I've heard in some lecture that the weak  implies the existence of $2^{32}$ fixed points in the algorithm, i.e. that $DES_k(x)=x$ for $2^{32}$  such $x$'s.  I couldn't find anything about that issue, can you give some reference or explain it here?
Can someone explain me this  please I am a bit confused?  It's from a texbook I am reading (block cipher companion)  it says that:      each weak key has  $2^{32}$ fixed points $m$ where $\operatorname{DES}_w(m) = m$ ( $w$ is a DES weak key). In other words encrypting a fixed point gives exactly the same point. To see this consider the $2^{32}$ ciphertexts after 8 rounds of encryption for which the two 32 bit halves are equal. For a weak  DES key, the round keys of the eighth and ninth rounds are equal. Thus the intermediate texts after seven rounds and after nine rounds of encryption will be equal. Since the round keys of the 7th and the 10th rounds are equal, the intermediate texts after 6 rounds and after 10 rounds of encryption will be equal. Continuing like this it follows that the plaintext will be equal to the ciphertext.    Now did not quite get :   Why are there $2^{32}$ fixed points?  And why after the eighth round of encryption the two halves must be equal? What does the author mean by the intermediate texts exactly? And how are they being equal?   P.S.: This question is not asking for what the weak keys are. I am asking for "fixed points" not exactly weak keys. For this reason I think it is not a duplicate of any other question. There is one similar question to what I've asked for which I have checked and saw that no satisfactory answer has been given there either.  Here is the similar question with no proper answers:
Analogy: "as if" vs. "as though" This is something that confuses me from time to time. When making an analogy in literature, is it better to use the phrase "as if" or "as though", or is it completely a style thing?  E.g.      She looked frazzled, as if she had just gotten off of a roller coaster.   or     She looked frazzled, as though she had just gotten off of a roller coaster. Often confusion arises when the  spell is used, an  is worn, or the 's anti-magic ray is employed, whether a given character's ability will still function.   The spell does give you some guidelines on how to adjudicate the effects of the loss of magic but says very little about how to determine what is affected.  In short, how do I tell if an ability is magical? See dashboard as non logged user in community Is it possible, as a non logged user, to see report and dashboard in community when it is open to non logged user? And  for analytic dashboards? How can I change the color of a part of a TextView? What is the proper way to re-throw an exception in C#? I have a question for you that stems from my partner doing things a different way than I do.  Is it better to do this :  try {     ... } catch (Exception ex) {     ...     throw; }   or this:  try {     ... } catch (Exception ex) {     ...     throw ex; }   Do they do the same thing? Is one better than the other? What's the "official" explanation for Harry's (and everyone else's) absence from muggle school? Intuition behind measurable random variables and $\sigma$-algebra I've been trying to understand $\sigma$-algebras and how it encodes information in context of filtration. While certain parts seem clear and logical, I can't say I get the whole picture.  I'll try to explain the counter-intuition I get with the classical example of : the probability space $\Omega = \{ HH, HT, TH, TT \}$ and a r.v. $X(\omega)$ equal to the number of heads.  At times $0$, $1$ and $2$ the available information is represented using $\sigma$-algebras $\mathcal{F}_0=\{\emptyset,\Omega\}$, $\mathcal{F}_1=\{\emptyset, \Omega, \{HH,HT\},\{TH,TT\}\}$ and $\mathcal{F}_2=\{\emptyset, \Omega,\{HH,HT\},\{TH,TT\},\{HH\},\{HT\},\{TH\},\{TT\}\}$.  One can notice that $X(\omega)$ is not measurable with respect to $\mathcal{F}_0$ and $\mathcal{F}_1$, because $X^{-1}((\frac{3}{2}; +\infty))=\{HH\}$. To me it is quite surprising: intuitively $X$ makes perfect sense at all times. In particular it has an expected value at time $0$, which I interpret as that the probability and value of all outcomes $\{\omega\}$ can be computed. How do you think of a non-measurable function?  Here's another way of expressing the same confusion. The most natural choice of $\sigma$-algebra in a finite discrete case is $\mathcal{F}=2^\Omega$, and it is implicitly used in all elementary probability problems. However, this choice of $\mathcal{F}$ does not reflect the fact that some information is known or unknown, conditional probability does. Does it mean that the statement "$\sigma$-algebra is known information" make sense only in conditioning? Why is it convenient then? \documentclass[11pt]{article}  \begin{document}   The above equations on application of Laplace Transform become:  $$sL_1\hat{i}_{L1}(s) = D\cdot \hat{v}_{C1}(s) + D\cdot \hat{v}_{C2}(s) - D\prime \cdot \hat{v}_g(s) + (C_{C1} + V_{C2} - V_g + V_D)\cdot \hat{d}(s)$$ $$sL_2\hat{i}_{L2}(s) = -D\prime \cdot \hat{v}_{C1}(s) + D\cdot \hat{v}+{C2}(s) + D\prime \cdot \hat{v}_g(s) + (C_{C1} + V_{C2} - V_g + V_D)\cdot \hat{d}(s)$$ $$sC_1\hat{v}_{C1}(s) = -D\cdot \hat{i}_{L1}(s)+ D\prime \cdot \hat{i}_{L2}(s) + (-I_{L1} -I_{L2} - I_1)\cdot \hat{d}(s) -D\prime \cdot \hat{i}_l(s)$$ $$sC_1\hat{v}_{C1}(s) = D\prime \cdot \hat{i}_{L1}(s)- D\cdot \hat{i}_{L2}(s) + (-I_{L1} -I_{L2} - I_1)\cdot \hat{d}(s) -D\prime \cdot \hat{i}_l(s)$$ $$sL_l\hat{i}_l(s) = D\prime \cdot \hat{c}_{C1}(s) + D\prime \cot \hat{v}_{C2}(s) - D\prime \cdot \hat{v}_g(s) +(V_g - V_D - V_{C1} - V_{C2})\cdot \hat{d}(s) - R_l \cdot \hat{i}_l(s)$$  \end{document}   Error:   181 Underfull \hbox (badness 10000) in paragraph at lines 181--181 181 Overfull \hbox (22.63293pt too wide) detected at line 181 My son is 5 years old and sexually active I am a 20 year old female student and I have a 5 year old son who is really sexually active. For instance, he kisses girls by force, he forces them to lick his penis, he "fingers" them, he speaks about sex openly with everyone except me, his father and my family.  There is a whole lot more that he does and the most frustrating part is that when teachers ask him where he saw those things, he says he learned it from me.  We do not even live together.  Currently I am in Cape Town doing my degree and he is in KZN with my mother.  Since he was 8 months old, he went to live with his father because I had to finish high school.  We never lived together. We have tried talking to him and it hasn't helped.  We've punished him but still he keeps on getting worse. What should I do?  I am really angry, embarrassed and confused. If I am not mistaken, this is his third year acting like this. I'm trying to write a small script to change the current directory to my project directory:  #!/bin/bash cd /home/tree/projects/java   I saved this file as proj, added execute permission with chmod, and copied it to /usr/bin. When I call it by: proj, it does nothing. What am I doing wrong? Why is gravity such a unique force? My knowledge on this particular field of physics is very sketchy, but I frequently hear of a theoretical "graviton", the quantum of the gravitational field.  So I guess most physicists' assumption is that gravity can be described by a QFT?  But I find this weird, because gravity seems so incredibly different from the other forces (yes, I know "weirdness" isn't any sort of scientific deduction principle).  For relative strengths:   Strong force: $10^{38}$ Electromagnetic force: $10^{36}$ Weak force: $10^{25}$ Gravity: $1$   Not only does gravity have a vastly weaker magnitude, it also has a very strange interaction with everything else.  Consider the Standard Model interactions:    No particle (or field) interacts directly with all other fields.  Heck, gluons only barely interact with the rest of them.  So why is it then that anything that has energy (e.g. everything that exists) also has a gravitational interaction?  Gravity seems unique in that all particles interact through it.  Then there's the whole issue of affecting spacetime.  As far as I'm aware, properties such as charge, spin, color, etc. don't affect spacetime (only the energy related to these properties). How can I run everything as root I have dual booted to lubuntu (with Windows XP) and everytime and then I'm getting asked for my password. How do I run everything as root and not ask a password again? Ideally I wanted to run nginx but it has permission denied issues:  apathetic@ubuntu:~$ service nginx start Starting nginx: nginx: [alert] could not open error log file: open() "/var/log/nginx/error.log" failed (13: Permission denied) 2012/08/03 20:06:25 [warn] 4762#0: the "user" directive makes sense only if the master process runs with super-user privileges, ignored in /etc/nginx/nginx.conf:1 nginx: the configuration file /etc/nginx/nginx.conf syntax is ok 2012/08/03 20:06:25 [emerg] 4762#0: open() "/var/run/nginx.pid" failed (13: Permission denied) nginx: configuration file /etc/nginx/nginx.conf test failed Subgroups of $\Bbb{R}^n$ that are closed and discrete I have a CSV with an address field (no lat/long), so I went to Google My Maps, imported the csv and geocoded of the records.  Now I export the layer as KML and added the KML to QGIS and Google Earth Pro, but in non of them I can see the markers. In fact in QGIS, the KML layer is added as datatable (no geometry).  What's happening? How can I export a geometry layer from google My Maps? How exactly does Knuth's Up-Arrow notation work? When should we use proximity rule in "either/or", and "neither/nor"? How to prevent upper case in page header? I just came across this small problem which I can't seem to fix. I have a section which looks like this:  \section{Properties of $C_q$}   The problem is that the header of the page uses upper case and so it displays this section title as if I had written  \section{Properties of $C_Q$}   How can I make sure the q is always lower case? How to draw line through group plots in pgfplots? This issue has been driving me crazy for the past 10-20h. I have a group plot (pgfplots) where I have 2 plots one under another. I'd like to draw a vertical line (e.g. a helpline at x=150e-6) that would go through all of them.  When drawing inside the axis or rather nextgroupplot environment I can use something like this: \draw[thin] (axis cs:150e-6,\pgfkeysvalueof{/pgfplots/ymin}) -- (axis cs:150e-6,\pgfkeysvalueof{/pgfplots/ymax}); but this allows me to draw only on one of the subplots. If I go outside the axis environment I am able to draw on the entire {tikzpicture}, but I can't use the axis reference.  \begin{figure*}[h] \centering \begin{tikzpicture}  \begin{groupplot}[       group style = {group size=1 by 2, x descriptions at=edge bottom, vertical sep=0.2cm},       ytick={0, 1.8},       ymin=0,       ymax=3,       xmin=0,       xmax=550e-6,       change x base=true,       x SI prefix=micro]  \nextgroupplot \addplot table [x index = {0}, y index = {1}, col sep=comma] {plots/plot1reduced.csv};  \nextgroupplot[x unit=s] \addplot table [x index = {0}, y index = {3}, col sep=comma] {plots/plot1reduced.csv};  \draw[thin] (axis cs:150e-6,\pgfkeysvalueof{/pgfplots/ymin}) -- (axis cs:150e-6,\pgfkeysvalueof{/pgfplots/ymax});  \end{groupplot} \end{tikzpicture} \caption{Case 1} \label{fig:case1plot} \end{figure*} Find whether the integral converges or diverges. $$\int_{0}^{1}\frac{\log(x)}{1-x^2}dx$$  I simplified it to $$\int_{0}^{1}\frac{\log(x)}{(1-x)(1+x)}dx$$  Here I have $2$ "bad" bounds (both $0$ and $1$).  I think, to prove such integral converges, I should find something that is equivalent to or less than $\log(x)$ so I can exchange them, maybe $\sqrt{x}$, so it may cancel out the fraction, and I can compare it to $$\int_{0}^{1}\frac{dx}{x^p}$$  Also I had an idea to substitute $x$ with $u+1$ so I can use $\log(1+u) \sim  u$ when $u \rightarrow 0$. I'm allowed(not familiar with anything else) to use direct comparision test, limit comparision test, Cauchy test and Abel-Dirichlet's test.   Any hints are much appreciated, I just want to get the big picture.   Thank you. First Question for me.  I have benefited from other answers here but have not fully resolved my problem.  Here's my problem...  The LM311 Comparator output does not drive the N channel MOSFET fully on.  The LM311 has a 110mv ref voltage on the inverting input. The non-inverting input is driven by an opto-isolator output. The Opto-isolator is driven by 1 of 4, AC 60Hz voltages.  &lt;1VAC, 9VAC, 55VAC and 115VAC.  The goal of the design is to have the FET fully turn on the 12VDC fan when there is appreciable AC voltage.  So fan fully On for the 9, 55, and 115VAC voltages.  It seems it should work because I measure voltages on the non inverting input of 1.6, 2.2 and 3VDC respectively for the 3 appreciable AC voltages.  And my ref voltage on the inverting input is 110mv.  So it "almost" works because the fan is off at the &lt;1VAC, but not fully "On" for the other AC voltages.  I believe this is because Vgs is 167mv, 6.1V, 7.7V and 9.9 Volts for the 4 AC voltages listed above.  And the fan is only getting 7, 8 and 10VDC instead of a desired 12VDC. Would really like to be able to fully turn this FET all the way on.  So Close!  Any input or guidance is appreciated.  Ps-Things I think I've done right? Flyback diode across the fan. Putting N channel FET to switch to ground on low side of fan. LM311 has Open collector output using correctly?  PSS-I realize that the pinouts for the LM311 are not right in my schematic.  I have them right in my protoboard circuit.  Its a single device.  Pin 2 is noninveritng and pin 3 in Inverting  Thanks Again.      The suggested duplicate question and answer doesn't help for the following reasons: Both diagram and schematic are"No file found" making it pretty difficult to follow the answers. I am not using any positive feedback in my circuit as I don't think I need one. I have a simple pull up resistor to my 12V supply, not a voltage divider to the power supply as is mentioned in the answer. Thanks.  Just for the record I wanted to post a more accurate schematic and thanks for the response on using a Cap to hold up the opto isolator output. I actually have a full wave input to the phototransistor but I'm sure the cap will help anyway. To answer the last question, the Pin 7 comparator output has a pullup resistor to 12v. The issue is still that Vgs is only 10v when the comparator has 3v on pin 2 and 110mv on pin 3. I would have expected the LM311 output to go much closer to 12v (Vcc) with those inputs. The intent is for the FET to go completely on for pin 2 voltage > pin 3.