If RSA Signature is not Decryption, why is it not advisable to use same keys for enc/dec and sig/ver? What is the proof where you may be exposing your private key when you use the same one for decryption and signing? This is for RSA OAEP.
The RSA signature operation is basically the same as encrypting with the private key.  In particular, both operations use the same kind of keys.  Is it safe to use the same RSA keypair both for encryption / decryption and for signing / verification?
Proof/Derivation of Closed form of Binomial Expression $\sum\limits_{k=0}^{2n}(-1)^k\binom{2n}{k}^2$ The binomial expression given as follows: $$\sum_{k=0}^{2n}\left(-1\right)^{k}\binom{2n}{k}^{2}$$ results nicely into the following closed form: $$(-1)^{n}\binom{2n}{n}$$ I wish to know how exactly is it being done? I haven't been able to make much progress in solving it. My approach: \begin{align} \sum_{k=0}^{2n}(-1)^{k}\binom{2n}{k}^{2} = \binom{2n}{0}^{2} - \binom{2n}{1}^{2} + \binom{2n}{2}^{2} - ... -\binom{2n}{2n-1}^{2} + \binom{2n}{2n}^{2} \\ = \binom{2n}{0}.\binom{2n}{0} - \binom{2n}{1}.\binom{2n}{1} + \binom{2n}{2}.\binom{2n}{2} - ... -\binom{2n}{2n-1}.\binom{2n}{2n-1} + \binom{2n}{2n}.\binom{2n}{2n} \\ \text{By Symmetry of binomial coefficients} \\ = \binom{2n}{2n}.\binom{2n}{0} - \binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n-2}.\binom{2n}{2} - ... -\binom{2n}{1}.\binom{2n}{2n-1} + \binom{2n}{0}.\binom{2n}{2n} \\ = \binom{2n}{2n}.\binom{2n}{0} + \binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n-2}.\binom{2n}{2} + ... +\binom{2n}{1}.\binom{2n}{2n-1} + \binom{2n}{0}.\binom{2n}{2n} - 2.\left(\binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n -3}.\binom{2n}{3} + ... + \binom{2n}{1}.\binom{2n}{2n-1}\right) \\ \text{By Vandermond's identity, first component, i.e.not enclosed within -2.(...) evaluates to C(4n, 2n)} \\ \binom{4n}{2n} - 2.(\binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n -3}.\binom{2n}{3} + ... + \binom{2n}{1}.\binom{2n}{2n-1}) \end{align} I'm lost beyond this point. It will be extremely helpful if someone can direct me in the right direction or provide the answer to this perplexing and challenging problem. Thank you. I tried to prove that $(p \to q) \lor (q \to p)$ is a tautology.  I used $p$ and $¬q$ as conditions. (Premises 1 and 5)  I managed to get to a solution, but I'm not sure if it's right.  Can you please check it?   Thank you!!!:) OR-ing power supplies (diode or mosfet) I have got two power supply options on my board:   USB 5.0V DC Power Jack of 5.0V   My intention is to design a or-ing power supply option, as either of them will be used at a time. Possible solutions are shown below:    Using oring diodes   or using two p-mos    Possible issues observed with the above options:   Using diode is cheap and good option but an ordinary diode gives a drop of 0.7V and a schottky diode will give a drop of 0.16 to 0.20 volts, which is also not acceptable because my processor  recommended operating power supply is 5.0V and minimum supply voltage is 4.75. With a 0.2V with schottky diode the 5.0V comes down to 4.80V and the space between recommended and supply fed is very less. Then I thought to use p-mosfets which serves the purpose of oring along with reverse polarity protection, but assume a case when both the supplies are fed together by mistake and there is minor difference between USB supply and DC jack say 4.9V for USB and 5.10V for DC Jack. Then, the DC jack supply may pump the current in USB Jack, which may damage USB.   Is there any other better and cheap option I should use for or-ing supply, or is there something wrong with my analysis? DNS settings on managed switches in LAN We have a Windows 2008 server that is connected to a managed switch with about 15 Cat5e connections and three fiber branches that go to different parts of campus. At each of these parts there is another managed switch that branches out to the LAN. Our cable Internet comes into one of these locations, goes through the supplied modem and into a Sonicwall.  Question: Under DNS settings on all the switches, should I list the LAN Server IP and/or just the Gateway IP (Sonicwall)? How do I install Java? How do I install Java, including in my browser, so websites with Java work? I often find  syntax very helpful, e.g. process substitution like in diff &lt;(sort file1) &lt;(sort file2).  