I just want to present to you, we need a new person just because you know who he is and I'm trying to take my life but for the rest we will continue on the user channel. It's my work. Yeah. No, no, no, no, no, if you are sending. So everyone and welcome to my talk. I am a senior software engineer who is maintaining the open SSL package for Reddit and price Linux and Fedora. So that makes me at least a little bit qualified to talk about this subject. So this is my agenda. So why we will be talking about the API changes in the new open SSL version. And so we start by the API changes. Why should we port to the new API and then we go through the actual changes and what what they mean for us. And also we at least a little bit talk about how to keep the backwards compatibility with the old API viewport. So why the API changes. The old API was extremely prone to API breaks. That means the reason is that it was actually prone to API breaks not only between versions but also depending on build time options. Here is the example. The problem is that the structure is public and there are parts of the structure which are conditional on build time options. And that really means that you don't have any single API. Various distributions can choose different build time options and then they are not compatible. So why do the changes because we need to bring better API stability in future without limiting internal changes. And so the new API in the new API the structures are opaque so the members are hidden. And this also brings like improved support for external crypto devices and tokens because the applications that use the API cannot have or cannot depend on the actual data inside the structures. And so the external devices can choose different implementations or things like that. So why should we port. Of course the main reason is that we should port if we want to access any new features from OpenSSL from any future versions. Because the old 1.0 branch is now only for bug fixing and there will be no features added. And there is no plan for any future release on this old API. For example the OpenSSL 1.1 X that means not 1.1.0 but 1.1.1 or what will be the name I don't know. So the next feature release should bring TLS 1.3 support. And I thought it's cool. Well not so much because it's a little bit tedious work but the support for 1.0.2 will also end by 2019 and then there will be even no bug fixes there.  So let's port. The first difference is in initialization before you need to call OpenSSL at all algorithms or SSL library in it. These functions you call one of these based on whether you want to use TLS stack or not. Now this is not needed except when you want to load configuration file which is actually useful because in the configuration file you can configure engines and other things. And also you need to explicitly initialize if you have some special requirements. These are mostly about low profile if you actually don't want to initialize something. This is example how you initialize if you want to load the configuration file. So the first one is equivalent to the OpenSSL at all algorithms and the second one is OpenSSL library with loading the configuration before you had to explicitly call out configuration. And the call is a little bit different now. There is also no need anymore to initialize log and threaded callbacks, thread ID callbacks because OpenSSL now by default uses the system native locking which for Linux is retards. Also clean up. Before there were multiple calls depending on what you called for example whether you loaded error strings and initialized EVP sub stack of the API. Now it's not needed because it's done automatically on exit handler. You can explicitly call it. You use OpenSSL cleanup function but you should never do that in library because the cleanup call will make the OpenSSL library unusable. There are some hacks in the library so you don't actually need to call the cleanup because if you load it for example by DL Open then it will take care of it and it won't unload. So it should not cause any leakage of data. So the hash is API. There is actually the old API or very old API is to use directly the actual algorithms. This is actually deprecated or it should not be used anymore for a longer time. It's not new in 1.1 but I have to extend it. You should use this general functions which is EVP interface and before you in 1.0 you used EVP MDCTX init and destroy or you could directly allocate the structure. Now you need to use of course because the structure is opaque. Now you have to use the new and free calls to allocate the handle pointer to the object. There is an unresolved problem and the serialization of intermediate hash state. This is usually not needed but there might be some special applications where you need this. This is being worked on for HMAX before you also directly allocated the HMAX structure. It's not possible anymore. There was actually no new and free or equivalent calls before so now you have to use them. Again there is no serialization and if you need to reuse the context you don't want to allocate it and allocate new one. You use HMAX CTX Reset. Here is some example. Of course there is no error checking in this example but basically all the calls now return positive value as a success or one and zero or negative values as error. Should the data be non-terminated and should the data you pass now terminate? No, no, no, no. It's actually no. So whether the data you pass to the update or function, whether it's not terminated, no. It's always the lines. You have to pass the lines. This is not no change from the one OIPI. For symmetric ciphers you again before directly allocated EVP Cypher CTX structure then initialize it by EVP Cypher CTX in it. Now you have to allocate it by the explicit new and free call for the allocation and again there is a reset call for context reuse. Here is the example. So basically you see that the new API is trying to get more regular than before. For public key cryptography the preferred interface is to use the EVP PK functions for the signing, public key encryption, decryption, signature verification. But you still need to work with the RSADS, ADH and EC key structures which are again made opaque. Now when you, for example, generate keys or manipulate the keys somehow, for example, initialize them, load them. There is some naming convention and that the set zero and get zero functions don't increment the reference counter inside these structures. And that means that the ownership is transferred for the set zero functions. If you, for example, set some public key component or private key component of the RSA key, then the ownership of the big num value is transferred to the EVP or the actual key structure. And in case of the get zero functions, you again just obtain the component and you have to, ownership is staying with the key and you don't reallocate it. Because it's a reallocate when the EVP PK or the key structure is reallocated. There is some detail that the public key components of the actual keys need to be set before the private key components. The library checks whether the public key is set when you are setting the private parts. In some cases it can be a problem, then you can work around it by setting zero big nums. Not now, because the now values are actually like unset the part. But this is of course a hack. You should not use the key by passing it to any function that needs the public key components in this case. There are the set one and get one functions which increase the reference count. And in this case you have to free the data that you pass into these functions or get it from them. These functions were already present in 1.0 API. For TLS, there are fairly minimal changes unless you do work things. Of course that means also that the misleading SSL API names are kept. It wouldn't make no sense to rename them because it would be just renaming of too many things. The most confusing name, the SSL V23 method which is like wildcard method for all the protocol versions supported. This was actually renamed because that was like most confusing and there were the most issues with this name. Again initialization without configuration is implicit but better it's to load the configuration files so use the initialization with loading the config. If you do some weird things you need to in some some accessors to the internal SSL data or TLS data that's not there in the API. Please report because missing accessors take care of SBOX and even in the 1.0 release the missing accessors can be added. When you report and you want to keep the compatibility with 1.0 API, the best way is to implement the missing calls in your application. You should try to keep the missing symbols hidden. You can either rename them via macros so the names are not confusing when you reduce source code. You can also use static functions if you for example use these calls only in single source code file. Sometimes of course ifs are inevitable but can be minimized their usage. Here's the example how you compare the SSL version number so for the version that I ordered in 1.1 you implement the functions in your code. So summary, porting is fairly trivial but it can be tedious because there might be multiple calls needed and so on. There are some examples on this URL, there are pitches for the 1.1 transition which are shared between various distributions. API manual pages are improving and upstream is now fairly responsive to documentation fixes and improvements. Thank you for attention and for eventually porting your applications or other applications to the new API. Any questions? You mean the sterilization? Yeah, unfortunately there is no way. I would say that there is a way to export the state of the crypto encryption for example. So this is considered bug but I would say that this won't get in 1.1.0. It might get into the next feature release because that would be a big change because it would require new APIs. How do PECs certificate directly into the application so that SSL doesn't load certificates from this but directly from the... Yeah, this is no change from before. You can basically, for example, there are many ways how to do that but one of them is for example that you have the binary data of the certificate inside some static... ...data and you load it via bio-memory, memory-bio which is initialized by... So you should see those same codepads to get from file... Yeah, you use memory-bio instead of file-bio. I have no idea. So the question was whether the other libraries plan to move to the new API. I have no idea, unfortunately, but I suppose that they won't, at least not in the near future. Maybe after some time when the advantages of the new API is clearly shown that they might. Definitely, there will be new APIs, new features. Yeah, so the question was whether there are any further plans for improvements of the APIs. I actually hope that there won't be any similar kind of API break in the future. Yeah, there can be some minor API breaks but I hope there won't be such a huge break in the future. And there certainly will be new APIs for new things. Like, for example, currently there is some thinking about better generalization of the certificates and private key-loading and things like that. Also including support for TPMs and so TPM keys and so on. So there will be improvements and I hope that they won't be incompatible backwards. The question was about the OpenSSL tool, whether there are changes. So there are some changes in the OpenSSL tool but they are much smaller than the changes in the API. Let's say I'm maintaining some old demon which uses 1.0 API and does it mean I won't be able to use TLS version 1.3 without migrating to the API? Yeah, so the question was whether applications of system demons which are using the 1.0 API, whether they will be able to use TLS 1.3, yeah, they won't be. There is no plan to backport TLS 1.3 support to 1.0. Even I don't think that as a downstream we would do such a patch because it would be basically a re-implementation. The thing is that there are many more changes in the library than the API changes. Especially on the TLS tag there are multiple changes which touch the whole implementation and backport would be basically a completely new implementation. Maybe it would be actually possible to take some backport from library as a cell code. I don't even know whether they follow the upstream OpenSSL changes of the TLS. Unfortunately I think that it's not possible to do it like shared library but you can copy. There are some efforts to create some like copy library which would help with this. You basically would include a source code. But I don't know if there is any definitive. We are full time. So thank you.


You may have heard about it. We at the ECB have decided to review our monetary policy strategy. So together with the 19 national central banks of the euro area, we will look at various issues, including in particular what we mean by price stability. But we also want to understand more about how other topics influence our job. Like employment, like climate change, like digitalization. The euro is a public good. The euro actually belongs to all of us. It belongs to you. So we need to hear from you in this strategy review. Stable prices help you make decisions in many aspects of your life, from saving to borrowing and from spending to investing. Please share your ideas and concerns with us. How can you do that? Visit our website and use our listening portal. I really look forward to hearing your views, which will help us serve you better.


