was \invneg from the MnSymbol package, but the suggestion given there does not seem correct to me. Or at least, it does not match what I'm familiar with. That answer results in an inverted negation symbol, whereas the contraction operator should have a stroke down at the floor of the line, and a long stroke going up almost the full height of the line. It can be found in many texts, but why can I not find the correct LaTeX markup? How do I pass the forest stage on Candy Box 2? I cannot pass the forest stage on Candy Box 2. The dogs and forest mosquitos kill me fast.  My stats and equipment are:   220 health Polished silver sword (7 damage, also have the trolls bludgeon; 12-16 damage) Leather boots and gloves (no stat bonus) Basic and advanced spells from the witch across the bridge   On another save I also increased my health to 1000 and still could not pass the forest stage, any advice?  Thanks. Code working in jsFiddle but not in browser Using custom tables in nft I have a Fedora 31 system on which I am using iptables-nft. I need this because there is still a bunch of software that expects the legacy iptables command line tools.  This means that my nftables configuration has the corresponding set of tables to match the legacy configuration:  table ip filter table ip nat table ip6 filter table ip mangle table ip6 nat table ip6 mangle   I use a containerized VPN service that, prior to nftables, would enable masquerading on my primary ethernet interface by running something like this when the vpn comes up:  iptables -t nat -A POSTROUTING -s 172.16.254.0/24 -o eth0 -j MASQUERADE   Since upgrading to Fedora 31 and iptables-nft, this no longer works. The container (running alpine) does not have the iptables-nft compatability wrapper, but it does have the nft command itself.  I can't use the nft cli to add rules to the existing tables, because this will break iptables-nft. But I can create new tables. I was hoping I could just apply a configuration like this:  table ip vpn {     chain postrouting {         type nat hook postrouting priority filter; policy accept;         ip saddr 172.16.254.0/24 oifname "eth0" counter masquerade     }      chain forward {         type filter hook forward priority filter; policy accept;         ip saddr 172.16.254.0/24 counter accept     } }   ...but this doesn't appear to have any impact. By setting the chains in this table to priority 0 I was hoping they would match before the legacy nat table, but that doesn't appear to be the case.  Is there a way to make this work? Why do shadows from the sun join each other when near enough? I was laying on my bed, reading a book when the sun shone through the windows on my left. I happened to look at the wall on my right and noticed this very strange effect. The shadow of my elbow, when near the pages of the book, joined up with the shadow of the book even though I wasn't physically touching it.   I have posted :The video seems to be the wrong way up, but you still get the idea of what is happening.  What is causing this? Some sort of optical illusion where the light gets bent?  Coincidentally, I have been wondering about a similar effect recently where if you focus your eye on a nearby object, say, your finger, objects behind it in the distance seem to get curved/distorted around the edge of your finger. It seems awfully related...  EDIT: I could see the bulge with my bare eyes to the same extent as in the video! The room was well light and the wall was indeed quite bright. Execute bash commands over SSH while staying in interactive mode afterwards Probably it is a weird thing I would like to achieve but: I would like to SSH to a remote host and then execute some bash commands like "alias" automatically to use them afterwards in an interactive mode. The remote host does not allow the personal environment.    Is this possible? Would "expect" do it? Or is there any other way to achieve this? Basically I would like to have my bashrc without modifying any files on the remote host. We are using .Net to make web service calls the Salesforce API. Is there a way to know for sure that the calls are using TLS 1.2? Are there any API logs or records of the protocol is use in Salesforce we can review to be confident the change is working? Java String split is not working improper integral calculation I do not know how to solve this integral and then expand the result in powers of $t$  $$(1-t)^{-1/2}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{2x^2t}{1+t}\right)}{1+x^2}dx=?,~~~~|t|&lt;1.$$  any idea? I am trying to install ia32-libs on Ubuntu 12.04 LTS (x86).  I enabled all the repository sources but I'm receiving the error message below:  ~$ sudo apt-get install ia32-libs   Reading package lists... Done   Building dependency tree          Reading state information... Done   Package ia32-libs is not available, but is referred to by another package.   