What is the effect of the birthday bound on the security of AES-GCM, when a very large number of messages are encrypted?  I am looking for attacks that are useful in obtaining information about the plaintext and/or creating forgeries, not just distinguishers.
In a  a doubt was voiced about my answer, which claims GCM to requires $2^{128}$ for a successful forgery. The doubt was that the square root needs to be taken meaning the security would be $2^{64}$.  So of course I immediately checked the relevant security theorem ():  $$\mathbf{Adv}^{\text{auth}}_{\operatorname{GCM}[\operatorname{Perm}(n),\tau]}(\mathcal A)\leq \frac{0.5(\sigma+q+q'+1)^2}{2^n}+\frac{q'(\ell_A+1)}{2^\tau}$$  with $\sigma$ being the total plaintext size in blocks, $q$ being the total number of encryption queries, $q'$ being the total number of decryption queries, $n$ being the size in bits of the underlying permutation, $\ell_A$ being maximum authenticated input length in blocks and $\tau$ being the tag size in bits.  Now clearly from this we can see that performing $2^{n/2}=2^{64}$ queries yields a strong enough advantage for a break.  Now my question is: When talking about "n-bits of security" or "needs $2^n$ operations to break", do we usually talk about "online" security, ie queries to an oracle can be done and cost $1$ or "offline" security where no oracles are available or do we need to make this decision depending on the situation?  TL;DR: Does GCM offer 64 or 128 bit security?
inequality on the unit disk $n$ points $z_1,z_2,\cdots,z_n$ in the unit open disk are given. Prove or disprove that there exists $z$ in the unit circle such that $\prod_{i=1}^n |z-z_i|^i \ge 1$.  I think it can be solved by elementary analytic methods but no idea. How can I change Win 10 protected registry entries? Limit of $2^{1/n}$ as $n\to\infty$ is 1 How do I prove that:   $\lim \limits_{n\to \infty}2^{1/n}=1$  Thank you very much. How many nonnegative integer solutions are there to the pair of equations  \begin{align}x_1+x_2+\dots +x_6&amp;=20  \\ x_1+x_2+x_3&amp;=7\end{align}  How do you find  non-negative integer solutions? Page range truncation with imakeidx This is a followup question to egreg's answer to  The answer runs into a problem if an index topic occurs in more than one page range. The following MWE, based on egreg's answer:  \begin{filecontents*}{subjind.ist} delim_0 ", \\checkrange{" delim_t "}" \end{filecontents*}  \documentclass{book} \usepackage{xparse} \usepackage{imakeidx}  \makeindex[title=Subject Index,options=-s subjind]  \ExplSyntaxOn \NewDocumentCommand{\checkrange}{&gt;{\SplitArgument{1}{--}}m}  {   \formatrange#1  } \NewDocumentCommand{\formatrange}{mm}  {   \IfNoValueTF { #2 }    { % no --, it's not a range     #1    }    { % it's a range     \egreg_range:nn { #1 } { #2 }    }  }  \bool_new:N \l__egreg_range_compare_bool  \cs_new_protected:Nn \egreg_range:nn  {   % print the start number   #1--   \int_compare:nNnTF { \str_count:n { #1 } } = { \str_count:n { #2 } }    {% same number of digits, remove equal ones at the start     \__egreg_range_compare:nn { #1 } { #2 }    }    {% different number of digits, print both     #2    }  } \cs_new_protected:Nn \__egreg_range_compare:nn  {   \bool_set_true:N \l__egreg_range_compare_bool   % do a loop on the digits in the first number   \int_step_inline:nn { \str_count:n { #1 } }    {     \bool_if:NTF \l__egreg_range_compare_bool      {       % if the digits coincide, print nothing       \str_if_eq:eeF { \str_item:nn { #1 } { ##1 } } { \str_item:nn { #2 } { ##1 } }        {         \str_item:nn { #2 } { ##1 }         % at the first differing digit, don't compare any longer         \bool_set_false:N \l__egreg_range_compare_bool        }      }      { \str_item:nn { #2 } { ##1 } }    }  } \ExplSyntaxOff  \begin{document} \setcounter{page}{95}\index{Topic0|(}Text\newpage \setcounter{page}{101}\index{Topic0|)}Text\newpage \setcounter{page}{124}\index{Topic1|(}\index{Topic2|(}Text\newpage \setcounter{page}{126}\index{Topic1|)}Text\newpage \setcounter{page}{134}\index{Topic2|)}Text\newpage \setcounter{page}{141}\index{Topic0}Text\newpage \index{Topic0}Text\newpage \index{Topic0}Text \printindex \end{document}   yields the following error message:  ! LaTeX3 Error: Too many '--' tokens when trying to split argument.  