Context:   Assume I wanted to use RSA as the basis for a hash function or DRBG. Also assume that my construct would be insecure if someone were to have the private key. Is there any way generate the a usable public key to be used such that it is provably just as hard for me to generate the private key as it would be for anyone to do so?  I'm willing to take for granted:   the existence of a suitable source of nothing-up-my-sleeve numbers the availability of significant computational resources.     The best idea I've come up with would depend on the ability to prove a lower bound the the size of the smallest factor in a number (e.g.: an $N$-bit number where all factors are at least of size $N*0.49$-bits) with significantly less work than it takes to actually factor it. (That and the ability to find such numbers.)
It is possible to find an efficient algorithm for constructing a provably hard to factor number $N$, together with a witness that shows that it is indeed hard to factor. EDIT, since it was not clear: I also want to be sure that even the person that constructed the number cannot factor it.  Of course, a random large enough integer is going to be hard to factor with high probability, but we might be unlucky.  There are also multi-party protocols for shared generation of RSA moduli (such as ), but I would like a non interactive technique.  Motivation: This was suggested by Poncho's answer to
Using the affine linearized polynomial $f(x)=x^p-x-b$ is irreducible over $\mathbb{F}_q$ if and only if $\operatorname{Tr}_{\mathbb{F}_q/\mathbb{F}_p} \neq 0$. But here $\mathbb{F}_3$ is not a field extension of $\mathbb{F}_5$.  Also we have that: Let $a \in \mathbb{F}_q$ and let $p$ be the characteristic of $\mathbb{F}_q$. Then the trinomial $x^P - x - a$ is irreducible in $\mathbb{F}_q[x]$ if and only if it has no root in $\mathbb{F}_q$. If $n\ne 4$ is composite, then $n$ divides $(n-1)!$. guess the color of the next card There are 26 red cards and 26 black cards on the table which are randomly shuffled and are facing down onto the table. The host turns up the cards one at a time. You can stop the game any time (even at the beginning of the game). Once you stops the game, the next card is turned up: if it is red, you get $1; otherwise you pay the host one dollar. What is the payoff of this game? I'm a bit confused about what Gstreamer does.   Does it come installed by default with Ubuntu ?  If so which version  is the default for 14.04?  Does it provide hardware accelerated video playback (decoding) ?   If so does it work with VA-API and VDPAU or is it an alternative to those?    Thanks for your help Why does this function return a different value every time? How to secure database passwords in PHP? My friend and I are planning a vacation to Ireland later this year. We'll be there for a week, flying into Dublin and then renting a car and driving around the country.  Both of us are 23. Are there any age restrictions on renting cars in Ireland? What would be the cheapest/best option for us? Am trying to Update Leads.  When Lead converted flag is true.  Lead Status is Contacted. I have to update all the contacted leads to Converted.  Am Getting this error on the data loader:     ERROR: cannot reference converted lead   Maybe because converted leads cannot be updated because they are locked out once they are converted?  Is there a workaround to update Lead Status to Converted from Contacted? I use a very custom style (not mine) and very frequently I get a heading  on the bottom of a page and the next paragraph on the top of the next page. Sometimes this paragraph may be a list or similar. The same is true for listings and the listing description.   Is there a general way how I can adjust the "badness" of these events? Or can I adjust when a section heading would be on the last, say, 3cm of a page it should go to the next page? Or can I say, if a *listing description" is on the top 3cm of a page pull 5 more listing lines from the last page onto this page?   The style I use does not use standard structure tags and environments. But lets call them section, subsection, listing and I will try to migrate them to my lingo. How to export variable for use with sudo? On Slackware, using sbopkg permits one to build a package from source. Repos is not big as Debian, but it's nice.  Some software can use environment variables, for example on the VICE c64 emulator, if the variable FFMPEG is set to yes, it will enable ffmpeg recording the emulator.  I tried to use  $ export FFMPEG=yes; sudo sbopkg -B -i vice   but ffmpeg is disabled.  Instead I had to use  $ su - $ export FFMPEG=yes $ sbopkg -B -i vice   which works.  How to use environment variables with sudo? Subject/Complement Agreement. How to describe problem with "The thing is the objects." In my , I provided the following, problematic,  wording (especially bold italic), and I need help to better understand this issue so I can fix my answer:1          The thing is the books. (Reduced form of Sentence 2.a)         Fundamentally, Sentence 2.a (the so-called "correct" answer), is grammatically defective. Recall there is another grammatical rule: the subject and subject's complement should match in number.    A clarification is needed here. I am not suggesting here, in this EL&amp;U question, that there is any such singular grammatical rule as indicated in the quote above.  The quote continues as follows:     The reduced sentence makes the disagreement between the subject's and complement's plurality obvious. What is one to do? Language is linear and we know "The thing is X" is better than "The thing are X", so we go with the former, which is the subject-verb agreement rule. But what is "the books" then? It must be thought of as a collective noun, even if it doesn't look or feel like one. Forcing something to be a collective noun is related to the idea of notional agreement.   Others noted that grammatically defective is too strongly worded.  There is no such "grammatical rule" for subject/complement agreement.  