Eric, I remember one of the first times we met. You told me you like my videos because I make it easy, healthy, and delicious, and delicious. That's right. Well, let me show you one with this air fried salmon. First thing we're going to do is slice this bad boy into two pieces. We're going to use sriracha as a binder. What I'm going to do here is I'm going to mix the everything bagel and the chipotle together. Have you ever tried that? No, let's give it a shot. We're going to mix about half and half, sprinkle generously on our salmon. Should look like that. Now we got it in the air fryer. We're going in at 400 for about eight minutes. Get a look at that. Look at that. Still got a little color to it. Let's see if it's ain't good. That's the yum yum, get you some. That part right there is my favorite part. That's the crunchy. Really good. Sriracha binder mixes chipotle and everything bagel. You'll think it's late. Yum yum, get you some.


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The Ending of One Piece. A monumental moment to tie up everything together. How will this epic series end? Hello, my Nakamitachi, this is Joygirl and today we're going to be discussing the ending of One Piece. The fact that we are discussing the ending, it becomes clear of how little time we have left to enjoy the world's greatest series, relatively speaking, and in that light, I do encourage you to subscribe to this channel if you have not already, so that we can make the most of our time whilst it lasts. But moving on from this slightly bittersweet emotional note, the reason for this video is due to a translation of a recent interview where Oda clarified a well-known fact, which is actually now a well-known myth regarding the ending of One Piece.  This idea became quite well-spread after Oda said something to this effect during an old interview, but this seems to have been misunderstood and is now confirmed to be untrue. According to a much more recent conversation on a TV talk show called Pirate King TV, which I might add is a One Piece-themed variety show which sounds really awesome, however, I not heard of it before. In any case, Oda clarified that the notion of him changing his ending, if a fan discovered it, is not true. And this fits with what we already know about Oda and the ending, which is that it came to him back in his college days, which he has been tweaking ever since. And there are a few things we should note. I don't think the myth came out of nowhere, and it's just something that was completely made up by fans. From my understanding, Oda did say something similar along the lines of changing the ending in an old interview, but it was more like, I avoid fan theories because if I found something that guessed it right, I would want to change my ending to the story. Or even if he did say that he would change his ending, he wasn't meaning it very seriously. Also, I want to thank Sandman AP and Arta the Library of Ahara for their translations of Oda's most recent interview and spreading the truth about his misunderstood statement. There is a point of difference between their translations, which is that according to Sandman, Oda said that he will never change the final chapter and panel just because a fan gets the ending right. As Oda declared as early as 1999, which was only a couple of years into one piece, he decided what the final chapter and panel to the series would be. And if this is the case, then it does seem that there is some room for the mangaka to make some changes to the story and how it arrives at its final chapter. Nevertheless, it is just highly unlikely that Oda would change something as monumental as the entire series' ending simply because a fan guessed it correctly. As someone as well planned out as Oda, who has dedicated literally half of his life to the series, it's hard to imagine a better ending than one he has already set up, and equally hard to imagine that someone would guess it 100% correctly. And even if they did, we wouldn't be robbed of the joy of watching Oda execute said ending because let's be honest, even if we somehow arrived at the correct ending, reading it unfold as Oda intended will be 100, 1000, 1 million times better than we ever could have anticipated. But this is still too big a challenge with all the unanswered elements left in the series. So then, what is the point of this video, you may ask? Well, the endeavor which we are undertaking today is to discuss the various loose ends which will need to be resolved before the series wraps up and how this will accumulate to a possible ending. So without further ado, let's get into it. We already know some tiny hints of how One Piece will end. For example, in the SBS corner of the most recent volume, SBS 97, Oda clarified that the approximate 5-year timeline for the ending of One Piece concerned only the juiciest part of the story. Which means that by the ending or at least the final chapter, Luffy and the straw hats will have most likely found the One Piece and have already defeated the final antagonists. But I guess this does still raise some questions of what the juicy final plot line entails. We are still facing off too Yonko in our current 1-0 arc and assuming that they are defeated, which in itself is still a large maybe, who will Luffy face off next and in what order? For example, there are the following individuals or groups that have been set up to be Luffy's foes. Blackbeard is the first that comes to mind. As the newest to join the rank of Yonko and has been portrayed as the polar opposite of Luffy, this pirate whose actions contributed to the death of his brother will be likely facing off Luffy in the future. There is also the world government, particularly Sakazuki, whose philosophy of absolute justice makes him hell bent on capturing our straw hat captain, as well as the mysterious figures of the Gorosei and Im, whose physical strength is not yet known, but they still seem to pose a huge threat to Luffy from reaching his goals. Even aside from these big names, there may still be even others whom Luffy may have to fight, such as against his fellow worst-generation member's most notably kid, as he has been portrayed to be Luffy's rival. And as we have discussed in a previous video, maybe even shanks will end up becoming Luffy's enemy. Just kidding. Maybe? In all seriousness, there are also many mysteries that we will still need to resolve. What is the meaning behind the will of D? What or who is the third ancient weapon? What happened during the void century? And of course, what is the one piece? And in light of the recent clarification made by Oda, it seems the mangaka has some general ideas on how to resolve all of these remaining battles and questions, even though some details may change. But even on top of this, there are so many other remaining loose ends which need resolving. Before we can continue, however, anime-only fans will have to skip ahead so as to avoid any spoilers. Prior to finding the one piece, our straw hat crew will need to find the Road Polyglyph, which Roger found on Fishman Island, but this has since disappeared, so where is it now and who moved it? And speaking of Fishman Island, what is the history behind the Noah and what will become of the prophecy which foresaw Luffy destroying their island? What is the meaning behind Joy Boy's apology on the historical ponderglyph still in the sea forest? Which brings us to questions concerning mystery characters. Who even is Joy Boy? Who is weevil, really? And what role will he play? When will we meet Vega Punk? And how did Dr. Kuroha know about the Will of D? And how will the revolutionaries play into everything going on currently? Aside from characters, there are other mysteries such as who does the giant straw hat we saw at Marijua belong to? What is the secret treasure of Marijua, which Dr. Flamingo is aware of? Does the all-blue exist? And if so, where is it? What is the voice of all things and where does it come from? What was Zunish's crime? What is the history behind the treasure tree Adam and sunlight tree Eve? What is the origin of devil fruits? As well as recent events which we have yet to see the result of such as what happened at Baltigo, what happened following the reverie, and what happened to Moria. And so many more because there are still other questions on top of these. Smaller mysteries such as deeper backstories behind characters which may be as simple as why Roger became a pirate and here is the manga spoiler to any anime-only fan so please skip ahead to the time on the screen. Or the truth behind Frank is allegedly pirate parents, but nonetheless, one's hinted to be answered in the future. And just because of the sheer volume of remaining mysteries, I do believe that many of these will be resolved together, most likely at the time of the discovery of One Piece which will reveal the truth behind many other questions, but we haven't received many more confirmations regarding these mysteries, but one thing we do know is that there will be a major war, one to make Marineford look cute. Odour has said on multiple occasions that following the One-O-Ark, we will be witnessing the greatest war the world has ever seen. So in my head, for a war to be greater than Marineford and greater than the God Valley incident, unless of course Odour was referring to drawing a flashback to the massive historic incident, this final war will have to be one which involves the Straw Hats, the Blackbeard Pirates, the World Government and the Revolutionaries all as the primary players in a huge chaotic battle where there are multiple conflicting interests and goals. And I wouldn't limit it to just these groups because I can see many different groups and individuals also participating as part of various alliances in this war. However, this does still leave us with a big question of how will all of this play out? And while that is still uncertain, I can at least say that if we can use Marineford and One-O as a measure of how epic this final war will be, then I have full confidence in Odour to deliver on this final saga. And as there are so many questions still remaining unanswered, this also does mean that there are various discussions still to be had, which personally makes One Piece all the more enjoyable. And on that note, please let me know of which remaining mystery you are most eager to have answered and please also let me know of what questions I didn't mention in this video by leaving a comment below. Please like and share this video if you enjoyed watching and please also subscribe as discussing the many questions of One Piece is truly my favourite pastime and I would love for all of you to join me in that journey. This is Joygirl and I'll see you again soon.


Hi, I'm Naomi Yamamoto. I'm the Minister of Advanced Education for the Province of V.C. Today is International Youth Day. The United Nations has dedicated the theme this year to change our world. That for me means the use of social networking tools. And I'm sure like many of you, you're on Facebook and Twitter. I'm a Twitter fan myself and I'll hope to schedule a tweet that you'll receive today. I hope that you use the social networking tools to enrich your lives and your studies. I use Facebook and other social media to plan my group projects when I have projects to do as a group for classes. That really helps it, I don't know, feel like it's less work and more play. It affords you a lot of opportunities that you might not have had or might not have known about just because of how daunting and how spread out the different social networks I mean in the traditional sense at a university are. Professors are really catching on to social media so some of them will communicate with students via Twitter so you can tweet them a question and they'll tweet you back an answer or they'll have online office hours where you can ask them a question using instant message instead of going to see them in person. I mean not only do students tell you about personal or social events but also events that are academic like I guess side team or class events that are happening and because it's online and it's on something like Facebook you can see that your friends are joining and it becomes more than just some academic event put on by the campus. Obviously these people exist on campus but if you're able to join Facebook groups or Twitter is an excellent way to keep in touch with people or finding people that you might not otherwise know about but who might have very similar goals and aspirations as you do. With social media it's such a global thing if you just put one thing on Twitter it can easily be trending like halfway across the world to somebody over there. So it's a great way to spread ideas and it makes it easier for students to express their opinions and show what they're really feeling and then gain some support for that. I think social media is a tremendous place to organize and it's a tremendous place to share information. It's not necessarily the place that fosters strong social bonds that are necessary to actually change the world but once you're committed to doing it social media is the first place to go because you have it's easily accessible and you just have a wide network of people to spread your message to. I think students can use social media tools to change the world to sort of create their own media and their own news. I think students are becoming their own media sources and less and less people are going to mainstream media and having to worry about if their messages are objective or not. Twitter and Facebook is not limited by geographical boundaries or social ethnic backgrounds or anything like that so it's just another medium for communicating and gives us greater access to people who wouldn't be exposed to our ideas otherwise.


