Finding Private Key $d$ using RSA
Calculating RSA private exponent when given public exponent and the modulus factors using extended euclid
Let $p$ be prime and $(\frac{-3}p)=1$. Prove that $p$ is of the form $p=a^2+3b^2$ Let $p$ be prime and $(\frac{-3}p)=1$, where $(\frac{-3}p)$ is Legendre symbol. Prove that $p$ is of the form $p=a^2+3b^2$.   My progress:  $(\frac{-3}p)=1 \Rightarrow$ $(\frac{-3}p)=(\frac{-1}p)(\frac{3}p)=(-1)^{\frac{p-1}2}(-1)^{\lfloor\frac{p+1}6\rfloor}=1 \Rightarrow$ $\frac{p-1}2+\lfloor\frac{p+1}6\rfloor=2k$  I'm stuck here. This is probably not the way to prove that.  Also tried this way:  $(\frac{-3}p)=1$, thus $-3\equiv x^2\pmod{p} \Rightarrow$ $p|x^2+3 \Rightarrow$ $x^2+3=p\cdot k$ stuck here too.  Any help would be appreciated. Is it possible to colour or put textures on Bevel curves? So with my model, I've been struggling to create hair because it always looked weird or off and recently I discovered the "Bevel Curve" addon and it's an absolute lifesaver.   So I converted the bevels into mesh to colour it, except it lags blender like crazy and I need this model for a video game I'm currently developing called Candles(so I can't upload the model, sorry) but I was wondering, is it possible to put a texture on a bevel curve like you would mesh or any other method or giving it colour? Estimate the population variance from a set of means Would you please explain the difference between these two sentences?     Spain ceded (or gave) Puerto Rico to the United States. Since then, Puerto Rico has been a U.S. territory.      Spain ceded(or gave) Puerto Rico to the United States. Since then, Puerto Rico was a U.S. territory.   I am stuck between 'has been' and 'was' in the above sentences. I'm using Apache JMeter 2.3, which now supports "attempt HTTPS spoofing" under the Proxy Server element.   I've tried this on several different servers, and have had no success.  Has anyone been able to successfully record from an HTTPS source with this setting?  Or barring successfully recording, can anyone share a work-around? When available, I simply have HTTPS turned off at the server level, but this is not always feasible. Thoughts? Can store bought salmon be used for sashimi? Is it safe to consume raw fish purchased at the fish counter of a typical suburban grocery store? If not, do you have any tips for finding sushi grade fish. Also, are there requirements for safe sushi/sashimi beyond the use of safe fish? How to burn a Windows .iso to a USB device? I want to burn a Windows ISO to a USB device in Ubuntu. How do I do this?   I know how to burn a Ubuntu ISO into a USB device, but with a Windows ISO it's not the same. How to find and return a duplicate value in array Suppose $A$ is a $n \times n$ constant matrix. How can I prove $\det(e^A) = e^{\displaystyle \sum_{\lambda_i\in\sigma(A)} \lambda_i}$,  where $\sigma(A)$ is the multiset of eigenvalues of $A$?  The following matlab code shows this is true in $n = 3$ case:  A = rand(3) detA = exp(sum(eig(A))) detmA = det(expm(A)) In Java, how do I convert a byte array to a string of hex digits while keeping leading zeros? A question about the idea of Laplace transform In Laplace integral transform equation one multiplies the function $f(t)$ by $e^{-st}$. I read in many tutorials that $e^{-st}$ decays much faster than any other function so the integral diverges.  But what I don't understand what makes one to think to multiply a function with  $e^{-st}$ to transform it in $s$ domain? What is the motivation behind it? Why would you suddenly come up with an idea as such: "Oh there is a function $f(t)$ in time domain what can I do to transform it to complex freq. ($s$) domain? Hmm let me multiply it with $e^{-st}$ and integrate it from zero to infinity" What would have motivated this idea of multiplying and integrating for transformation? Consider the smallest number in each of the $n\choose r$ subsets (of size $r$) of $S=\{1,2,\ldots,n\}$... Consider the smallest number in each of the $n\choose r$ subsets (of size $r$) of $S=\{1,2,\ldots,n\}$.Show that the arithmetic mean of the numbers so obtained is ${n+1}\over{r+1}$. I have no idea, how to solve it.It would be better if the solution has bijection. Please help! Why more (and varying number of) logical reads with read ahead (prefetch)? After creating tpch database in my SQL Server, I tried below query:      set statistics io on     DBCC DROPCLEANBUFFERS;             select top 100 * from dbo.lineitem order by l_partkey;   The table lineitem has a non-clustered index on l_partkey. I issued the above queries for several times and found out that the logical reads vary each time:      Table 'lineitem'. Scan count 1, logical reads 1019, physical reads 4, read-ahead reads 1760, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.     Table 'lineitem'. Scan count 1, logical reads 1007, physical reads 4, read-ahead reads 1720, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.     Table 'lineitem'. Scan count 1, logical reads 1030, physical reads 4, read-ahead reads 1792, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.   From the post here: , I know it could be caused by read ahead behavior.  BUT exactly why read ahead could cause more logical reads? How does it change the SQL Server behavior? Like SQL Server may read more index page since it's in cache anyway?  Anyway, I disabled read ahead and issue the above query again. Now it reports the same amount of logical reads each time. BUT the logical reads is much smaller!!      Table 'lineitem'. Scan count 1, logical reads 404, physical reads 160, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.   So my question is, why the read ahead feature could cause many more and various logical reads count?  Out of curiosity, I tried another query without the "order by":      select top 100 * from dbo.lineitem   Here's the result without read ahead:      Table 'lineitem'. Scan count 1, logical reads 5, physical reads 3, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.   Here's the result with read ahead:      Table 'lineitem'. Scan count 1, logical reads 15, physical reads 2, read-ahead reads 3416, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.   The one with read ahead still has more logical reads. So, why? How can I SELECT rows with MAX(Column value), PARTITION by another column in MYSQL? Passing a varible from jQuery to PHP I just learned that using CBC encryption with an IV which is predictable is not secure.  From what I understand, using certain plain texts, and then guessing the IV that it uses, the attacker can verify if the IV he guessed was right. How does this qualify as an attack? Is there a license / royalty free DRM available for protecting and distributing video content? Can CBC ciphertext be decrypted if the key is known, but the IV not? How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$? $f(x)=x/1$ is not injective? In chapter 3 of Atiyah and Macdonalds Intro to Comm Algebra there is the statement  "We also have a ring homomorphism $f:A\rightarrow S^{-1}A$ defined by $f(x)=x/1$. This is not in general injective." (Note: $A$ denotes an integral domain and $S$ a multiplicatively closed subset of $A$.)  How could this possibly not be injective? If $f(x)=f(y)$ then $x/1=y/1$ so $1x=1y$ hence $x=y$. What am I missing here?