The multiplicative group mod $p$ is isometric to the additive group mod $p-1$, yet computing discrete logarithms in the additive group is easy and completing discrete logarithms in the multiplicative group is usually hard.  What is it about group representation that makes group representation matter so much for the complexity of computing discrete logarithms?
I am trying to construct a theoretical Diffie-Hellman key exchange protocol. However, i cannot understand the difference between choosing an additive group or a multiplicative group. I believe an additive would make it much more simple, but would a multiplicative group make it more difficult to break? What would be the difference between the use of those 2 kinds of number groups?
My sisters and I all have iPhones under the same account. I also have an iPad under the same account. I am trying to get the iMessage on my iPad compatible with my phone number, not email address.   The only phone number choice to choose from is my sister's iPhone number. How can I choose my phone number for iMessage on my iPad? How to copy a dictionary and only edit the copy Function whose inverse is also its derivative? What are some good examples of a function  $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ and showing that $f^{-1}(x) = f'(x) \implies b-1=\frac{1}{b} \implies b=\phi$ and got $$f(x) = \frac{x^\phi}{\sqrt[\phi]{\phi}}$$ Which seems to work according to WolframAlpha, but I'm having trouble double-checking it. Any other ideas? NullPointerException accessing views in onCreate() If I enter my password correct first time, I login as normal.  If I enter my password wrong, it says invalid password. If I attempt to enter the correct password now then the screen freezes and doesn't go to dock as normal. I must restart using a physical button or shutdown to get out. Login works normally after restart, assuming I get it right first time.  EDIT:  After entering a wrong password, if I click cancel and then go back to my user to login, it works fine. Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice? How to populate options of h:selectOneMenu from database? Real world uses of hyperbolic trigonometric functions How do you add a timer to a C# console application Just this - How do you add a timer to a C# console application? It would be great if you could supply some example coding. Assume a p x n data matrix  (e.g. p independent samples of n variables). The n variables are correlated according to a correlation matrix $\rho$. Now if you consider the data matrix as some state t (t=time)  and based on this state I would like to calculate the next timestep (t+1). Therefore the correlation matrix would be allowed to have values &lt; 1 in the diagonal, which means that the diagonal elements would indicate the influence from each variable $x_{p,i}[t]$ on $x_{p,i}[t+1]$. So I would be interested to find a function f like this: $x_{p,i}[t+1] = f(x_{p,i}[t],\rho)$ so that the p x n data matrix  consisting of the $x_{p,i}[t+1]$ replicates the $\rho_{ij}$ (except the diagonal element). f should be something like this: $x_{i,j}[t+1] = \sum_{(k=1\ldots n)}\rho_{i,k} \cdot x_{kj}[t] + e_{i,j}.$ This means specifically, I not only want to calculate a data matrix, that replicates the correlation matrix as such, but also to let each observation be depending on its own state and linear dependent on the previous state of all the intercorrelated variables of its own state. For ease one can assume that the p variables are distributed normal and $e_{ij}$ are independent and distributed with mean$(e_{i,j}) = 0 $. Then for a numerical simulation it would be interesting  to find a formula to calculate the variance/standard deviation of each of the j variables (sd($e_{[.,j]} = ?).$ Thank you for any help! I have attached an image as an example below. I want to twist a rectangle. Do I have to convert that rectangle into a curve or into mesh and then twist it with ctrl+T?   Please explain with an example. Interpretation of ROC curve Can \includegraphics be used to change an image color? I'm including a number of PDF images in my document using \includegraphics.  I'm wondering if there's a way for LaTeX to adjust their color profile, hopefully the same way it can flip images upside down, resize, and crop, as here:   Let me detail a little more my desired outcome, in case there's another way to achieve this: The images are brightly colored---I'd like to reduce color levels, or possibly render them grayscale, and I'd prefer to not have to do external pre-processing, if this is possible.  I'm using XeTeX. I have recently been reading about Quantum Electrodynamics which I found very interesting, but even more confusing.  I understand photons mediate the electromagnetic force and interactions between charges can be described by virtual photon exchange. And that in Quantum Mechanics particles paths can be evaluated by sum over all paths...  An electron travelling from one point to another will take all paths, emitting and reabsorbing virtual photons. Does the electrons take all paths including those that involve faster than light velocities?  Is this question null, Does it mean anything to ask about properties of virtual particles?  Do I have the wrong idea about what virtual particles are? Text editor to edit large (4.3 GB) plain text file Is there any text editor, which can edit such file?  I've  tried:   gedit kate nano vim mcedit   without success. In the experiments for photoelectric emission, the light is incident on one face of the emitting plate, for example the anode, when determining the stopping potential. The electrons are emitted by this face of the plate. Why are the electrons emitted in the direction of the incident light, and not opposite to it?  I have read a little about photons (the key words being, a little), and from what i gather, they have an intrinsic momentum $p=h/\lambda$, which is transferred entirely to the particle that absorbs it (along with the energy $h\nu$). This should imply that the electrons try to move further into the cathode, and not escape it! What is the fault in the reasoning?  Thanks! Prove $GL_n(\Bbb{R})/SL_n(\Bbb{R}) \cong \Bbb{R}^\times$ $$GL_n(\Bbb{R})/SL_n(\Bbb{R}) \cong \Bbb{R}^\times$$  This is trivial to prove with the first isomorphism theorem - by using the determinant as a endomorphism, then as $SL_n(\Bbb{R})$ is $1$ under the determinant, it is the kernel, and by FIT, the above is proved.  I was wondering how to prove this without the isomorphism theorem ? Where can I get freely available audio, graphics, and other resources for games? How do i prove that continuous linear transformation between normed spaces is bounded? Let $(V,||\cdot||_V),(W,||\cdot||_W)$ be normed spaces over $\mathbb{C}$.  Let $T:V\rightarrow W$ be a linear transformation.  Assume $T$ is continuous.  Then, how do i prove that $\exists c&gt;0$ such that $\forall v\in V, ||Tv||_W&lt;c||v||_V$? What harm can be done with a copy of one's passport? In one of the questions on this site, it was suggested the use of  to save a backup copy of your passport, in case you need to proof who you are if your original document is lost.   I like the idea, but what if someone gets access to your Dropbox folders and is able to acquire the copy of your passport.  The same goes for when you leave a copy of your passport at a hotel receptions desk. Once someone has the info printed on that copy, would that make it easier to steal your identity?