I can't love you down the telephone line Please stop me going out of my mind When the sugar don't make my coffee taste sweet Send some home for me Little honey Little honey, honey Little honey, little honey, honey No substitute I'll fix For the feeling of your sweet lips Where my engine won't start hot while my heart Send some home for me Little honey Make a sane man crazy Make me come find you Sweating up these blues When the whiskey test sign has lost its power Send some home for me Little honey


everyone. It's your girl, Kenty here from Layton Sports Cards and I'm ripping 2022 Panini Spectrum Football Hobby 4 box half case break number 14. Pick your team. Got Barry G. Falcons, Kristen S. Packers, Daniel D. Bills, David F. Chiefs, Dennis F. Chargers, Dylan J. Titans, Edgar B. Ravens, Edward P. Cowboys and Dolphins, Felix N with the Cardinals, Panthers and Browns, Henrik J. Broncos, Jacob H. Giants, James C. Niners, Jeffrey H. Steelers, Jeremy H. Patriots, Jonathan L. Jets, Jordan T. Washington, Justin S. Colts, Max K. Bangles and Vikings, Mitch M with the Bears and Texans, Mitchell B. Jaggs, Morgan L. Seahawks, Taylor W. Rams, Tim S. Raiders, Tyler K. Saints, Valerie P. with the Bucks, thanks for closing us out, Valerie. Walter S. Lyons and William L. with the Eagles. All right, best of luck, everyone. Here we go. Again, we've got number 15 up next. All teams are on sale. Let's get those, let's get those teams filled up. There is the link right there for you. Here we go. It is Victor. I like the look. I like the look of them. Oh, it's really nice. Let's see a Nebula, though. A Nebula or a Gold Vinyl. It's fine with me. Sins of Falcons for Break 15, says Luke. Hold some nice ones out of the last one. Let's see what we can do. What do we have here? How about a Tori Holt on the Super Bowl insert auto? There you go, Rams. Tori Holt. We have a Kevin Harris on the Silver Rookie and Sunday Spectacle 56 of 75, Austin Eckler Chargers. Man, this guy was like heaven sent from my fantasy this year. Oh my gosh. Oh, Eckler? Yeah, he's a monster. We have John Lynch for the Bucks that is 54 out of 75 and a Jameson Williams rookie Lions 98 of 99. We have Kyle Pitts on the base. Trilance 52 of 75 on the catalyst patch. 49ers and two out of nine. Wow. Illustrious legends. Jerry Rice. Boom. There you go, Niners. Wow. Who's got San Fran? There you go, James. James C. And that is on card too. Very nice. Two out. Jeez. That's sick. We'll mag that one for you. Congrats on that, James. Yeah, Holt and Rice. That's true. We have a de-hop on the die cut. Fiti of Fiti. Wavelength, Josh Allen. That's sick. 53 of 99. I like that insert. That's really cool. Yeah. And on card. 67 of 99 monumental memorabilia of Breeze Hall. Uh-oh. Go away, Luke. Go away, Luke. 20 of 20. Dual patch auto drink London and Desmond Ritter for the Falcons. Wow. Barry G. There you go, Barry. Three color and two color. That's sick. That is really nice. And bookend too. Wow. What a box. We're going to mag that one too. There you go, Falcons. Whoa. I love these. Yeah, me too. Wow. Joey B on the solar eclipse. That's nasty with the bangles orange too. That is sick. Sinsenasty Max K. Wow. Wow, wow, wow. That is so cool. Very nice. Were these considered K-Sinses here? It doesn't say anything. That's the first one I've seen. And we opened a lot of K-Sinses. Yeah, I was about to say I didn't see any of these yesterday. It doesn't say anything about them on the checklist, but I know they're pretty short print. I don't see anything. Do you think it's like, like 75? This? Yeah. Um, yes. Wow. That's sick. We're going to mag this one for you, bangles. And we've got, wow, what a box. Two out of four building blocks. A ModSauce Gardener. This is a heck of a box. Nice patches on there too. All three colored. What's up, T-Po? I don't know. I don't know. We've got 47 out of 75 RPA Skymore on the Radiant Rookie. Chiefs? That solar eclipse, Joe Garo is one of my favorite carvings. And speaking of the devil, Joe Scheiste back at it again. 36 out of 60. Wow. That was one heck of a box. All right, box number two. My favorite football player. Current or like all time, T-Po. What's up, Joshua? We've got Danny Dunn. One out of 60. Nakobi Dean base rookie. We've got 31 out of 75 Max Impact. Justin Herbert, he's going to have to make a Max Impact tonight. Beat those Jags. Look forward to watching some football later. 41 out of 60. Nice. Christian Watson for the Packers on the two color. G10. It's my favorite football player of all time. I'm probably going to have to go with your own bettas. Wow. That is sick. It's a great player. All right, this cart is pretty damaged. It looks like it got caught on, what's that thing called? That closes the packs? Oh, we won't yet. Yeah, the crimp. Or that's just what it's called. Yeah, we're going to send a box here with the Seahawks on this one as it's clearly damaged there. Make sure that we go at Seahawks. Then we've got a Kyler Murray, 90 of 99. Hopefully he comes back. Good as new for next year. ACL injuries are tough recovering. 45 out of 99 Max Impact of Justin Tucker. Justin Tucker. And we've got for the Bears aspiring. This is five out of five. Bealus Jones Jr. for the Bears. Nice one there. That is the Psychedelic parallel. Three color patch there. There you go. Bears. Tom Brady or Joe Barrow. Nice. That's one of the one-to-one manning for the Colts. Shoot. Let's do it. We've got Christian Watson on the green die cut, 30 of 30. Josh Allen base. Four out of 25. Alec Pierce for the building blocks. Colts. Two color on each one of those. Nice. And we've got for the Browns. That is trying to do number 88 out of 99. RPA David Bell. Inspiring RPA. Barry Sanders all time for you, Victor. That's a good one too. We have Dak Prescott, 23 out of 40. That's going to be a good game tomorrow. I'm looking forward to that one too. Rookie Spectress of Tarjan Davis Price. Niners. Sunday Spectacle one out of 75. Jonathan Taylor for the Colts. And we've got how about 18 of 20. Rex Ryan. Well coach card here. That's different. There you go Jets. Rex Ryan. 18 out of 20. Nice. I think he became more liked after he was a coach. Yeah. He's a pretty good analyzer actually. Five teams left in this next break. Number 15. Cardinals, Panthers, Browns, Broncos, Texans. There is the link. All those teams are on sale. Let's fill her up. Let's get a one on one manning. Do you imagine a one on one manning and that Jerry Rice in the same case? Can't be nutty. Got Justin Fields, 88 of 99. And Josh Jacobs, man. We've got 98 of 99. Peyton Manning. This is in a Broncos uni.   Sorry about that one, Justin. We pulled a manning, but it's the Bronco. Player worn on that one. And uh-oh. The Zap Attack, 66 to 75. RPA, Bailey Zappie on the four-colored patch. It's a nice patch for a Patriots. Bailey Zappie. There you go, Pats. Were you on the Zappie train? No. Plays like no. We've got Tony Romo on Pro Bowl uni on that one. One of 15. I was like, what jersey is that for a second? And we've got Type 1 Thornton on the Silver Rookie. Oh, this is gonna be cool. Seven of 10. Max Impact Hines Ward. That's really cool. He was another one. I was, it's him, Jerome Betis and Troy Palamallo. I like my top, like, all-time favorites. Nice one there, Steelers. And 68 of 125. Pierre Strong Jr. on the RPA Patriots. Patriots heating up here. I have an authentic Reebok home jersey. Jerome Betis that's signed. Yeah, we've got Rich Gannon, 54 of 60. Raiders. Derrick Johnson base. Over, we've got 900 points. We'll be random off. 40 out of 75. Monumental memorabilia of Romeo Dobbs. She's Zappie that was hit that had a dolphin's name patch. No? Really? No, I didn't see that. I mean, I respect Big Ben, but he just, he wasn't always my favorite. Nope. Wes Welker on the Silver. 15 of 75. Rookie Aura of Traylin Berks for the Titans. This one is going to be one out of five. There you go. Tennessee. Nice one there, Titans. Again, that's the psychedelic. I like the psychedelic in the background. We've got 78 of 99 building blocks. Traylin Berks. Back to Back Berks. And Colt 70 of 75. Ricky Otto Jelani Woods. Jelani Woods. All right, last box here. Last box. And we've got two more half cases of this. Then we're going to move into some mosaic. Again, all teams on sale for this next break. Let's get that filled up. That breaks right after this. Down to two teams. Cardinals and Browns. Thanks, Tim. Best of luck. We haven't pulled any corrals yet, so hopefully we hit you. A big one. Tyreon Davis Price. Rookie 17 of 99. Sam Fran. Rookie Aura. John Menchie III. Texans. That is five out of 50. 88 of 99. I have a little bow nose on the Epic Legends patch. That is player one. And Closer Mojo here. 14 of 15. I loved watching him play too. Mike Allsdott. That's sick. There you go, Bucks. He's always had a very nice signature too. Very nice autograph. There you go. What's up, Donald? How's it going? Doing good. Speaking into existence. First rave with me. Well, best of luck, Tim. Let's see if we can get you. We've got Tyler Linderbaum. Die cut 17 of 50. Ravens. Randy Moss for the Vikings. Got 90 of 99. Championship gear. Steve Young for the Niners. And we've got 25 of 35 aspiring on the green ink. A Mod Sauce Gardener. This is sick. I love the green ink to match the jets. Isn't that sick? Yeah, that's filled. Nice one there. Everything about that color match. That is green patch, green card, green ink. That's awesome. Master. We've got Nick Chubb. That is two out of eight. Nice one there. Sorry, those numbers are so hard to read on these for so reason. This is the Neon Splatter. That's a cool one. I haven't seen that yet. Chris Johnson Silver. 71 of 99. Brilliant. Michael Pittman Jr. for the Colts. And throwback here. 46 out of 60. Cardinals. Jake Palmer. A little pulmarotto there. Zona. Last pack. We have the Marcus Ware. 20, 28 of 60. Dallas. Ooh, chili sounds good. Givante Williams Base. Nine out of 10. Brilliant patch. Zeke Elliott for Cowboys. And the last set of the break. Going to the Pats. 82 of 99. Taekwon Thornton RPA. All right. Again, we do have 900 points to random off. Give me one second to set up the random here. All right. Again, top team after the random will take home the 900 points. Good luck, everyone. Swirl our dice. Ten times. Here we go. One, two, three, four, five, six, seven, eight, nine, 10. Going to the Bills. Bills taking home the 900 points. Do a quick recap here. You got Bills, right? Yes, ma'am. Okay, cool. Zeke Elliott Cowboys. Pittman Jr. Colts. Steve Young Niners. Bo Jackson Raiders. Traylin Burks Titans. Romeo Dobbs Packers. Hindsboards Steelers. Pink Manning. Broncos. Jonathan Taylor Colts. Alec Pierce Colts. Justin Tucker Ravens. Justin Herbert Chargers. Brees Hall Jets. Amon Sos Garner Jets. Tralance Niners. Austin Eckler Chargers. Taekwon Thornton. These are all of our autos. Patriots. Jake Plummer Cardinals. Amon Sos Garner. That's Zeke on the green for the Jets. Mike Allstatt Bucks. Jelani Woods. Colts. Pierre Strong Jr. Patriots. Bailey Zappie Patriots. Rex Ryan for the Jets. David Bell. Browns. Belis Jones Jr. Bears. Christian Watson Packers. Skymore Chiefs. Tori Colts-Rams. And then our Macked Cards. Such a sick. I love this. Solar Eclipse with the orange. The Bengals. Joey B. Joe Sheissy. Joe Burr. We have the Dual Patch Auto. Desmond Ritter and Drake London. That is 20 of 20. Congrats Falcons on that one. And then the illustrious legends Jerry Rice on card auto. Two out of nine. Heck of a half case there. Thanks again everyone. We got more coming right up.


So number theory will, just a number of concepts about addition, subtraction, multiplication, division, which you all know, but we'll adjust it a little bit from what you, what you commonly think about and then see a few equations which were useful and some other properties of numbers which are useful when it comes to public key cryptography. We'll start very simple. We'll talk first about divisibility in prime numbers and I think you learn even in primary school now that we can talk about one number divides another number. So we can say B divides A if there's some M, some integer M such that when we multiply by B we get A. That is if we can divide and there's no remainder we say B divides A and we talk about divisors. So we can say B is a divisor of A if we can take A and divide by B with no remainder we say B is a divisor of A. Sometimes we may write that in a short form of B vertical bar A. So we know about divisors. We can then also, given a number has a set of divisors we can talk about the greatest common divisor amongst two numbers. So A has a set of divisors, B has a set of divisors, what's the greatest of that set? The greatest common divisor or GCD and there are algorithms for finding the greatest common divisor. We'll go through a couple of short examples and a new thing that you may not have heard of or you can't remember. If two integers A and B have a greatest common divisor of one we say those two integers are relatively prime. A couple of examples. What are the divisors of 16? It's that simple. Two, is there a number before two? One, alright. We can divide by two. We cannot divide by three. Okay, so simply the divisors. Easy. Twenty-four, we can divide by three. Four by six, three by eight, two by twelve, one by twenty-four. So that's the divisors. Greatest common divisor of sixteen and twenty-four, well the greatest value there is eight. That's all. We've done this many times in high school, primary school maybe. Fifteen divisors, one, three, five and fifteen. Greatest common divisor of sixteen and fifteen is one. If we look at those sets, one. So we say there that fifteen and sixteen are relatively prime. They're not prime numbers, we'll see them later, but we say that those two numbers are relatively prime to each other. Fifteen is relatively prime to sixteen and vice versa. Because the greatest common divisor is one. And another thing you probably know is that we can often write, well, that's all we've gone there. The next thing we know is about prime numbers. So instead of writing the divisors, we often think about the prime. So what are the primes on the slides? What's a prime number? Any integer greater than one, if and only, if it's only divisors, a plus or minus one and plus or minus p itself. It can be divided by one and itself only. And we can say any integer can be factored into the prime. So the divisors can be written as a set of primes multiplied together. So the divisors, we also say the factors, the factors of fifteen, one, three, five and fifteen. So we can talk about prime factors of a number. So some integer a can be written as multiplying a set of primes together, prime numbers only. Sometimes there will be multiple primes multiplied together. So we can talk about the prime factors of a number. What are the prime factors of sixteen? How can we write sixteen as multiplying primes together? And let's forget about one. So what's the first prime? Two. Two is a prime. It can only be divided by one and itself. One, our definition we consider greater than one. So two, two to the power of three, two to the power of four, sorry. Sixteen is two times two times two times two. So its prime factors are two and so the factor is two but it's four times there. 24 is what? What are the prime factors of 24? One's not, six is not prime, two and three, two to the power of three is eight times by three is twenty-four. So two and three are the prime factors. We can say 24 is, so in fact what we can generally do, we can do it for fifteen as well, three and five are the primes there. In general any integer can be expressed as multiplying primes together and sometimes we just list it in a general form such as here. And there's a mistake here, it's not P2, it should be P1 as the first subscript. That is some integer is prime one to the power of some positive integer, zero, zero one or above times the second prime, so prime one is two, the second prime is three, the third prime is five and so on. So we can multiply the primes and give some exponent and that will create any integer. So we'll often come back to prime factors. Some number can be written as multiplying primes together or the other way, given a number we can find those factors. What are the prime factors of a number? It will come up as an important problem. What's the prime factors of twenty-two? Two and eleven. Hundred and forty-five, there are two as a hint, five and five times what? Twenty-nine, five and twenty-nine, so a little bit more work in the head but we can find prime factors and there are algorithms for doing so and we'll come back to it but an important point for security is that for a very large number, not one hundred and forty-five but maybe a number with one hundred and forty-five digits, a very large number, it takes a long time to find the two prime factors. That is if we multiply two primes together to get a very large number, given that large number it's hard to work out what the primes are and that will be a feature of some of the encryption algorithms. What are the primes? You should start remembering some of them, not all of them of course but for the simple problems you can remember the first five or ten, so there are lists of primes you can find, you'll quickly remember them. Alright, easy, we can divide numbers, we know something about primes, let's do some arithmetic, some addition, subtraction, multiplication and division. Of those four operations, well are there any other operations in arithmetic? Maybe we can expand exponentiation, we add numbers, subtract, multiply, divide, raise to the power, exponentiation, A to the power of B and the opposite of raising to the power, logarithm. The logarithm of a number is the opposite of exponentiation. So think of those six operations and you know they're related so they all go in pairs. Subtraction is really addition. Subtraction in our arithmetic is adding a negative number. Alright, to subtract a number we add the negative. The negative of a number we will say is the inverse, the additive inverse to be specific. We'll list that in a moment but we'll introduce and say if we add the negative of a number we get subtraction. So subtraction is just addition where we add the additive inverse of a number. And similar, multiplication and division go together. Division is just the inverse of multiplication, that is to divide we multiply by the inverse of the number. Ten divided by three is ten times one divided by three, one over three, the inverse of three. So we can talk about multiplication and division is just multiplication times by the multiplicative inverse. So we'll introduce that and similar, their exponentiation and logarithms go together. And that will become important again when we look at cryptographic algorithms. So I know you can do all those six operations. Now we're going to do it mod n, modular arithmetic. And most of it's the same as your normal arithmetic but we just mod by n at the end. Well we know what mod n means, well sometimes it's not so clear. Do we have a positive or a negative number? So we will say a mod n, a is an integer, n is a positive integer, is the remainder when a is divided by n. We refer to n as the modulus. We can say that two integers a and b, we can think are equivalent. We call it congruent modulo n if those numbers mod n are the same. Twelve mod ten, two mod ten, twenty two mod ten are all congruent modulo ten. They're all the same in mod ten. Two mod ten is two, twelve mod ten is two, twenty two mod ten is two. So we say that congruent modulo n and we'll try and write that as three lines in the equals. So we'll think of mod n as an operator, some number mod n. And it returns an answer in the set of zero up to n minus one. So mod ten, the answer was always between zero and nine. No negative values, always positive, zero to n minus one. So we'll sometimes write that as the set z n. When we perform modular arithmetic, we do addition, subtraction, multiplication, division, exponentiation, logarithms, where the answers or the operations are within the confines of that set z n. So we'll go through some examples of those six operations mod n. Start simple. Just yell out the answer so we can move along quickly. There's no need to follow the lecture notes, just some examples today. And to go through these examples, how about the five or six students at the back come down to the front so we can all here come down to the front seats. Let's turn off these two monitors. Plenty of spare seats down the front. Then I can hear you when you tell me the answer. We know 13 mod ten, the remainder is three when we divide by ten. 13 divided by ten, the remainder is three. We can say 13 is congruent modulo three, congruent modulo ten, they are equivalent. In mod ten, 13 is the same as three. In the set z ten, the answers go up from zero to nine. So when we're doing mod ten, the answer is always between zero and nine. So let's do some addition. And in these examples, we'll do everything in z ten. Everything's mod ten. I'm not going to write the mod ten. So some examples, addition, what's four plus three? Easy. Four plus seven, not eleven. Eleven mod ten. So everything's mod ten. So in addition, it's easy. We just do our normal addition and then mod the answer by ten. So four plus seven is one. So addition in modular arithmetic is very simple. It's natural for our perspective. We just mod the answer by our modulus n. Subtraction. Well, let's do subtraction in, over here I'll write the normal, normal arithmetic. That is our no modulus, just normal arithmetic. In normal arithmetic, what's seven minus three? Four or seven plus minus three. Subtraction is just addition, where we add the negative number. So seven minus three is really seven plus minus three, where minus three we say in our normal arithmetic minus three is the additive inverse of three. The additive inverse of a number is the number we add to it such that we get zero. What's the additive inverse of three? Three plus minus three gives us zero. So the additive inverse of three is minus three. This is our normal arithmetic. Additive inverse since, and I'll write the sign as well, plus three, add minus three equals zero. Therefore, just in brief, the additive inverse, A i, additive inverse of three is minus three. The point being is that we don't need to do subtraction, we just do addition. Subtraction is addition plus the additive inverse. So we need to find the additive inverse to do subtraction. That's normal arithmetic. Let's return to Z ten, everything mod ten. What's the additive inverse of three? Write it down. Additive