This may mean that the package is missing, has been obsoleted, or   is only available from another source    E: Package 'ia32-libs' has no installation candidate   output of sudo apt-cache policy ia32-libs    ia32-libs:     Installed: (none)     Candidate: (none)     Version table:     output of cat /etc/apt/sources.list  (removed similar hyperlinks)    deb http://archive.ubuntu.com/ubuntu/ precise restricted main multiverse universe   deb-src archive.ubuntu.com/ubuntu/ precise restricted main multiverse universe #Added by software-properties   deb security.ubuntu.com/ubuntu/ precise-security main restricted universe multiverse   deb-src security.ubuntu.com/ubuntu/ precise-security main restricted universe multiverse #Added by software-properties   deb archive.ubuntu.com/ubuntu/ precise-updates main restricted universe multiverse   deb-src archive.ubuntu.com/ubuntu/ precise-updates main restricted universe multiverse #Added by software-properties I find myself constantly looking up the syntax of  find . -name "FILENAME"  -exec rm {} \;   mainly because I don't see how exactly the -exec part works. What is the meaning of the braces, the backslash and the semicolon? Are there other use cases for that syntax? Effective way to find any file's Encoding Draft mode for pgfplots I'm working on a fairly big document that has a lot of graphs of functions- so far it has around 200, by the end it will have around 1500. I use pgfplots for each graphic, and the functions are pretty basic: exponential, rational, polynomial, and trigonometric.  Once I am happy with each graph, I don't necessarily need to see it every time I compile, and would like to minimize compilation time for future drafts. Naturally, the document is already chopped up into chapter files, but each chapter contains around 100 figures.  Having studied the pgfplots documentation, I haven't found a draft mode, but I'm interested in knowing if one could be created.    I'm envisaging something similar to the draft mode of the graphicx package, in which graphics are not included, and blank boxes are put in their place.  Some things I've considered:   the  documentclass; but with about 1500 figures, wouldn't this mean over-modulization?  tikzexternalize, but this still requires an initial 'monster' compilation to get the images externalized before the draft option could be invoked   Here's a MWE to play with  \documentclass{article} \usepackage{pgfplots}  \begin{document}  \begin{tikzpicture}     \begin{axis}         \addplot {x^2};     \end{axis} \end{tikzpicture}  \end{document} I am monitoring the number of pages indexed by Googlebot for one of my websites .  I noticed that since last week, the number of pages has dropped by about 60 or so. Now to be fair, I have not added that much over the last week. (Actually I don't think I added any new content.) But at the very least, I would have thought that the number of indexed pages would be static (if I was not adding anything new.)  I don't understand why the numbers have dropped.  Does anyone have an official explanation for this?. I have checked the Google Webmaster Tools, and have not come across anything that addresses this issue directly. Subversion Obliterate feature I was just thinking of writing a shell script to implement the obliterate functionality in an easy to do way (externally, using the suggested way, but automated).  Here's what I had in mind:  On the client   svn list -R &gt; file-list. filter file-list in several ways like grep to create a file "files-to-delete", something like a set of grep XXX  file-list&gt;&gt;files-to-delete. transfer files-to-delete to the server using scp.   On the server   Dump the repository svnadmin dump /path/to/repos &gt; repos-dumpfile, this can be kept as a backup too. Filter the dump file, for each word in "files-to-delete", do: cat repos-dumpfile | svndumpfilter exclude $file &gt; new-dumpfile Create a new repository and load the new file to it svnadmin create new-name; svnadmin load new-name &lt; new-dumpfile   Would this work? How can it fail? Any other ideas? We're trying to debug some applications performing broadcast.  What is the difference between the broadcast address 255.255.255.255 and as e.g. reported by ifconfig,  Bcast:192.168.1.255 Is there a proper way to handle large combat engagements with party members involved? For instance, a party of four player characters raise a force of 30 militia men to go take a bandit camp. When they get there the bandit camp has 40 men, and none of them notable.  For the sake of simplicity, both the 30 men and the 40 bandits are armed with long swords, and wearing leather armor.   Is there a proper way to handle combat between the 30 man militia and the 40 bandits while also incorporating our four player characters? If no "proper" way exists, is there a generally accepted "best" way? Don't invite people when moving their messages