The result of the cipher should be indistinguishable from random, but can it happen that cipher produces the ciphertext that is equal to the plaintext as it also lies in the corresponding space? I guess it's just the probability is extremely low, probably equal to guessing the clear text on the first attempt, but just out of curiosity: is it theoretically possible and/or happened in real life?  The same goes to other types of cipher and e.g. hash function on small inputs.
Is there any pair of 128-bit strings M and K such that AES$_K$(M) = M?  If yes, how do I go about (efficiently) finding such a pair?
Is it appropriate to put a comma before "which"? On Server Fault, we were discussing  in chat. There are non-native speakers in the chat and somehow we got on the topic of grammar and proper use of commas.   The sentence of interest:     I have just got a new Ubuntu LAMP VPS server, which I am trying to configure.   Some people thought that the comma was superfluous. Others did not.   When is it appropriate for a comma to preceed the word which, and is the example sentence above correct?    Also, before you vote to close this question as a dupe of , please consider the differences between the two questions and ignore the similarity of the titles. The OP's example sentence was restructured in that question instead of addressing the proper use of commas. I have found a strange behavior in TikZ. When using the \draw ... to ... syntax and trying to place nodes along the line with the node[...]{...} syntax, the nodes do not turn up at the expected location.  \documentclass{standalone} \usepackage{tikz} \begin{document} \begin{tikzpicture}   \draw(0,0) to[out=60,in=120] (1,0) node[midway]{x}; \end{tikzpicture} \end{document}     How is this done properly? Does shoving a creature break its grapple on another? A situation came up in my session where a fighter used his Shield Master Shove feature to shove a Giant Crocodile that was currently restraining another PC using it's bite+grapple ability.  I did a quick check at the time and did not find anything related to shove, although in the  condition description it states:     The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.   I ruled at the time that a shove would count in a similar way to being "hurled away" however I wasn't sure and said I would find more concrete information for the future.  I have searched high and low for somebody else asking this specific question but have yet to find it. I saw some reference to a Jeremy Crawford tweet about involuntary movement and grappling however I am unable to find it.  If a third party "shoves" a creature that is grappling another, does it only shove the grappler away, or does it move both the grappler and grappled?  I drew a diagram:   (; this question relates to if a third party shoves the grappler instead, so it's not a duplicate.) Add a "back to top" link The left-nav is great - I like it, users who don't can disable it. Perfect world. Now it's stickied it's always there too, which is really useful.  When I'm navigating in areas not directly linked to in the left nav and have scrolled down, I often want to re-use the other navigation. This means scrolling up. Which is fine, of course it's how websites work.  However, if the left-nav had a "Top" link which took the user to the very top of the page it would be great for getting back to specific contextual menus and things that are not linked to in the left-nav.  For example, if I'm viewing questions on a specific tag and I've scrolled down, I sometimes want to get back up to the top to remove a tag, or change from "Latest" to "Featured" etc. Although clicking the "Questions" link in the left-nav takes me back to the top, it removes the tag(s) I was searching with.  There are other use cases where this would be useful, like long questions and answers (Etc). So I think a "Top" link in the left nav would be very useful.    I think this is different to the  as it doesn't mention the left-nav, and my question is suggesting this idea because we have a good place to put it and it'll be a useful addition to the left-nav. Translation of these Cirth runes What is the translation of what's written below?  I think it may be the poem of the One Ring, but I'm not sure. Algorithm to find the irreducible polynomial What algorithm can be used find an irreducible polynomial of degree $n$ over the field $GF(p)$ for prime $p$. The reason I ask is I want to make a program for finite field arithmetic, but creating a field $GF(p^k)$ requires finding a irreducible polynomial. (I'm mostly interested in $GF(2^k)$.) Difference of Gamma random variables Any Set with Associativity, Left Identity, Left Inverse is a Group Related Link:    Reference: Fraleigh p. 49 Question 4.38 in A First Course in Abstract Algebra     I will present my proof (distinct from those in the link) for critique and then ask my question.  $G$ is a set and $ \times $ is an associative binary operation.  Suppose that there exists a $e \in G$ such that, for all $a \in G$, $ea = a$ and $a^{-1}a = e$ for some $a^{-1} \in G$.  Show that for the same $e$, $ae = a$ and $aa^{-1} = e$.  $a^{-1}(aa^{-1})a = (a^{-1}a)(a^{-1}a) = ee = e = a^{-1}a = a^{-1}(a^{-1}a)a$  Since $a^{-1} \in G$, it has a left inverse; apply it to both ends, and we have $(aa^{-1})a = (a^{-1}a)a$.  As a result, $ae$ = $a(a^{-1}a) = (aa^{-1})a = (a^{-1}a)a = ea$.  For the right inverse, start with $aa^{-1} = a(a^{-1}a)a^{-1} = (aa^{-1})(aa^{-1})$.  Since $\times$ is a binary operation, $aa^{-1} \in G$ and has a left inverse; apply it to both ends, and we have $e = aa^{-1}$.  In second comment following the question in the link, Mr. Derek Holt pointed out that the requester did not word his/her question correctly.  Specifically, the identity in the second axiom is not well-defined.     Let $(G, *)$ be a semi-group. Suppose    1. $ \exists e \in G$ such that $\forall a \in G,\  ae = a$;    2. $\forall a \in G, \exists a^{-1} \in G$ such that $aa^{-1} = e$.    How can we prove that $(G,*)$ is a group?    This formulation makes the same technical error as many textbooks. The $e$ in your second axiom is not well-defined. "But obviously it's intended to be the same $e$ as in the first axiom" you reply. But the first axiom does not necessarily specify a unique element $e$. So should we interpret the second axiom as meaning "for some $e$ as in 1" or "for all $e$ as in 1"? –  Derek Holt Sep 17 '11 at 15:31   Was he saying that if, in axiom 1, we have $ae_1 = a, ae_2 = a$, but $e_1 \neq e_2$,  when we get to axiom 2, do we have $aa^{-1} = e_1, aa^{-1} = e_2$, or two different inverses so that $aa_1^{-1} =  e_1, aa^{-1}_2 = e_2$?  I think my wording eliminated the ambiguity.  It does not imply that $e$ is unique, but if $e$ is a left identity and produces left inverses, then it is also a right identity and produces right inverses. I tried really hard on this one; please kindly point out my mistakes. I have a set of predictors in a linear regression, as well as three control variables. The issue here is that one of my variables of interest is only statistically significant if the control variables are included in the final model. However, the control variables themselves are not statistically significant.  Here is how the multicollinearity of all my variables look like (including control variables):   &gt; vif(lm(return ~ EQ + EFF + SIZE + MOM + MSCR + UMP, data = as.data.frame(port.df)))        EQ      EFF     SIZE      MOM     MSCR      UMP   3.687171 3.481672 2.781901 1.064312 1.438596 1.003408   &gt; vif(lm(return ~ EQ + MOM + MSCR, data = as.data.frame(port.df)))        EQ      MOM     MSCR   1.359992 1.048142 1.412658    My variables of interest are EQ, MOM and MSCR, and the control variables are EFF, SIZE and UMP. EQ is only significant if the three control var are included, and becomes insignificant when they are not:   Here are the coefficients (1rst row) and t-stats (2nd row) when control variables are included (notice that EQ is statistically significant)         intercept           EQ          EFF        SIZE         MOM       MSCR          UMP [1,] 0.005206246 -0.006310531 0.0001229055 0.004125551 0.007738259 0.00473377 5.838596e-06 [2,] 1.866628909 -1.746583234 0.0388823612 1.178460997 2.145062820 2.08131100 1.994863e-01  Now, here is the result of the regression when the control variables are excluded (notice that EQ is NOT statistically significant anymore)         intercept           EQ         MOM       MSCR [1,] 0.007313402 -0.002111833 0.007128606 0.00668364 [2,] 2.652662996 -0.595391117 2.036985378 2.80177366    The problem is that when I include my control variables, all my variables of interest are significant, but my control variables are not.  Which variables should I include in my final model? How should I structure my final model then, given the fact that the model will be used for forecasting?  Thank you, I have a dataset that has both continuous and categorical data. I am analyzing by using PCA and am wondering if it is fine to include the categorical variables as a part of the analysis. My understanding is that PCA can only be applied to continuous variables. Is that correct? If it cannot be used for categorical data, what alternatives exist for their analysis? How to find the index of an element in an array in Java? Is three hours enough for an international-domestic transfer at Delhi? My International flight will be reaching New Delhi IGI at 2:45 AM and I am planning to book a domestic flight at 5:45 AM.  Is three hours sufficient for me to move from International to Domestic?  I believe there must be a shuttle service inside the airport to transfer me from International to Domestic but still I don't know if Customs time at IGI could cause any delay or not. Link2SD not working completely Pluralization rule for "five-year-old children", "20 pound note", "10 mile run" Why are year, pound and mile in the singular form in the phrases below?   five-year-old children  20 pound note 10 mile run   Is that because they're acting as adjectives, which are always invariable in English?  Is it incorrect to say...   five-years-old children? 20 pounds note? 10 miles run? Why is the Descriptor not getting called when defined as instance attribute? We're all familair with this beautiful proof whether or not an irrational number to an irrational power can be rational. It goes something like this:  Take $(\sqrt{2})^{\sqrt{2}}$  If it's rational, then you proved it, if it's irrational, take $((\sqrt{2})^{\sqrt{2}} ){^\sqrt{2}} = 2$ and you've proved it.  I'm wondering if you can raise $\pi$ or $e$ to a certain non-trivial real power to make it rational? And if not, where is the proof that it can't be done?   p.s. - I almost left out the real part, but then I realized that $e^{i\pi} = -1$. Can we prevent titles with an unnecessary tag in them? So we have lots of questions that are titled like this:  Specifically, questions that have one of their tags in front of their title. Thankfully, these silly users usually use a hyphen or a colon afterwards, which makes it really easy to . Can we clear up this clutter?  Perhaps with a filter that strips off the excess when posting? EDIT: As an added bonus, see  for another style of question title that bugs the heck out of me. I'm working on the Fourier transform, but I don't know how to evaluate the integral:  $$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$ Blending colors in photoshop and illustrator Display only relevant reputation points