Is it possible to use such Bash commands in a Makefile? I'm thinking of something like this:  file-differences:     diff &lt;(sort file1) &lt;(sort file2) &gt; $@   In my GNU Make 3.80 this will give an error since it uses the shell instead of bash to execute the commands. I need to package my Python application, its dependencies and Python into a single MSI installer. The end result should desirably be:   Python is installed in the standard location the package and its dependencies are installed in a separate directory (possibly site-packages) the installation directory should contain the Python uncompressed and a standalone executable is not required How can a file be copied? What is your favorite method to handle errors in Bash?  The best example of handling errors I have found on the web was written by William Shotts, Jr at .   He suggests using the following function for error handling in Bash:  #!/bin/bash  # A slicker error handling routine  # I put a variable in my scripts named PROGNAME which # holds the name of the program being run.  You can get this # value from the first item on the command line ($0).  # Reference: This was copied from &lt;http://www.linuxcommand.org/wss0150.php&gt;  PROGNAME=$(basename $0)  function error_exit {  #   ---------------------------------------------------------------- #   Function for exit due to fatal program error #       Accepts 1 argument: #           string containing descriptive error message #   ----------------------------------------------------------------       echo "${PROGNAME}: ${1:-"Unknown Error"}" 1&gt;&amp;2     exit 1 }  # Example call of the error_exit function.  Note the inclusion # of the LINENO environment variable.  It contains the current # line number.  echo "Example of error with line number and message" error_exit "$LINENO: An error has occurred."   Do you have a better error handling routine that you use in Bash scripts? Player A taps  targeting Player B; its "deals 1 damage to target player" ability is placed on the stack.  Player B gets priority and responds with , targeting Goblin Fireslinger.  Both players pass priority. Snap resolves, and Goblin Fireslinger is returned to its owner's hand.  Does Player B still take 1 damage from Goblin Fireslinger's activated ability?  I know that if the creature had been killed/sent to the graveyard, the ability would still resolve, but I was told recently that because a player's hand is a private zone it couldn't. If anyone could confirm or deny this for me, it would be most appreciated. es.SO questions aren't excluded from the Hot Questions list As figured out in  localized SE sites must not be displayed in the HNQ's. But  actually is.        Has to be fixed I guess? Been trying to find graphic novel for decades - at least I think it's a graphic novel When I was a kid I had a book that I loved, as I think about it I'm pretty sure it would be characterized as a graphic novel although I was unfamiliar with that term at the time. It was a large hardcover comic book but a full-length story, probably a few hundred pages long.  This would have been about 25-30 years ago.  I recall a little about the plot: It starts with a spaceship crashing somewhere, I remember one of the first panels had the phrase that the ship "yawed wildly". I always remembered that because I had to look up the word "yawed".   The rest of the story is told in flashback form, with the rise of an empire, and I'm pretty sure there's a guy who's a slave or otherwise in the lower strata of society who rises to be the ruler of the world or something like that.    I thought that the title had the word "titan" or "empire" in it (Empire of the Titans, or Rise of the Titans, or something similar), but Google searches have been unfruitful. What recourse do I have if I believe a moderator has abused their privileges? Did the date of Frodo's rescue out of the fire of Mordor have any particular meaning in The Lord of the Rings? Did the date of Frodo's rescue out of the fire of Mordor have any particular meaning in The Lord of the Rings?  We read in The Return of the King (Book 6 Chapter 4) that Gandalf states to Frodo and Sam:     The fourteenth of the New Year, or if you like, the eighth day of April in the Shire reckoning. But in Gondor the New Year will always now begin on the twenty-fifth of March when Sauron fell, and when you were brought out of the fire to the King.   