I'm having trouble understanding this Let $x=(x_1,x_2,...x_n)∈\{0,1\}^n$ for $n∈\mathbb N$. Prove that if one-way functions (OWFs) exist, then there exists a one-way function f such that for every bit $i∈[1, n]$ there exists an algorithm $A_i$ such that: $$\Pr[A_i(f(x))=x_i]≥\frac12+\frac1{2n}$$ I find some information from this post , but its in power
One way function existence Let $x = (x_1, x_2,...,x_n)\in\{0,1\}^n$ for $n\in\mathbb{N}$. Prove that if one-way functions (OWFs) exist, then there exists a one-way function $f$ such that for every bit $i\in[1,n]$ there exists an algorithm $A_i$ such that: $$\Pr[A_i(f(x)) = x_i] \geq \frac{1}{2} + \frac{1}{2n}$$ I'm having trouble understanding this.
I'm new to d20. I recently acquired a bunch of D&amp;D books and I'm sure they are all either 3.0 or 3.5, but it doesn't say in the book which version it is. How can I tell which version the book is?  For instance, I have "Complete Warrior - A Player's Guide to Combat for All Classes". It's hard back, published in '03. Is it 3.0 or 3.5?  Another example, I have "Sword and Fist - A Guidebook to Fighters and Monks". Perfect Bound, published in '01. Is it 3.0 or 3.5? In this case what is correct?        Keep Tom and I updated.      or         Keep Tom and me updated.      I understand that me is an object pronoun. And therefore I feel the second option is correct.  Especially, because I would say "Keep him updated" and also "Keep me updated". Instead of "Keep he updated" or "Keep I updated". But something about saying "Keep him and me updated" feels wrong.  However I am not sure, because several people have told me that in this case I is the correct option because it is being used in conjunction with a name (Tom) and not a pronoun. Is that correct? Can I run Ubuntu OS on a Android 4.0 tablet? What is a "loop device" when mounting? I am using texlive 2013, and just ran the following command to update the packages.  sudo ./tlmgr update  -all   After it finished, I tried to compile tex documents that uses tikz package as the following MWE shows:   \documentclass{article}  \usepackage{tikz}  \begin{document}       aaaa  \end{document}   Then pdflatex complains; it said:   (/usr/local/texlive/2013/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty  (/usr/local/texlive/2013/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex  ! I can't find file `pgfutil-common-lists.tex'.  l.556 \input pgfutil-common-lists.tex   The problem is that, pgfutil-common-lists.tex is indeed in the directory:   /usr/local/texlive/2013/texmf-dist/tex/generic/pgf/utilities/   Why is that? pdflatex should be able to find it, right? I even download the  again, and replace the pgf directory in   /usr/local/texlive/2013/texmf-dist/tex/generic/   Nothing changed at all........... Any idea? Thank you for your critics.  %=====================================================================  Then followed the comments, I type the following command in the terminal:    sudo mktexlsr gives me command not found;  mktexlsr gives me:  mktexlsr: /usr/local/texlive/2013/../texmf-local: directory not writable.     Skipping...  mktexlsr: /usr/local/texlive/2013/texmf-config: directory not writable. Skipping...  mktexlsr: /usr/local/texlive/2013/texmf-dist: directory not writable. Skipping...  mktexlsr: /usr/local/texlive/2013/texmf-var: directory not writable. Skipping... mktexlsr: Done; Forkpsewhich pgfutil-common-lists.tex, nothing pops up. Why do people go to the Wall in Game of Thrones? Is the wall a bad or a good place to be? Some people are sent there as punishment, others go because they want to (Jon Snow), others to get away from harm (Arya).  If I am not completely correct, it is because I have only started watching the series recently, and it is a bit confusing!  Please also correct me if I am wrong with any facts! Where did Hogwarts get money for salaries, food and other purchases? We know that you can't transfigure food, so you have to grow it or buy it. Ditto money.  I'm sure that aside from some medicinal plants Herbology Professor had there were no farmlands at Hogwarts. So they must have bought food (as well as potions ingredients for Snape etc...). And they needed to pay the teacher salaries.  Where did all that money come from according to canon?  Ministry funding?  Tuition?  Patent money on Dumbledore's inventions?  Philanthropic donations by Malfoys and co?  Other? I'm having troubles with one of the problems in the book Introduction to Commutative Algebra by Atiyah and MacDonald. It's on page 11, and is the last part of the second question.  Given $R$ a commutative ring with unit. Let $R[x]$ be the ring of polynomials in an indeterminate $x$ with coefficients in $R$. We say that a polynomial $f = \sum\limits_{i=0}^n r_ix^i \in R[x]$ (with coefficients $r_0, r_1, \ldots, r_n$) is primitive if $\langle r_0,r_1,...,r_n\rangle = R$, i.e., the ideal generated by the coefficients of $f$ is $R$. Let $f$ and $g$ be two polynomials in $R[x]$. Prove that $fg$ is primitive iff $f$ and $g$ are both primitive.  