Please help me understand/improve upon my line of thought here.  I'm stating that Sentence 2.1 presents a fundamental "conflict" or "error" (however one might define error) at some level (morphological, syntax, grammatical, semantic):     The thing is the books.  (Or, "The thing is the objects.")   Based on the "obvious" morphosyntactic plurality conflict, the above sentence feels so wrong that I think many native speakers would think it "simply" grammatically incorrect. However, @F.E. gives counter examples:     And here are some more grammatical examples, where the number of subject and predicative complement don't agree: "They were a problem to us all", "That so-called work of art is simply four pieces of driftwood glued together" (CGEL, page 254-5). As CGEL says: "What is required is semantic compatibility, not syntactic agreement . . .". There are more examples on page 512: "The only thing we need now is some new curtains", "The major asset of the team is its world-class opening bowlers", "Our neighbors are a nuisance", "This gadget is five different tools in one".   I think all the examples given above can be explained as notional agreement.    Question 1: If we explain this phenomena in terms of notional vs syntactic agreement, where does "grammatically correct/incorrect" fit in?  Question 2: Can a simple sentence such as "The thing is the objects." be grammatically incorrect while more complex sentences such as "The thing is four pieces of driftwood glued together." be grammatically correct?  (The answer, in my mind, must be no; this is my conundrum.)  Question 3: Would it be better to say that Sentence 2.a has a low level of (linguistic) ?   Question 4: How about a low level of  and that it's semantically difficult to be (linguistically) ?    Question 5: Do we simply draw a hard line and say "The thing is the objects." is 100% grammatically correct?  I didn't particularly like this option based on intuitive notions of "grammatically correct".    1. Please help me and be kind; I'm trying to improve my understanding. I'm looking for cool, objective advice on how to look at this.  I numbered my questions so they can be easily referenced, if desired.  My preference is the final answer would sufficiently answer all the questions, but of course that can be done explicitly or implicitly. What is mathematical existence? Split $n$ into nontrivial factors via a nontrivial square-root of $1\!\pmod{\!n}$ Coming from an understanding of Fermat's primality test, I'm looking for a clear explanation of the Miller-Rabin primality test.  Specifically: I understand that for some reason, having non-trivial square roots of 1 mod p means that p is definitely composite; and I gather that you can find these non-trivial square roots by squaring x, but I don't really understand what these reasons are. Specific examples of non-trivial roots of a composite number would be helpful.  Cheers! (clickable)  Mainad has a valid height and width, however it isn't shown like subad1/subad2. Which are in essence exactly the same! (just a different background image).  Firebug shows my div as greyed out for some weird reason. Replacing the contents of mainad with just some text doesn't solve the problem (problem isn't related to inner elements)..   What's going on?! :( Let $R$ be a commutative ring with finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_n$. Let $M$ be a finitely generated module. Then there exists an element $x\in M$ such that $\frac{x}{1}\not\in\mathfrak m_iM_{{\mathfrak m}_i}$ for every $i=1,\dots,n$.    I cannot prove that such an element there exists. I was trying to prove it by induction on $n$. If $n=1$ it is true by Nakayama, so suppose it is true for $n-1$. Then for every $i$ I can find an $x_i\in M$ such that $\frac{x_i}{1}\not\in \mathfrak m_jM_{\mathfrak m_j}$ for every $j\neq i$. If $\frac{x_i}{1}\not\in \mathfrak m_iM_{\mathfrak m_i}$ for some $i$ we are done, so suppose $\frac{x_i}{1}\in\ m_iM_{m_i}$ for every $i$. I don't know how to go on, could you help me? Lightning component scripts loading issues since Spring 17 update We are planning a two-night trip to the Canary Islands. The cheapest options include Lanzarote and Las Palmas for flights. Travel dates are last week of November, so I guess weather should be okay. I have never been to these islands. I'd like to sleep on the beach in a tent or mattresses. Is that possible, safety, legality, and weather? It would also be nice to know how populated with tourists the islands are. Is there a way of setting culture for a whole application? All current threads and new threads? Is there a way of setting culture for a whole application? All current threads and new threads?  We have the name of the culture stored in a database, and when our application starts, we do   CultureInfo ci = new CultureInfo(theCultureString); Thread.CurrentThread.CurrentCulture = ci; Thread.CurrentThread.CurrentUICulture = ci;   But, of course, this gets "lost" when we want to do something in a new thread. Is there a way of setting that CurrentCulture and CurrentUICulture for the whole application? So that new threads also gets that culture? Or is it some event fired whenever a new thread is created that I can hook up to? I was shooting the lighting of the city that is visible at the top of the trees if you shoot landscape.  No matter what I do, the image is blurry. What have i done:   ISO 100 Manual focus towards infinity Using a tripod Using really sharp prime &amp; good camera   Basically, everything I could. The only example i have is here, but it is not good, because image is underexposed:    The main question: Why can't night photographs be as sharp as daily photographs (with approperiate exposure times of course)? Evaluating the indefinite integral $ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $ I have been having extreme difficulties with this integral. I would appreciate any and all help. $$ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $$