I've read several times that modern cryptography is based on the fact that multiplying two primes is easy, whereas getting the prime factorization of a random big integer is very hard. (see )  Does it mean that the largest prime numbers known are kept secret, for cryptography purposes ?  If they were not, one could compute the product of every pair of known prime numbers and decode the public keys ?
I have some thoughts on this. First, I want to say I am no expert on cryptography, I just know some stuff, and I took a cryptography class in University. I am very interested in this topic.  I understand that many security experts out there might get angry just by the discussion of these things, but I hope you can spare me :)  My Questions:  Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing prime keys up to 2^2048 ?  I understand this would take an immense amount of time to do, but it would only be done once.  Given the public key, you would be able to look up what the two primes are, and hence retrieve the private key instantly.  When I consider who might break such a code, I am primarily thinking about NSA and other Governments that might have spent years in research and have the resources and interest in doing so.  Breaking an 256 AES would then be easier, as AES keys are often generated using the RSA private key.  Would it be possible to build such an index ? How long would that take in permutations, and perhaps with the fastest computer in the world ?  I hope we can have a nice chat about this. We are just discussion theory here. I hope you can offer me some insight on this topic.  Thanks!
How do I add arrowheads to circular arcs? How did the Swooping Evil Venom infused rain affect the magical community? I didn't see any wizards drenched in the rain at the end of Fantastic Beasts and Where to Find Them (don't recall, actually). But would the rain equally affect the wizards as it did no-maj's?   This event is something that doesn't necessarily need to be forgotten by the Wizarding community. How would they avoid being obliviated by the rain?  (They could avoid the rain by Apparating, but what about the showers and drinking water?)  This question is picking at my brain. I use a Canon EOS Rebel T3i (600D) with the kit lens. I have experimented a lot with different settings and modes, but I feel like I have wasted many good shots that could have looked amazing, if only they were a little bit sharper.  I use auto-focus on most of my pictures and make sure to choose the right focus point, but all of my photos are still a little bit out of focus. When I zoom in, I can see that the pixels are not very crisp or well-aligned.  I am not sure if this issue is caused by my equipment or skills. Splitting words into specific fragments Subsurf Modifier artifacts So I was messing with Subsurf modifier to create a certain sword and then some weird artifacts began to appear (cape-like). I've tried adjusting the Crease, but couldn't achive what I wanted. Basicly, what I whant is the as on the screens, but wo any artifacts :)   Also, here's the .blend file, cause I assume I didn't make myselt 100% clear. When I'm not exploring fresh caves or lands in minecraft, I'm busy trying to find deposits of diamond, gold and iron deep in the earth. So far I've just been digging around just above bedrock on a whim to look for these rare minerals but I've started thinking that I should be taking a more controlled, efficient approach to mining.  What is the optimal mining strategy that yields the most rare minerals while minimizing blocks removed and time spent digging and traveling?  Edit: The existing answers are great for mining, but no one has addressed travel time to and from the mine which is usually ~60 blocks downward and incredibly tedious. Solutions? Cardinality of equivalence relations in $\mathbb{N}$ I came across this long proof on this site:    But I would like to know whether my direction can work.  Say we want to find the cardinality of all equivalence relations in $\mathbb{N}$. Since it is a subset of all relations in $\mathbb{N}$, I conclude it has a cardinality smaller or equal to $\aleph$. Now, define an injective function from $P(\mathbb{N})$ to the set of equivalence relations by matching each subset of $\mathbb{N}$ with the identity relation (which is an equivalence relation in $\mathbb{N}$.  Therefore the cardinality of all equivalence relations in $\mathbb{N}$ is greater or equal to $\aleph$ and using CSB we get the desired result.  Seems legit? NSF IIS status date changed, status still pending shade plane with a color function color(x,y) I have read the Rules of a competition. The text of the Rules include a sentence as follows:     As per stated in the Rules the entrants will be notified by May 30th 2010.   Does the sentence above mean that May 30th is included in the term of notification or not? I just need to check the reasoning in my proof is correct, I think it is valid although I'm not totally convinced because I can't follow the logic; does proving that $x$ is an integer prove that $ac|bd$?  Theorem: Let $a$, $b$, $c$, $d$ be integers. If $a|b$ and $c|d$, then $ac|bd$.  Proof: $aj=b$ and $ck=d$ for integers $j$, $k$. Then, $ac|bd$ implies $acx=ajck$ and thus $x=jk$ for some $x$. Since the product $jk$ is an integer, $x$ is an integer and thus $acx=ajck$ and thus $ac|bd$. How do I instantiate a string array object into a Spring.Net container? How do I integrate Github files to Orange for ML? I am currently trying to make a facial recognition workflow with Orange but since the widgets are still prototypes they are not all available in Orange. I found the Github files for the widgets but don't know how to install this so that I can see and use the widgets in Orange. How does entropy depend on location and scale? Book recommendations How to convert a 3 column csv file into a table (or matrix) I have an CSV input file format like this, with a nucleotide sequence in field 1, text in field 2, and an integer in field 4:  ATGC,CD3,56 ATGC,CD4,67 ATGC,IgD,126 ATGC,IgM,127 AGTC,CD3,67 AGTC,CD4,78 AGTC,IgD,102 AGTC,IgM,89 TCGA,CD3,334 TCGA,CD4,123 TCGA,IgD,456 TCGA,IgM,80 CGTA,CD3,54 CGTA,CD4,32 CGTA,IgD,82 CGTA,IgM,117   I opened this CSV file using Numbers in Mac it's display as 3 columns format, however, I want to convert it to the table(or Matrix) format (also a CSV file), making the first column, the nucleotide sequences, into a header, and want the result to also look like a table (or Matrix):       ATGC  AGTC  TCGA  CGTA CD3  56    67    334   54 CD4  67    78    123   32 IgD  126   102   456   82 IgM  127   89    80    117   Below is a section from my real input CSV file (sample input.txt):  AGAATAGTCTGATTCT,-,,38 AGAATAGTCTGATTCT,AnnexinV,,51 AGAATAGTCTGATTCT,CD127,,39 AGAATAGTCTGATTCT,CD138,,3 AGAATAGTCTGATTCT,CD14,,2 AGAATAGTCTGATTCT,CD16,,4 AGAATAGTCTGATTCT,CD19,,10 AGAATAGTCTGATTCT,CD20,,6 AGAATAGTCTGATTCT,CD24,,21 AGAATAGTCTGATTCT,CD25,,4 AGAATAGTCTGATTCT,CD27,,87 AGAATAGTCTGATTCT,CD3,,235 AGAATAGTCTGATTCT,CD34,,5 AGAATAGTCTGATTCT,CD38,,18 AGAATAGTCTGATTCT,CD4,,412 AGAATAGTCTGATTCT,CD43,,99 AGAATAGTCTGATTCT,CD5,,430 AGAATAGTCTGATTCT,CD56,,3 AGAATAGTCTGATTCT,CD8,,7 AGAATAGTCTGATTCT,IgD,,4 AGAATAGTCTGATTCT,IgM,,2 TGTGGTAGTTCGTCTC,-,,9 TGTGGTAGTTCGTCTC,AnnexinV,,42 TGTGGTAGTTCGTCTC,CD127,,6 TGTGGTAGTTCGTCTC,CD138,,4 TGTGGTAGTTCGTCTC,CD16,,40 TGTGGTAGTTCGTCTC,CD19,,7 TGTGGTAGTTCGTCTC,CD20,,2 TGTGGTAGTTCGTCTC,CD24,,24 TGTGGTAGTTCGTCTC,CD25,,2   How can I do this using Linux text formatting commands? It is me who am to blame.    It is I who am to blame.      Which one is correct here? I am bewildered about these two sentences. When we canonically quantize the scalar field in QFT, we use a Lorentz invariant integration measure given by $$\widetilde{dk} \equiv \frac{d^3k}{(2\pi)^3 2\omega(\textbf{k})}.$$  How can I show that it is Lorentz invariant? Create bootable CD from ISO Is there a problem with defrosting meat on the counter? I generally defrost meat on the kitchen counter. A friend of mine suggested that this was dangerous and suggested that I defrost meat in the refrigerator. I am no biologist but it seems to me that as long as the meat doesn't get warm defrosting it on the counter should be safe.  Generally, I remove the frozen items from the freezer and place them on a plate on the counter in the packaging they were frozen in. When they are mostly / completely thawed I place the meat in the refrigerator.  Am I wrong and should I be defrosting in the refrigerator?