Did Tolkien have any particular reason for choosing this date for Sauron's fall and Frodo's rescue out of the fire of Mordor, either allegorically or otherwise?  We are aware that Tolkien was not a fan of , but preferred the term applicability instead. How to keep a constant baselineskip when using minipages (or \parboxes)? When a minipage is used to box a text paragraph, the first line of the minipaged text and the last line of the previous text paragraph are closer than two consecutive lines in a standard text paragraph; analogously, the last line of text inside the minipage and the first line of text after the minipage are also closer.  The following MWE illustrates this:  \documentclass{article} \usepackage{lipsum}  \newcommand\Text{Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. Curabitur dictum gravida mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna.\textbullet}  \begin{document}  \Text  \noindent\begin{minipage}{\linewidth} \Text \end{minipage}  \Text  \end{document}   How can I guarantee a constant baseline-skip even if minipages (or \parboxes) are used? According to  , gold badge holders can now unilaterally close questions as duplicates. I hold seven gold badges for tags on SO, which means I can, on my own, close questions as duplicates for those tagged questions.  Simply put: DO NOT WANT.  However, I do not see any ability in SO (the only SE site I spend much time in) to opt out of any privileges, let alone this one.  Is there a spot that I'm just not seeing that allows me to turn off this new privilege? Self-Studying Measure Theory and Integration As the title says, I'm interested in reading about measure theory and integration theory on my own time, and was hoping for book recommendations, prerequisites, and things like that.  As for my background, I've taken introductory analysis, where we covered the "typical" subjects up until some theorems about differentiable functions.  I'm taking intermediate real analysis now; we talked about metric spaces in general for a while, but lately we've focused solely on $\mathbb{R}^n$. Now we're working towards a proof of the implicit function theorem in $\mathbb{R}^n$.  I haven't learned any of the theory behind Riemann integration, unfortunately. Is it important to be familiar with this first? How do I change permissions on a FAT32 formatted drive? I was trying to dual boot elementary os with windows on my laptop (Asus Zenbook UX305). To do that I shrank the windows partition using the windows disk manager, and then used the Elementary installer's "install alongside windows" option to set up the linux partitions automatically. When I rebooted after installing I was put into the grub console. I was able to boot linux manually with the commands  grub&gt; set root=(hd0,gpt4) grub&gt; linux /boot/vmlinuz-3.19.0-26-generic root=/dev/sda4 grub&gt; initrd /boot/initrd.img-3.19.0-26-generic grub&gt; boot     and I found that running the command  grub&gt; configfile (hd0,gpt4)/boot/grub/grub.cfg   would load the normal grub menu as is supposed to happen - so there don't seem to be any problems with the config file itself. I also found that the root and prefix are on (hd0,gpt1), so running  grub&gt; set root=(hd0,gpt4) grub&gt; set prefix=(hd0,gpt4)/boot/grub grub&gt; insmod normal grub&gt; normal   also loaded the normal grub menu. So possibly the problem is that grub is looking on the wrong partition?  In my BIOS, I have a choice between using grub and the windows bootloader to boot. Selecting the windows bootloader boots me directly into windows without any problems.  However, I can't find any way to get any fixes to stick after a reboot. I've tried rerunning grub-install /dev/sda, recreating the config file with grub-mkconfig, and using the Ubuntu boot repair gui tool - none of these have worked. Is there any way to fix this short of reinstalling either or both OSs? I'd rather not have to reinstall windows if possible. Sorting by multiple fields at same time in QGIS? Is it possible to sort multiple fields at the same time in QGIS?    For example, I'd like to sort a table by state name, and then within each state, sort by county name.    This would be the same as the Advanced Sort option available in ArcMap.