For immediate help type H &lt;return&gt;.  ...                                                l.3   \item Topic0, \checkrange{95--101, 141--143}   Entering non-stop yields the following output:    which omits "-3" from the second index entry of Topic0. The question is how to get around this.  Edit following up on egreg's ingenious solution.  I don't know if I should ask a new follow up question, so I'm posing it first here in this edit.  egreg's solution doesn't seem to handle index sub-topics.  If one adds the following 4 lines  \setcounter{page}{135}\index{Topic0!sub0|(}Text\newpage \setcounter{page}{136}\index{Topic0!sub0!subsub0|(}Text\newpage \setcounter{page}{139}\index{Topic0!sub0!subsub0|)}Text\newpage \setcounter{page}{140}\index{Topic0!sub0|)}Text\newpage   after  \setcounter{page}{134}\index{Topic2|)}Text\newpage   in the solution, one gets the following errors:  Output written in mwe.ind. Transcript written in mwe.ilg.  (mwe.ind ! Extra }, or forgotten \endgroup. l.4     \subitem sub0, 135--140}  ? r OK, entering \nonstopmode... ! Extra }, or forgotten \endgroup. l.5       \subsubitem subsub0, 136--139}  ) [145] (mwe.aux) )   and this output:    Alas I don't understand egreg's code well enough to make the required changes. why does loop-cut-and-slide ignore some faces but not others? Voting to Close after Accidentally Voting to Leave Open When reviewing on SO, I accidentally pushed the "Leave Open" button on a post that clearly should've been closed. I pushed the back button and saw that in the review panel my vote to keep open had been locked and it said:     PhpMyCoder voted to leave open 1 minute ago   So I clicked on the question heading to go to its page and I clicked the close link on the question page and voted to close for the relevant reason. However, when I returned to the review page (and did a hard refresh) it still said:     PhpMyCoder voted to leave open 2 minutes ago   Is it possible to vote to leave open and vote to close at the same time? I assume not. In which case, I'd say this is a bug because it should've updated to say:     OtherUser and PhpMyCoder voted to close 2 minutes ago Does a Nikon DX lens on a FX camera get subjected to the 1.5X crop factor? Einstein equation  where $E$ is energy, $m$ mass, and $c$ the speed of light in vacuum. So does it mean objects without any mass does not posses energy for eg lights photons does not have mass but how can they posses energy How do I install GNOME Shell extensions?  Is it possible to update, remove and manage the extensions? If $f$ is a continuous function with $\displaystyle \int_{0}^{x}f(t)dt\rightarrow \infty$ as $|x|\rightarrow \infty\;,$ Then show that every line $y=mx$ Intersect the curve $\displaystyle y^2+\int_{0}^{x}f(t)dt=2$  $\bf{My\; Try::}$ Put $y=mx$ in $$\displaystyle y^2+\int_{0}^{x}f(t)dt=2\;,$$ We get $$\displaystyle m^2x^2+y^2+\int_{0}^{x}f(t)dt=2.$$ We can write it as $$\displaystyle 2\int_{0}^{x}2m^2tdt+\int_{0}^{x}f(t)dt=2$$ So $$\displaystyle \int_{0}^{x}\left[2m^2t+f(t)\right]dt-2 =0$$ Now Let $$g(x) = \displaystyle \int_{0}^{x}\left[2m^2t+f(t)\right]dt-2.\;,$$ When $x\rightarrow \infty\;,$ Then $g(x)\rightarrow \infty.$ (Where $g(x)$ is an Continuous function.) Now I did not Understand How can i solve that, Help me Thanks References in externalized pgfplots I was writing on an document with a lot of pgfplots in it. Because of the memory limitation in LaTeX and in order to save some compiling time I tried out the externalize command. With most of the plots it perfectly worked out. But when I use the \ref command, namely \ref{&lt;legendname&gt;} the legend gets lost and instead there is a ??.  I think this question in a way is similar to  and . But no answer to these questions could solve my problem. Even the use of xr and zref packages showed no effect.  Am I just missing something?    