How would I attempt to brute force a RSA encrypted message with a 64bit computer if it the memory can't put up with calculations bigger than 64bit? Without going that far to attempting to crack it, how does my computer encrypt a message with a public key that's bigger than 64bit (e.g 2048)?
Whilst I understand how the RSA algorithm works, I don't understand how the CPU operates when it needs to use the mod function with a huge number $n$, for example.  $n = 8192$ bits; $c = m^e \mod n$;  In essence my question is how does the CPU deal with this huge modulus?
I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more 'rigorous'. My understanding of a proof is that a proof is some explanation to convincing others that a statement is true. If some proof is not rigorous, does it mean that while the proof is acceptable for many but it is not for everybody? If so, the proof is not correct since it going to be rejected by some people and it is not a proof at all. Or, does rigorous mean it uses more advanced tools? If so, why not any alternative (but simpler) proof is also acceptable as rigorous (which is more 'valuable' as well since it's simpler!)?   Saying those limited insight of mine, my questions are:   $1-$ What does rigorous proof mean, in general definition?  $2-$ In order to understand your answers, as an example I am re-writing three of the proofs for the compactness of the torus. Which of the following proofs are most rigorous, 'normal' and least rigorous; and why so?    Theorem: The torus $T^2$ is compact.  First proof: The torus is homeomorphic to the product space $S^1 \times S^1$. The circle $S^1$ is compact. Therefore the torus is the product of two compact spaces, and thus it is compact.   Second proof: The torus $T$ is the subspace of $\mathbb{R}^3$ obtained by rotating a circle about the $z$-axis. The torus is closed since given any point in its complement, there is an open ball of sufficiently small radius centered at that point and disjoint from the torus. Also, the torus is bounded since it is contained in the ball of radius $4$ centered at the origin. Since the torus is closed and bounded in $\mathbb{R}^3$, it is compact.   Third proof: The torus is homeomorphic to a space obtained as a quotient  space of the square by identifying opposite edges. The square is a closed and  bounded subset of $\mathbb{R}^3$ and therefore is compact. Furthermore, a quotient map on the square is a continuous function, and the image of a compact space under a continuous function is compact. Therefore the torus is compact.   Thanks a lot for your guidance. Experiment to find structure of water Who first determined the structure of water (two hydrogen atoms stuck to an oxygen atom at approx 105 degrees), and, more importantly, how was this done? How to solve this limit: $\lim\limits_{x\to0}\frac{(1+x)^{1/x}-e}x$? I tried to evaluate $$\lim_{x\to0}\frac{(1+x)^{1/x}-e}x$$but all my efforts were in vain . L'Hospital rule becomes very complicated. Let $\mathbb{R^{ n×n}}$ be the space of n × n real matrices. Show that if $ A(t) \in \mathbb{R ^{n×n}}$ is a differentiable function of $t$ then $\frac{d(det A(t))}{dt}$ is the sum of the determinants of $n$ matrices, in which the $i$th matrix is $A(t)$ except that the $i$th row is differentiated. Use this result to prove that the directional derivative of the determinant function at the matrix $A \in \mathbb{R ^{n×n}}$ along the direction $B \in \mathbb{R^{ n×n}}$ is given by $$(det)^{'} (A; B) = tr(Adj(A)B) = \langle Adj(A)^{T} , B \rangle $$ where $Adj(A)$ is the adjoint of $A$, and where the inner product on $\mathbb{R^{n×n}}$ is the trace inner product given by $\langle X, Y \rangle = tr(X^{T} Y )$ Conclude that $$D(det)(A) = Adj(A) ^T $$. What is the difference between align and equation environment when I only want to display one line of equation? I am typesetting a physics document, and sometimes I need to type equations of several lines while in most cases I only need to type one-line formula. So is it OK if I always use align environment instead of switching between align and equation What are their differences? Log-transformed variable is not significant, while variable itself is I'm doing a Logistic Regression on company level using the total assets to control for company size. Due to the skewedness of the data, I do a log-transformation of the asset data. While I get no significance for the log-transformed variable, the not transformed variable is highly significant (p &lt; 0.01). What could be a possible explanation of such a result? Does it mean that the relationship is not diminishing for extremely high and low values or does it show an outlier problem and should not be used for interpretation of the data. Check for column name in a SqlDataReader object How do I check to see if a column exists in a SqlDataReader object?  In my data access layer, I have create a method that builds the same object for multiple stored procedures calls.  