inverse of three. This is in Z ten. Z ten means the numbers, the answers between zero and nine. So don't give me a negative answer. What's the additive inverse of three? What number do we add to three such that we get zero? Same as here in the normal arithmetic. What number do we add to three such that we get zero? Negative three. Three plus seven equals zero in mod ten. Three plus seven is ten, mod ten. So yes, the additive inverse of three is seven because in mod ten, of course, I haven't written. Three plus seven, mod ten is zero. So we can talk about an additive inverse. If we can find the additive inverse then we can do subtraction. Another one. What's four minus seven? Four minus seven is seven. Four minus seven is four plus the additive inverse of seven. What's the additive inverse of seven? What number do you add to seven to get zero? Well, actually we just did that. Three, but it's the opposite. That is, the additive inverse of three is seven. Three plus seven equals zero. Therefore, seven plus three also equals zero. So the additive inverse of seven is three. Four minus seven is four plus the additive inverse of seven. Subtraction is just addition where we add the additive inverse. We know the additive inverse of seven is three. So it becomes four plus three. And now we just have normal addition. Mod ten, seven. Four minus seven is seven. So in modular arithmetic, addition is natural to us. That's easy I think for most people. Subtraction is a little bit different from how we think. We need to find the additive inverse. But it's in fact the same principle of our normal arithmetic. Two minus six, find the answer. Two plus the additive inverse of six. Six plus four equals zero. So it becomes two plus four equals six. Five minus three. Five plus the inverse of three. Five minus three. We found that before. The additive inverse of three is seven. So it becomes five plus seven. Two. Everything's mod ten. Five minus three is two. That makes sense. Two minus six is six in mod ten. Any questions so far? Yep. There are no negative numbers. All the set z ten, mod ten means answers are always between zero and nine. So this is the difference between our modular arithmetic and our normal arithmetic. We define everything is within that set of zero to nine. So there are no negative numbers. Back to the additive inverse. Let's consider all possible values in our set. We're in z ten. All possible values. A can be zero. The other possible values that we can operate with is z ten. What are the additive inverses? Zero plus what gives us zero? Mod ten. These are easy. So in this case, every number in our set has an additive inverse. And in general, in any set and any mod n, every number has an additive inverse. Doesn't matter at z ten, z eight, z one million. That is whatever the set, whatever the modules, will always have an additive inverse of each number. Meaning we can always do subtraction. Because subtraction is adding the additive inverse. Any questions before we move on to the next operations? Addition and subtraction. Easy so far. What's next? What's the next operation? If we can add numbers, we can subtract numbers. We need to multiply numbers. Multiplying is what? Adding multiple times. So there's a relationship between multiplication and addition. So let's look at multiplication and then division. And let's do it in z eight. Just to make it more fun. Three times two. Are you sure? What's the answer? Six. Okay, good. Three times four. Three times four in z eight is four. Remember everything's mod eight. Remember everything, we're just omitting writing the modulus. And the same with three times two equals six. Mod eight, we still get six. Multiplication is easy in modular arithmetic. It extends upon addition. We just do naturally our normal multiplication mod by our modulus n. What's the next operation then? We can do multiplication. Next is division. What is division? Multiply by the inverse. Is that? In our normal arithmetic, we say ten divided by three is ten times one divided by, one over three, we can think. The inverse of three. Not the additive inverse, but we say the multiplicative inverse. So we introduce the multiplicative inverse. Again, this is our normal arithmetic. What we know. Division eight divided by three, in fact, is the same as multiplication, eight times one over three. So we can convert division to multiplication where we say since three times one over three equals one, therefore the multiplicative inverse of three is one over three. This is our normal arithmetic. No modulus yet. The multiplicative inverse is defined as the number we multiply by such that we get one. Three times one over three gives us one. So three and one over three are the multiplicative inverses of each other in our normal arithmetic. So division divided by three is eight times the multiplicative inverse of three. Eight times one third. So that holds also in modular arithmetic. Back in Z eight, five divided by three, what do you get? Find five divided by three in Z eight. Fine calculator may not have this answer.  Five divided by three, five times the multiplicative inverse of three. Let's do the full steps. Division is multiplying by the multiplicative inverse. The multiplicative inverse is the number that we multiply by such that the answer is one. What times three gives us one in mod eight? So three times three gives us nine. Mod eight gives us one. So the answer is three. The multiplicative inverse of three is itself because three times three equals one. Therefore, five divided by three continuing is five times the multiplicative inverse of three, which is also three, which is seven. Five times three mod eight. Remember all mod eight at the end. For multiplication and addition, just do the mod eight at the end. The easy to solve. Five divided by three equals seven. Any questions before a couple more examples? So think about additive inverse, the number we add to get zero, multiplicative inverse, the number we multiply by to get one. Subtraction is adding the additive inverse. Division is multiplying by the multiplicative inverse. Yes, this. This is from our definition. We want to find the multiplicative inverse of three. What does that mean? Similar to our normal arithmetic, it's the number that we multiply by that gives us one as the answer. Everything's mod eight, remember. Something times three equals one in mod eight. So something times three mod eight equals one. What is that something? It's actually three. It's itself. Three times three mod eight gives us one. So we say three is a multiplicative inverse of itself. Yes. What other answers are there? Do we need to define this though? Right. The answers, remember, also always within the set z eight. So three times three is one mod eight. There's another answer which is, can I think of it, something times three mod eight. Yes, there are numbers greater than eight that will give the answer, and I can't think of one right now, but there are. But if those numbers are mod by eight, it will come back to three. So the answers or the numbers are always within the set z eight, zero to seven in this case. Three times 11 is 33, mod eight is one. Right, you found one. 11, three times 11. But what is 11? It's three in z eight. In mod eight, 11 and three are the same. They're congruent modulo. Okay. So yes, three, 11, probably 19 and so on. I think when we mod by eight. The reason we're doing this is we're working our way up. Addition and subtraction are not so, we don't see them so often in cryptographic operations, but we've moved from addition and subtraction up to multiplication and division, because they are based upon addition and subtraction. And then the next step is the exponentiation and logarithm, and we'll see that they are commonly used in cryptographic operations, public key cryptography. So that's why we need to learn about the principles here. Do you need to show the operations? Yes, sometimes there's a small question solved by hand, one of these, but usually not such small ones like addition and multiplication. Six divided by four, find six divided by four. There are no fractions, there are no negative numbers. The answers are between zero and seven. The integers between zero and seven. How do you find the answer? Well, six times the multiplicative inverse of four. So we need to find the multiplicative inverse of four. Something times four, mod eight, we'll write it down in full here, we're still in z eight, equals one. Where that something is zero to seven. No answer. There is no such number. So four does not have a multiplicative inverse. There's no number between zero and seven when we multiply by four and mod by eight and get one. So there is no answer. Not every number has a multiplicative inverse. Every number has an additive inverse, but not every number has a multiplicative inverse. Four does not. So we cannot do that. There's no answer. There's no multiplicative inverse of four. Staying in z eight, so let's list them. What are the multiplicative inverses of the eight numbers? Zero times something equals one. There's no such number. One times something equals one. Nine is not in our set. Remember our answer is zero to seven, so one times one. One is its own inverse. Two times something equals one. Well, we can't get an odd number when we multiply by an even number here, so no such answer. Three we saw is itself. Four we don't have an answer. Five, five times five. Mod eight is one. Six won't work when we mod eight. Seven, seven times seven. Mod eight is also one. In z eight, it turns out one, three, five and seven are multiplicative inverses of themselves. But it's not always the case. Let's go back to z ten to show another case. Zero cannot be multiplied by anything to get one. This is mod ten. One will always be a multiplicative inverse of itself. Doesn't matter what the modulus is. Two, two won't work. Two times something, mod ten gives us a remainder of one and even multiplied by something else will always give us an even number. We can't get a remainder of one. So the even numbers are not going to work. Three, three and seven. Three, seven, twenty one. Mod ten is one. Five, no. There won't be a remainder of one when we mod by ten. Seven, we've got that one. Nine, nine, nine's 81. So just a different set of multiplicative inverses. In z eight, they were inverses of themselves. But that doesn't have to be the case. As we see with z ten, three and seven are multiplicative inverses. And another important point, not all numbers have a multiplicative inverse. So we can't divide by just any number, only by sum. Which numbers do have a multiplicative inverse? When the remainder is one, if the two numbers are relative, if the number is relatively prime with the modulus, it will have a multiplicative inverse. What's the greatest common divisor of one and ten? One. One and ten? The greatest common divisor is one. Three and ten? The greatest common divisor is one. Seven and ten? One. Nine and ten? One. Eight and ten? It's not one. Six and ten is not one. Five and ten is not one. That is, a number has a multiplicative inverse in mod n if that number is relatively prime with n. Relatively prime means the greatest common divisor is one. That is, we're not listing them all, but the greatest common divisor of one and ten, we can list them all three and ten, seven and ten. They're all one. That is, all those numbers are relatively prime with ten, and therefore they have a multiplicative inverse in mod ten. So when we want to do division, we need to know which numbers have an inverse, and we can find them by checking if they're relatively prime with a modulus. We can do addition, subtraction, multiplication, division. We can't do division in all cases, only when we have a relatively prime number to the modulus. Let's come back to the slides, see what else we've said. The properties or the laws of our normal arithmetic, there are similar laws of modular arithmetic, and some of them are obvious. The three, for example, a plus b mod n is the same as a mod n plus b mod n, all mod n. And the third one is especially useful, a times b mod n, we can expand out, the a mod n times b mod n, and then all mod n. Remember, if you mod a number mod n multiple times, if you add a mod n mod n, all mod n is the same as a mod n. If you add another mod n at the end, it makes no difference. 12 mod ten is two. Mod ten again is still two. Mod ten again, it's still two, so we can add mod n's at the end, making no change to the answer. This third one is sometimes used to simplify multiplication. When we have large numbers, we can expand out two large numbers mod n multiplied together, gives us a very large number, then mod n. An easier way to solve that is to take those two large numbers, a and b, and mod n first, which makes them smaller numbers. Multiply two smaller numbers is easier, and then mod n. So that's a common way to both in our heads do calculations of multiplication, but also computers can use algorithms or laws like that to speed up the calculation. And there are other laws there. Let's see an example of this third rule, and then we'll move on to some other concepts. Staying in z eight, a simple case, we could probably do it in our head, but let's see if you can expand to solve it a little bit easier. 132 mod eight, well you could try and calculate what times eight was the range, but the property that we can take advantage of to make it a little bit easier, if we recognize anything about 132, what are the divisors of 132? Something times something is 132, maybe if you know your times tables. There are small numbers, but 12 times 11, there are other ways to do it, but 132 is 12 times 11, so we can expand that. 12 mod eight times 11 mod eight, all mod eight. 12 mod eight is four, 11 mod eight is three, 12 mod eight, four. That's just an example of doing this expansion to maybe make it a little bit simpler. Now these numbers you could do it either way in your head, but as the numbers get larger, especially even for computers, it makes it easier if you expand because these numbers become smaller before we multiply. Try that one, a couple of minutes, try to expand and do it without a calculator. No calculator, solve that. 11 to the power of seven, mod 13, and try and use the expansion to solve it, just to see how powerful it is. So one approach to this is think, right, let's make this number smaller. 11 to the power of seven is 11 to the power of four times 11 to the power of three, or even better. Now let's stick with that. 11 to the power of three, I'm not so good at my multiplication, so even that's too big for me. Maybe we can split that up. We'll come across here. I know my squares, I know 11 squared, but I don't know 11 to the power of three, so we can think of that as, 11 to the power of four is 11 squared squared, mod 13 times, 11 to the power of three, I don't know, but I know that's 11 squared mod 13 times 11 mod 13, 11 to the power of one, and let's not forget mod 13 at the end. Some of these numbers I can do in my head now. 11 squared, 121, mod 13, can someone do that for me? Four, 11 mod 13, 11 squared mod 13, 11 squared is, mod 13 is four, and then square that again is 16, mod 11, let's leave it in here just to be complete. Four, 11 squared mod 13 gives us four, squared mod 13, 16 mod 13, we get three times four times 11, we could probably do that in our head. 12 times 11, actually that was before, 132 mod 13, divide by 13, the remainder is going to be two. 10 times 13 is 130, right? The idea of this is that this expansion makes large calculations much simpler. Not just for you in a quiz or an exam, but for a computer that has to deal with very large numbers. Very large numbers, maybe hundreds of digits. Not hundreds, but hundreds of numbers is the length of the number. Multiply them together, calculate the modulus, takes time, so break them into smaller numbers. What's next? That was an example of this property that we could use, and that's the main one we'll see in the cryptographic algorithms. Division we've talked about, we've defined additive inverse, multiplicative inverse. Some number has a multiplicative inverse in n if it is relatively prime to n. So we can do division. In the last 10 minutes, before we won't introduce the theorems, we'll do that next week, let's consider another thing, relatively prime. How many numbers are relatively prime to 10? Back to our examples. How many numbers are relatively prime with eight? One, two, three and four. How many? Just counting them. How many numbers were relatively prime with ten? One, two, one, two, three, and four? Four? It's not always four, but let's consider counting the number of numbers relatively prime with some number. Let's get a different colour. Start simple. For example, how many numbers are relatively prime with four? And less than four. We'd need to test three numbers. One, two, and three are one and four relatively prime? Two and four, three and four. So relatively prime, remember, means the greatest common divisor of four and one, four and two. So we'd like to count how many numbers less than four are relatively prime with it. Four and one, greatest common divisor is one, four and two, divisor is two. Four and three, divisor is one. These are relatively prime. This one's not. So two numbers less than four are relatively prime with four, the count of numbers. And we have a special name for this. The number of numbers less than n, which are relatively prime with n, is called Euler's totient. Euler's totient function is spelled E-U-L-E-R-S, the top part here. Euler's totient function. Written, so the totient we say, easier to say, the totient of n is the number of numbers less than n, which are relatively prime with n. The totient of four equals two. We did it before, the totient of eight, we had, from our previous example, we had four. The totient of ten, from our previous example, we counted four numbers. What's the totient of nine? Euler's totient function is defined as the count of numbers relatively prime with n and less with n. So the totient of nine, consider the numbers from one to eight, check if they're relatively prime with nine and count how many are. Well, let's, one and nine are one and nine relatively prime. Yes, one in any number is always relatively prime. Two and nine, greatest common divisor of two and nine is one. So yes, they are relatively prime. Three and nine, greatest common divisor of three and nine is three, so no. Four and nine, greatest common divisor of one, so yes, I'll give it a tick, it is relatively prime. Five and nine, six has a divisor of three, so it is nine, so no. Seven's okay, seven and nine, and eight is also okay. One, two, three, four, five, six numbers. Euler's totient of nine is six. Just to make sure you know what you're doing, let's try another number. We'll not write it down. One and thirteen, yes, one and n always relatively prime. Two and thirteen, think of the greatest common divisor, greatest common divisor between thirteen and two. Is it one? If so, relatively prime. Two and thirteen are relatively prime. How many numbers less than thirteen are relatively prime with it? All of them, twelve. Okay? Because, because of what? Thirteen is a prime number. By definition then all of the numbers must have a greatest common divisor with that number, the prime number of one. So the numbers less than thirteen, which are relatively prime with thirteen, one through to twelve, so there are twelve numbers. So that's a nice shortcut. The Euler's totient of a prime number is the prime number minus one. If you recognize twenty-nine, it is a prime number. Totient is twenty-eight. So if we know n is a prime number, we can quickly calculate the totient. Very easy. If it's not a prime number, then, well, we saw the algorithm on our small numbers, we could check the numbers. We go through and check them. If it's a prime number, we don't have to check them. We know straight away that if twenty-nine is prime, the answer of Euler's totient of twenty-nine is twenty-eight. But if it's not a prime number, we need to check. And it turns out that there are no fast algorithms for checking. The algorithms that people know of for checking these, basically you need to try them all. And that takes a long time when n is very large. That will become another security property we're using cryptography. A couple more to finish for today. Totient of seven, it is prime. Totient of five, five is prime. Totient of thirty-five, thirty-five is a prime? No. So what are you going to do? You could try numbers one compared to thirty-five, two, three, up to thirty-four and compare them all. But we do notice thirty-five has two prime factors of seven and five. And it can be easily shown that the quotient, Euler's totient, not quotient, the totient of two prime factors multiplied together is the totient of those numbers multiplied, running out of space. Totient of seven we know is six. Totient of five is four. The answer is twenty-four. So we have n, n is thirty-five. If we can factor it into the two primes, in this case seven and five, then it's easy to calculate and you can check the math. It's not hard to show that the totient of two numbers multiplied together is the same as the totient of each number and then multiply. And since it's easy to find the totient of the two primes, seven and five is six and four, it's easy to find the totient of seven times five of thirty-four, thirty-five. This is a property that will make use of insecurity as well. And it will come down to how easy is it to find the totient of a number, a large number. If it's a prime number, yes, very easy, just the prime number minus one. If the number is made up of multiplying two prime numbers, also very easy. In this case, it's made up from multiplying two prime numbers so we can quickly find the totient. If not, it's very hard. We basically need to try all the numbers. And if we have to try a large number of numbers, it will take too long to try them. And that will be a security property we see in public key crypto. Those properties listed here, the totient of one is one. That's the definition. For prime p, the totient of p is p minus one. We saw that. And this property we just saw. The totient of n, when n is made up of multiplying two primes, p and q, is p minus one times q minus one. We'll take advantage of that as well. Next week we'll see some further theorems that take advantage of some of these properties. And then we'll move on to public key cryptography and see an algorithm that uses these properties to make it secure.