MWE of the maindocument:  \documentclass{article} \usepackage{zref-user} \usepackage{xr-hyper} \usepackage{pgfplots} \usetikzlibrary{pgfplots.groupplots} \usepgfplotslibrary{external}  \tikzexternalize \externaldocument{part1}  \begin{document} \input{part1} \end{document}     With part1.tex looking like this:  \begin{tikzpicture}     \begin{groupplot}[       group style= {  columns=3,xlabels at=edge bottom,                 y descriptions at=edge left,                 horizontal sep=2cm,group name=plots},     width=0.4\textwidth,     xtick=data,     legend columns=3]     \nextgroupplot[xlabel=site 1,legend to name=grouplegend]     \addplot coordinates {(1,20)(2,23)(3,24.5)};     \addlegendentry {AaAaA,BbBbB,CcCcC}%     \nextgroupplot[xlabel=site 2]     \addplot coordinates {(1,14)(2,16)(3,17)};     \nextgroupplot[xlabel=site 3]     \addplot coordinates {(1,10)(2,11.5)(3,12.25)};     \end{groupplot}   % Legend     \node at (plots c2r1.north) [anchor=south] {\ref{grouplegend}}; \end{tikzpicture}     I'm writing with MikTeX 2.9 and the latest updates of the packages used. For compiling I used the command pdflatex -enable-write18 &lt;file name&gt;.  I hope the MWE is short enough ;) and some knows how to solve this problem. Size of files in Windows OS. (It's KB or kB?) One "kilobyte" (KB) is 1024 bytes in JEDEC-standard, whereas the definition has shifted, in most contexts, to mean 1000 bytes (kB) in accordance with SI. To resolve this difference, binary prefixes (kiB) are used.  So we have 3 choices for using prefixes - JEDEC, IEC (both in Binary), and Metric (in Decimal).  My questions are-   What prefix standard does Windows use in showing the file size? (surely it's not IEC standard) Why does Windows OS show sizes of files in KB (using a capital alphabet "K") when it's a small alphabet "k" for a Kilo in SI units?   A capital "K" represents Kelvin in SI system of units.  Am I missing something here in understanding? How to make a Rigid Body fall off a conveyor belt? I've created conveyor belt and and two objects with rigid body physics. But the first object won't fall and second object falls backwards.  How can I fix it? Python, Unicode, and the Windows console When I try to print a Unicode string in a Windows console, I get an error . UnicodeEncodeError: 'charmap' codec can't encode character .... I assume this is because the Windows console does not accept Unicode-only characters. What's the best way around this? Is there any way I can make Python automatically print a ? instead of failing in this situation? Edit:  I'm using Python 2.5.  Note: @LasseV.Karlsen answer with the checkmark is sort of outdated (from 2008). Please use the solutions/answers/suggestions below with care!!  is more relevant as of today (6 Jan 2016). Why is Modus Ponens valid? I am having trouble understanding what defines Entailment operator. On Mathoverflow I posted this question on what I perceive to be .  Consider: :     P therefore Q  P            Therefore, Q   My question is could we have arbitrarily defined it such as:     P therefore Q  P            Therefore, P   However, if an advanced civilization receives information in the form of following nested structures/containers or packets, how will they interpret Modus Ponens if in an alien land Modus Ponens is not defined?    Therefore, it raises the validity of MP. How do we define the implication or entailment operator? Because if we use an argument that relies on implication or entailment operator ipso facto, then it is trivial that logic is circular.  Is there a flaw in my rationale? I have two Xbox 360's, one WiFi and the other wired through the same Linksys E3000 router with DD-WRT (v24-sp2 (04/13/11) std-usb-nas firmware).   Mine (the wired one, though hers has been wired in the past and it didn't appear to help) always seems to be open (green), and hers is either moderate or closed (oj/red)  Even with the original router firmware I've never been able to get both consoles into "open nat" mode, and it makes playing certain games together almost impossible for me and my wife  because we can't seem to get into the same game.   I have uPnP enabled and the DMZ disabled, and some minor port forwarding to allow for a Minecraft server (Ports 25565 and 25566 for a testing copy of the server), but other than that the settings are mainly at their default, on the xbox's and the router.   Are there any settings I can configure to allow this to work properly? Why Wi-Fi is getting turned-on automatically in iOS 11 Proving that $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots + \frac{1}{n(n+1)} = \frac{n}{n+1}$ How would we go about proving that $$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots +\frac{1}{n(n+1)} = \frac{n}{n+1}$$ Is there any way to increase the drop rate of items?