One of the stored procedures has an additional column that is not used by the other stored procedures.  I want to modified the method to accommodate for every scenario.  My application is written in C#. Is there a recursive version of the dict.get() built-in? How to recenter an object's origin? So, I'm working creating a lamp... yes, newbie here.  After modeling for a bit, I noticed that my point of origin was no longer the center of my object:    How to I reset an object origin point to the center of it's geometry? What's the difference between using a 50mm f/1.8G and a 50mm f/1.8D with a Nikon D80? Recently, Nikon announced their , as revamp of the old 50mm f/1.8D. :     The main difference between this new   "nifty fifty" and the almost   decade-old ƒ1.8D is that it has an   autofocus motor inside. This means you   can use it with any current or recent   Nikon SLR. The older lens lacks this   motor and is instead driven by one in   the camera, which cheaper bodies don't   have.   I currently use a Nikon D80 (and I don't plan on changing that any time soon), and I've been considering getting such a lens. Is there any difference between the D and the G lenses when used on a D80? I believe that the D one works just as well, since D80's have internal autofocus motors - is that right?  Besides the autofocus motor aspect, is there any other big difference between them? Since the D is priced at around $120 and the G at $220, should I just get the D? How do I delete data for one game in the iPad Game Center? I have one particular game that saves data through the game center. Somehow the saved game data got messed up which causes the game not to work properly when I a connected to the internet (I.e. When the game center is accessible).   Is there some way to delete all the data belonging to that game only? Why am I getting 50V at a light when the switch is off? I am trying to put in a switch and a light it controls in a storage building and  is how I wired it.   Click for larger view  When I checked the wiring it registered 120V when the switch was on, and right around 50V when the switch was off.  Why would power still be getting to the light (I'm guessing there is a problem with the switch) and why would it only be 50V?  Oh, and where everything branches off at the beginning I have a junction box there where all of the wires are joined there. Are console games locked to a certain region? I'm going on a vacation from Europe to the US and was planning of bringing along some Xbox360 or PS3 games, but now I'm curious if they will be compatible with my European Xbox.  In that past there would be a clear difference between NTSC and PAL games, but is this still the case with the current gen of consoles? Add review queues to Android app I know there are some similar posts on adding review queues to the mobile web page. However, this is regarding the Stack Exchange Android app. I really would like to review posts and edits from the Android app, however this isn't included in the app yet. I'm not sure how it would be implemented; maybe there could be a button or something for review. Could we please see this? Wrong kernel version returned by uname Ubuntu 16.04.1 LT uname -a returns:    Linux decebal-mobile 4.2.0-35-generic #40-Ubuntu SMP Tue Mar 15 22:15:45 UTC 2016 x86_64 x86_64 x86_64 GNU/Linux   but E: Package 'linux-headers-4.2.0-35-generic' has no installation candidate  and my machine says linux headers version is at linux-headers-4.4.0-21-generic.   I have seen a similar problem described in a .  How can I update that? I noticed that if I transfer a few large files between two hard drives, it's pretty speedy, at around at least 30&nbsp;MB per second, but if I transfer thousands of files less than 5&nbsp;KB, it is pretty damn too slow.. around 1 to 2&nbsp;MB per second.   Is there a way to speed up the copy/paste process with thousands of small files on Windows 7? What is the difference between the words which and that?  For example I have the following sentences:   “I have a car which is blue.” “I have a car that is blue.”   Are there any rules specifying usage of which and that? Identify story: Overpopulation problem causes people to live one day a week This story is about a man that is able to live every day of the week, while everybody else has a single day of wakefulness assigned to them. Due to overpopulation, each human spends 6 out of 7 days of the week in stasis, and has jobs assigned to them in daily shifts. 1/7 of the population works on Monday. 1/7 works on Tuesday, &amp; so on. The main character has found away to circumvent the process and illegally live out his days consecutively by skipping the stasis program that is imposed on the general population and by working a different job everyday. Can somebody name this scifi story? Proving the geometric sum formula by induction Avoiding IF ELSE due to variable been NULL