Almighty God states in the Holy Qur'an Bismillah ar-Rahman ar-Raheem وَمَنْ قُتِلَ مَظْلُوْمَنْ فَقَدْ جَعَلْنَا لِوَلِيِّهِ صُلْطَانَةَ فَلَا يُسْرِفْ فِي الْقَتْلِ إِنَّهُ كَانَ مَنْصُورَةَ The one who is killed unjustly, then we have given his heirs or his guardians power authority to seek retribution therefore let him not slay others excessively for he is supported My respected brothers and sisters illuminate your hearts and minds with a very loud salawat As we begin our exploration into the first general visit of Imam al-Hussein alaihi salam we are introduced to the concept of revenge divine revenge revenge is a very powerful driving force in our lives sometimes you have wars sparked because of revenge sometimes entire communities are ruined because of vengeance and revenge sometimes families are ruined relationships are ruined because of revenge it's a driving it's a powerful driving force in our lives we all know that feeling when you feel as if you have been oppressed you feel you've been wronged you feel others have committed an act of injustice against you that feeling of revenge we've all experienced it it's powerful the energy that it generates in you it says if sometimes you feel like destroying the whole world because of this feeling of revenge and vengeance that you get now the greatest act of injustice that occurred in the history of humanity was the killing of Imam al-Hussein alaihi salam his blood was spilled unjustly this was the greatest act of injustice the greatest act of oppression that was done in the history of humankind what we see in the opening passage of this first general Ziyara the Imam al-Hussein alaihi salam in mafatiha al-jinan as we explained last night there are several about seven general visitations to the Imam alaihi salam in these nights we are exploring the first Ziyara to uncover its meanings to delve into the depths of this amazing Ziyara that has been passed down to us by Imam al-Sadiq peace be upon him in the first passage opening passage of this Ziyara we are introduced to a common theme that not only do you find in this Ziyara but in other Ziyaras of Imam al-Hussein alaihi salam and this theme is the idea of vengeance that the death of Imam al-Hussein alaihi salam will be avenged the divine vengeance from those who killed Imam al-Hussein alaihi salam unjustly now there are several questions here that are worth analyzing number one who is the one who will seek the vengeance from the killers of Imam al-Hussein number two how will revenge be sought for the death of Imam al-Hussein through which means and from who because we know obviously that the killers of Imam al-Hussein alaihi salam perished 13 centuries ago so if anyone is going to seek revenge from the killers of Imam al-Hussein how is that possible when they have died 1,300 years ago and finally we will examine the difference between divine revenge or let's call it social revenge as well and personal revenge what is the difference between these two and how is it necessary for us as human beings to protect ourselves from personal revenge in the opening passage of the Ziyara before the idea of vengeance is mentioned we see two introductory phrases the Ziyara of Imam al-Hussein alaihi salam starts by this phrase peace be upon you oh proof of god and the son of his proof the highest status that the imams of Ahlul Bayt peace be upon them carried was that they were the proofs of Allah subhanahu wa ta'ala on this earth through them Allah subhanahu wa ta'ala is testing the humanity Allah created us for what purpose isn't it to test us Allah is testing us through the Ahlul Bayt peace be upon them because in order for you to have a valid test you need the proofs right when you go to school or college and you're given an exam doesn't the professor give you the proof before that either in his lectures or in the books or in the extra readings the proof has to be somewhere if you go to a class and there's no way for you to find the answer and the proof then that's not a fair test a fair test is one in which you're given the opportunity to search for the truth Allah has created us human beings to try us and test us and he has sent the proofs and the signs with the prophets and the successors to the prophets so al-Imam al-Hussein al-Salam is the proof of Allah subhanahu wa ta'ala because through him Allah sends the proofs to humanity this is the highest position that he carried as an Imam and all the prophets of God now those who killed al-Imam al-Hussein al-Salam they actually transgressed against the proof of Allah subhanahu wa ta'ala Allah has created you to test you and he sent you the proofs is there a crime greater and worse than messing with the very proof of Allah subhanahu wa ta'ala then killing the very proof of Allah subhanahu wa ta'ala so this introductory phrase demonstrates to us the magnitude of this crime because it's preparing us for the vengeance when you have a huge crime that's committed then people expect vengeance justice necessitates that there is an act of revenge taken from the perpetrators the following statement says peace be upon you you know when someone from your family members let's say god forbid your father your brother is killed in the arabic language this person is described as being a when you lose someone someone is unjustly killed that is your because he's been killed and he belongs to you he belongs to your family or he belongs to your tribe this phrase describes the imam as being the of Allah subhanahu wa ta'ala because he belongs to Allah and if you belong to Allah subhanahu wa ta'ala Allah will take it upon himself to seek the vengeance from your killers because when someone is killed from your family isn't it your obligation to seek vengeance to seek retribution from the one who unjustly killed let's say your father or brother that is your obligation as a family member you need to defend the one who was killed from your family and imam el hussein belongs to Allah subhanahu wa ta'ala therefore he has taken it upon himself to seek the vengeance from the killers of imam el hussein and this is a lesson for us if you die in the way of Allah subhanahu wa ta'ala it's as if you symbolically become a part of god's family and he takes it upon himself to defend you and to seek revenge from those who killed you and he will take it upon himself to seek vengeance from those who killed him then we are introduced to the idea of vengeance peace be upon you oh vengence of god and the son of his vengeance Allah subhanahu wa ta'ala reveals in the holy quran that whoever is unjustly killed woman we have given his heirs let's say a father is killed Allah has given authority and power to his children to his sons for example to seek retribution Allah subhanahu wa ta'ala has given this right if someone unjustly willingly knowingly kills one of your family members you have a right to go to the judge and ask the judge for this person to be punished for the death sentence to be passed on to this murderer because he knowingly and deliberately committed an act of murder and justice must be served now when it comes to Imam al-Husayn alayhi as-salam who is the one who see who will seek his vengeance Allah subhanahu wa ta'ala the ziyarah says as-salamu alaika yathar Allah you're the vengeance of God so Allah subhanahu wa ta'ala is the one who will seek the vengeance now why Allah subhanahu wa ta'ala number one he died in the way of Allah subhanahu wa ta'ala so Allah subhanahu will take care of that number two the blood of Imam al-Husayn alayhi as-salam is so valuable is so sacred only God can seek the vengeance for it only God can do justice to it because we cannot come close to grasp the greatness of Imam al-Husayn's blood the sanctity and the value of his blood only Allah can grasp that and number three so many groups were involved in the killing of Imam al-Husayn alayhi as-salam so many factors were at play only God can do justice to them we can't some say well al-mukhtar killed many of the killers of Imam al-Husayn didn't he seek revenge no only Allah knows how many people were involved throughout history that caused the bloodshed in Karbala and besides al-mukhtar couldn't you know seek revenge from all the killers of Imam al-Husayn he tried to seek revenge from some of them but not all of them Allah subhanahu wa ta'ala is the only one who's capable of seeking this revenge no one else in history can do that Allah subhanahu wa ta'ala is the only one who knows exactly what happened who was responsible for what and how to seek the revenge therefore Allah subhanahu wa ta'ala will seek the revenge from the killers of Imam al-Husayn alayhi as-salam and he will avenge his blood this also applies to all the imams of Ahlul Bayt in one of these applications that we recite and honoring lady Fatima to the Zahra as-salamu alayhi alayhi it says Allahumma kun al-taalib laha mimman zalamaha was taqaffa bi haqqaha oh Allah you be the one who will seek revenge from those who oppressed Fatima and those who took her right lightly they denied her right and then it says wa kun illahumma tha'ira bidami awladiha oh Allah you be the one who will avenge the blood of the imams from her progeny now when we speak about divine revenge Allah subhanahu wa ta'ala seeking revenge what does it mean for God to seek revenge one of the names of Allah is al-Muntaqim in Arabic the Avenger the one who seeks vengeance as we know vengeance is it starts with a feeling we've all experienced that feeling right it's this feeling where you want to harm someone because that person harmed you it's this powerful feeling that we experience it's this hatred that we have toward someone whom we think oppressed us Allah subhanahu wa ta'ala is above feelings Allah does not experience these feelings that you and I do so what does it mean for Allah subhanahu wa ta'ala to seek vengeance besides Allah subhanahu wa ta'ala doesn't have any personal feelings with anyone with us human beings it gets personal i have this personal hatred vendetta against someone and i want to seek vengeance but Allah subhanahu wa ta'ala is above personal vengeance so when we have the holy Quran speaking about Allah subhanahu wa ta'ala seeking revenge in many parts of the Quran Allah speaks about revenge what does that mean in Quranic literature vengeance means the establishing of God's justice when Allah subhanahu wa ta'ala establishes justice that is one way of us understanding his revenge let me give you an example you have a criminal in society who commits a crime he commits an act of homicide an act of murder now what does our law do our law enforcement agencies they take this criminal and they imprison this criminal he's sentenced to a life you know sentenced in prison to a life term in prison now when this criminal is sentenced to a prison to a sentence a life sentence this in itself is a type of social vengeance we can say that our society has sought revenge from this criminal and he's being punished for that crime so the meaning of vengeance when it comes to Allah's actions means justice we have to implement justice this