The $\Rightarrow$ part is easy. Say, $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$; then $fg = \sum\limits_{i=0}^{m+n} c_ix^i$, where $c_k = \sum\limits_{i + j = k}r_is_j$. Since $fg$ is primitive, there exists a set of $\{\alpha_i\} \subset R$, such that $\sum\limits_{i=0}^{m+n} \alpha_ic_i = 1$, to prove $f$ is primitive, I just need to write all $c_i$'s in terms of $r_i$'s, and $s_j$'s, then group all $r_i$ accordingly, rearranging  it a little bit, and everything is perfectly done. And the proof of the primitivity of $g$ is basically the same. The $\Leftarrow$ part is just so difficult. Say $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$ are both primitive, then there exists $\{\alpha_i\}; \{\beta_i\} \subset R$, such that $\sum\limits_{i=0}^{n} \alpha_ir_i = 1$, and $\sum\limits_{i=0}^{m} \beta_is_i = 1$. At first, I thought of multiplying the two together, but it just didn't work. So, I'm stuck since I cannot see any way other than multiplying the two sums together. I hope you guys can give me a small push on this. Thanks very much in advance, And have a good day. Could it be that Goldbach conjecture is undecidable? The result closest to Goldbach conjecture is Chen's theorem [Sci. Sinica 16 157–176], the proposition ``1+2''. It is natural to ask if it is likely that under our arithmetic axioms the Goldbach conjecture is an undecidable proposition. Should one create slides similar to those that one uses in a good powerpoint presentation? Or are there things that a poster should include that a powerpoint should not include? (and vice versa) Are there well known HTTP-only sites? Captive WIFI portals suck.  So often when I open in a browser (Desktop Chrome or Mobile Chrome) a HTTP site, I get the captive portal, but with auto-completion and so quickly I connect again to WIFI.  The problem is that after the captive portal redirects, I'll have also a HTTPS redirect and Chrome remember the certificate and to use only HTTPS.  So I cannot use the same site twice (in a session).  A well-known public HTTP only site will resolve this.  Well-known sites usually work, causing some less debugging of the WIFI connection. How strongly can a 1mo be rocked? My 1 month old baby only seems to quiet down when I vigorously rock her (while I carefully hold her head). Can I cause her some (brain?) damage if I rock her too hard? Filter the score of the questions on the homepage before loading them At the moment questions on the homepage are loaded by the top N most recent and then ones with a score of -4 or less are filtered out. This doesn't make much difference for a user who is logged in but consider anonymous users; their home page is only 15 questions in length. If someone was to edit several posts that bump to the top with a score of -4 or less then the home page looks tiny. At the moment on SFF, for example, the homepage for anonymous users only shows 9 questions:  At one point it only showed 3 questions (a few weeks back):  Can the filtering logic be changed to before the questions are loaded to appear on the homepage? Taking Seats on a Plane No matter how I do it, I always end up with $x = 0, 90, 270$ and $360$. All of those except $270$ is right, but I can't quite figure out how to get the $270$ degrees out of the answer. I've tried using trig identities, I've tried squaring both sides, but I always end up with $$2\sin x\cos x$$ which then leads me to $x = 0, 90, 270, 360$. But $$\sin (270) + \cos (270) = -1$$ so I'm doing something wrong. How to finish current activity in Android Where does the word "torsion" in algebra come from? Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry (which relates back to the ordinary use of the word). So how did it acquire this usage in algebra?  I'm interested to understand the intuition behind why the word "torsion" was chosen for this notion, as well as when it was first used. My regex is matching too much. How do I make it stop? I have this gigantic ugly string: J0000000: Transaction A0001401 started on 8/22/2008 9:49:29 AM J0000010: Project name: E:\foo.pf J0000011: Job name: MBiek Direct Mail Test J0000020: Document 1 - Completed successfully  I'm trying to extract pieces from it using regex. In this case, I want to grab everything after Project Name up to the part where it says J0000011: (the 11 is going to be a different number every time). Here's the regex I've been playing with: Project name:\s+(.*)\s+J[0-9]{7}:  The problem is that it doesn't stop until it hits the J0000020: at the end. How do I make the regex stop at the first occurrence of J[0-9]{7}? Auto-mounting network shares per user I have a server that has a number of CIFS shares that need to be accessed on a per-user basis. For example, I have a Music share which I have full access to, but my wife has read-only access.  When either myself or my wife log into our Ubuntu 11.04 laptop I would like these shares to be automatically mounted per user. Now I understand that if I mount as -t cifs without specifying a user then it will use the USER environment variable. However, I also need to specify a password, so how can I do that when each user has a different password?  I think my questions are:   Is there a way for me to have a per-user /etc/fstab? If not, is there a way to specify that a mount is only applicable to a certain user? Also, the share password is always the same as the local password. Is there a way to specify that this password should just pass through from the client to server rather than having to specify it in a credentials file somewhere?   Or maybe I'm missing something and there's a completely different solution. Can anyone help? In most fiction I've read or encountered, the change occurs as a simple consequence of the lunar cycle. Though from what I know of Harry Potter, the change that takes place is due to sighting the full moon (as occurs in the third movie, Prisoner of Azkaban) where the gang, plus Lupin, Severus, Sirius and Pettigrew emerge from the Whomping Willow and upon sighting the full moon he begins to change.  Is this the trigger for werewolves in the Potterverse? If not, why did he (Lupin as an example, though for any werewolf in general) not change whilst in the shrieking shack that night?  Edit: This  addresses Lupin's change but I'm after more general information about werewolves.