criminal must be met with justice and when we implement justice and take this criminal and throw him into prison this is an act of vengeance it's called social vengeance it's not personal it's not that we have a personal hatred with this person in fact the judge or the police they don't even know this criminal they have nothing personal against him but justice must be served and this is how we come to understand the revenge of Allah subhanahu wa ta'ala Allah subhanahu wa ta'ala only seeks revenge when he established the clear signs and the proofs and those evil ones knowingly deliberately stubbornly reject the signs of Allah for example the Holy Quran tells us about the people of Musa A.S Allah sent them all those clear signs and one verse Allah says nine signs Musa was equipped with and he took to the people of the Fura'un but they rejected all those signs knowingly they were certain that Musa was truthful they yet they rejected all those signs what happens next? Allah says we took our revenge from them from the people of the Pharaoh how did Allah seek revenge? Allah caused them to drown in the waters Allah established justice he punished them and this is divine revenge this is divine vengeance so Allah subhanahu wa ta'ala is the one who will seek the vengeance for the death of the Imam of Hussain A.S now the important question here is how will Allah subhanahu wa ta'ala seek the vengeance we know Allah subhanahu wa ta'ala based on his wisdom he simply doesn't you know seek vengeance directly usually he doesn't Allah can if he wants to but Allah usually uses means Allah can say be and it is but Allah subhanahu wa ta'ala usually uses means in this life in order to seek vengeance how will Allah subhanahu wa ta'ala avenge the death of the Imam of Hussain A.S when narration tells us that on the day of Ashuwa when Imam Hussain A.S was killed in that brutal way the angels of the skies could not restrain themselves they broke into this severe warning in weeping upon seeing what happened to the beloved of Rasulullah A.S and the angels could not help it they said oh Allah how can you let the killers of Hussain get away with such a huge crime how can you allow that when will you seek revenge from them Allah subhanahu wa ta'ala revealed to them Allah created an image from them you remove the curtain from their eyes and they saw this image Allah says to the angels look what do you see they saw nine imams from the progeny of Imam Hussain A.S and one of them was Qaim standing Qaim in Arabic means standing Allah subhanahu wa ta'ala told the angels you see that Qaim one the one was standing the last of them the 12th or the ninth from the project Imam Hussain they said yes we see him Allah says I shall seek revenge from the killers of Hussain using the sky and I will seek the vengeance and I will avenge the death of Imam Hussain A.S therefore Imam Al-Madee A.S is the one who will seek the vengeance from the killers of Imam Hussain A.S now Imam Al-Madee has two rights in seeking the vengeance number one he is the heir of Imam Al-Hussain according to Islamic law the heirs have the authority to seek vengeance grandfather of Imam Al-Madee the only one he is from Allah we have the right we have the personal right to seek revenge because his grandfather was killed because Imam Al-Hussain is the representative of God he is the leader for humanity representative of God therefore he will seek like a president such as the killed government taking upon itself its responsibility to seek justice and retribution because Imam Al-Madee has the right to seek the vengeance from two ways the personal right resides for example to act with the man the of you are familiar with this phrase and it is extremely recommended that you should always stop the case and pray to Allah to have the most important prayers is to ask Allah to give you with the honor we offer each other condolences and we offer condolences to take this prayer and we will participate in seeking that for the blood of Imam Hussain does not have the slogan or the motto of the companions of Imam Al-Madee's words you are the one that we will seek the vengeance for the blood of Imam Hussain arms that were amputated from Ab El-Fadl Al-Abbas for which of for which of those or do we seek revenge for what happened for Zaynab alayhi salam being harassed by the enemies being paraded from one city to another city or for those orphans who were separated from from their fathers on such a night al-Imam Al-Hussain alayhi salam had to leave one of his orphans back in the city of Medina on such a night remember that scene in which al-Imam Al-Hussain alayhi salam when he wanted to depart the city of Medina and head towards Karbala he called for his brother Ab El-Fadl Al-Abbas the horses ready we have to move on we have to go to Mecca and from Mecca to Karbala the women and the children who stayed behind they came to farewell al-Imam Al-Hussain al-Wida one of those girls one of those orphans who stayed behind was Fatima one of the daughters of al-Imam Al-Hussain Fatima Al-Aleela she was ill she was sick Imam al-Hussain alayhi salam told um Salama I want you to take care of her we can't take her with us the journey is too long and she's too ill to join us Fatima al-Aleela she sees her father Hussain along with her uncle Ab El-Fadl Al-Abbas she sees her brother Abdullah al-Razi all of them leaving her she crawls towards her father Hussain and she tells him oh my father where are you going why are you leaving me behind he told her there's a treacherous journey ahead of us a long journey and you're sick you cannot join us you're too weak to be with us stay with um Salama then if we settle on the land of Karbala if we get there successfully and nothing happens to us then I may send Abbas to come and pick you up she tells him oh father Hussain then at least keep Abdullah al-Razi keep him with me so at least I have someone to see consolation in someone to remind me of you don't leave me all alone over here without any of my siblings the Imam told her oh Fatima he's too young he's too young he breastfeeds how can I separate between him and his mother they farewell Fatima and they go on their way to Karbala on the second day of Muharram the caravan of Imam al-Hussain alayhi salam reaches the land of Karbala they reach an area al-Hussain realizes that the horse is no longer moving the Imam would tell the horse move let's go on towards Kufa but the horse refused it stayed in that land Imam al-Hussain realized that something is not right over here Zaynab alayhi salam was standing close to Imam al-Hussain she tells him my dear brother I feel something heavy in my chest something very heavy in my heart what's happening I feel this sadness this sorrow this depression in this land what is the name of this land al-Imam al-Hussain asked his companions he gathered them he told them my dear companions what is the name of this land one of the companions came forward he says oh Aba Abdullah one name for this land is Al Ghadria the Imam says is there another name he was looking for that name which he heard from Rasulullah sallallahu alayhi wa ala because the Imam alayhi salam would constantly hear this name umsalama she narrates one day when Hussain and Hussain were little children the Prophet went into his room and he told umsalama I want you to make sure no one disturbs me the Prophet wanted to worship wanted to rest make sure no one comes into my room to disturb me umsalama says I saw Hussain running towards the room of his grandfather I held Hussain I told him no your grandfather is resting and then suddenly he ran out of my hands and he went inside the room of Rasulullah she says I went close to the room of Rasulullah sallallahu alayhi wa ala you know what umsalama says she says I saw Hussain sitting on the stomach of the Prophet the Prophet was laying down and then Hussain was sitting on the chest on the stomach of the Holy Prophet and I heard the Prophet wailing and weeping I went inside I told him Ya Rasulullah why are you wailing and weeping Sunni historians and scholars have narrated this event brothers and sisters umsalama says Ya Rasulullah why are you crying what happened don't you love Hussain he told her umsalama just now jibra'il descended upon me and he told me ya Muhammad do you love your grandson Hussain I said yes I love him so much then he told me oh Muhammad know that your grandson Hussain will be massacred on the plains of Karbala do you want to see the soil of Karbala Ya Rasulullah Musnad Ahmad he narrates this he said yes I want to see it jibra'il he gets a handful of the soil of Karbala and he shows it to the Prophet the Prophet takes the soil he puts it in a small container and he gives it to umsalama he tells her umsalama whenever you see this turning into red whenever it becomes you know like blood red realized that my grandson has been killed and indeed on the day of ashura umsalama when she looked at this small jar container she saw the blood turning into red and she saw the dust turning into red she realized that Imam al Hussain had been killed so Imam al Hussain was looking for this word which he had heard from the Prophet the word of Karbala then one of the companions he came and he stood up and he says oh Aba Abdullah the name of this land is Karbala al Imam al Hussain alayhi salam he broke into tears he said Allahumma inni a'udu beka minal karbi wal bala oh Allah I seek refuge in you because the meaning of the word Karbala it's a combination of two words of trials and tragedies and tribulations oh Allah protect us from the tragedies and the tribulations then he told his man he told the companions oh my companions here we shall settle let's camp out here my dear companions this is where we shall settle my dear companions it is in this land that our men shall be massacred and killed it is in this land that the children shall be orphaned and it is this land which the woman shall be taken as prisoners of war Zainab could not take it she broke into tears and Imam al Hussain came down from his horse and he told her oh Zainab you have to be patient and then he showed her he told her Zainab do you see that low-lying area this is the place that I shall be massacred then he takes her to the river of Furato believers and he tells her oh Zainab in this area your brother Abel Fazl al Abbas he shall be struck with an iron rod his hands will be amputated and this is the area that he shall be killed brothers and sisters raise your hands in dua with broken hearts and tears Allah has promised that he will forgive us our sins everyone together recite this holy verse five times together so Allah answers our prayers anyone who's in distress anyone who's sick that you know this is the moment of dua brothers and sisters raise your hand in dua everyone together Bismillahir Rahmanir Raheem O Allah O Allah O Allah hates the reappearance of our master Imam al Mahdi O Allah make us amongst his sincere followers and dedicated servants O Allah grant us the ziyara of Imam al Hussain O Allah grant us the shafa'a of Imam al Hussain Muhammed and the people of Muhammed