Welcome back to our lecture series Math 4230, abstract algebra 2 for students at Southern Utah University. As usual, be a professor today, Dr. Andrew Misseldine. Lecture 6 represents the start of our coverage of chapter 15, which is going to be about the Seeloff theorems and their applications towards simple groups. In section 15.1 of Tom Judson's textbook, we, following his table contents, I should say, we're going to develop and prove the three Seeloff theorems. In this lecture, we're going to prove the first Seeloff theorem, and in the next lecture, lecture 7 will prove Seeloff theorems 2 and 3. But before we get to see the first Seeloff theorem, I actually want to prove in this video something called Cauchy's theorem. Now, in abstract algebra 1, we saw Lagrange's theorem, which told us that the order of any subgroup divides the order of the whole group, at least for finite groups, right? Infinite groups, the arithmetic, it's a little bit more messy. So just delving into finite groups, the order of a subgroup always divided the order of the group. So if H is a subgroup of B, then we have that the order of H divides the order of G, something like that. It's a very, very useful statement. But this is Lagrange's theorem. It's tempting to hope, or at least it's hopeful, that you can have a converse to Lagrange's theorem. That is, for every divisor of the group, there's a subgroup that obtains that order. But that's actually not the case, right? You have groups like, say, A4, the alternating group there. Its order is 12, and it does have a subgroup of order one, the trivial subgroup. You do have a subgroup of order two, because you could take a subgroup generated, a cyclic subgroup generated by a 2-2 cycle. It does have a subgroup generated, I should say a subgroup order of order three, because it's generated by a three cycle. You do have a subgroup of order four, because if you take the identity in the three 2-2 cycles, it forms a client four group, and you go from there, you also got a subgroup of order 12, because it's the whole group itself. But what about six? Six divides 12, but A4 does not have a subgroup of order six. Try and try as you might, you can't get one. And in fact, we can prove that there is no subgroup of order six. We're not gonna do that in this video. But this is sort of like the simplest counter example, at least one of the simplest ones, if there's one smaller, but I'm pretty sure this is gonna be our simplest counter example of the converse of the Granges theorem, that there do exist groups where you don't have a subgroup for some divisor of the order of the finite group. And that might seem a little too bad, but that's the reality of it. Now, while we don't get the full converse of the Granges theorems, we do get partial converses. For example, when it comes to cyclic groups, for cyclic groups, you actually have unique subgroup for every divisor of the group. That's pretty easy to form. And then also when you get abelian groups, we have a lot of structure on abelian groups. I mean, by the fundamental theorem of finite abelian groups, we can write every abelian group as a direct product of cyclic groups. And as such, we can basically get a subgroup of any divisor order we want. So it's actually pretty easy to get a subgroup of a fixed size for abelian groups, but not abelian groups like A4, things get problematic. So this is where the Seeloff theorems come into play in particular Cauchy's theorem as well. Because these give us some partial converse to Lagrange's theorem. It gives us some conditions which will guarantee the existence of subgroups of specified orders. But instead of a general factor, we can only sort of guarantee it for prime powered or prime powered ordered subgroups. So that's the issue with six. Six is the only divisor of 12 other than 12 itself. That's not a power or a prime. And so that's why we don't get six, but two, three, four, they're all powers of primes. By combining the Seeloff theory here, we can actually guarantee that there's groups, subgroups of that order. So with that, let's start with Cauchy's theorem. This is extremely useful theorem here. Cauchy's theorem says that if G is a finite group and P is a prime, such that P divides the order of G, then there is an element of G whose order is P. So every prime divisor of the order of the group, that group will have an element whose order is that prime divisor. And the proof of Cauchy's theorem is gonna follow by induction. And we're gonna play induction on the order of G, like so. So if the order of G is equal to one, it's the trivial group. This is vacuously true in that situation.  If the order of G is a prime number, we've seen previously that the only groups whose order is a prime is a cyclic group. And like I mentioned, with cyclic groups, we've studied those in abstract order one. Cyclic groups have a unique subgroup for every order. In particular, if you take a prime divisor, it's sub, there will be a subgroup whose order is P and it'll be generated by an element of order P, right? So it's pretty trivial when you're prime, all right? So then let's assume we maybe have composite order. Let's suppose that the result holds for all groups whose order is less than G. So that's the induction hypothesis right now. So considering the order of G, every group less than G, its order, if its order is smaller, which it does because it's smaller, it satisfies this property. So the basis of Cauchy's theorem is induction plus the class equation. Remember the class equation, it tells us that the order of the group can be decomposed as the order of the center of the group. The center of the group is all of the elements of the group that commute with everything. And then we add to the order of the center a bunch of indices. These are indices of centralizer subgroups for which you'll remember that a centralizer subgroup, what's C of X1 here, the centralizer subgroup, this is all of the elements in the group G that commute with XI right there, X1 in this case. So the centralizers are always subgroups. We take their indices, this is what the class equation gave us because if you do the conjugation action of a group onto itself, then the stabilizers, the isotropy subgroups of that group action are the centralizers. There that is the isotropy of the conjugation action. And then you can apply the class equation for any group action. If you apply it using congisly classes, you get exactly this right here. The center is exactly those elements of the group who are only conjugate to themselves, just as a reminder of this vocabulary and terminology and such. So in particular, those elements that are showing up that are represented over here, the X1, the X2, the XK, these are elements which are non-central. So they are conjugate to multiple elements in the group besides themselves. In particular, since they are non-central elements, their centralizers will be proper subgroups and thus these indices, they're gonna be relevant to consider, okay? So continuing on here, suppose that there is an index G colon C of Xi for some Xi that's not divisible by P. Well, by Lagrange's theorem, the order of G is factorable as the order of the centralizer with then the index of the centralizer. Well, since G by assumption divides the order of G by Euclid's lemma, since it's a prime number, it must divide either this factor or this factor. By assumption, it's not this factor. Therefore, P then divides the order of the centralizer where that centralizer is a proper subgroup. Therefore, we see that we've now produced a subgroup who has an order divisible by prime by P. It's a proper subgroup so it's a strictly smaller order by the inductive hypothesis. The centralizer has an element of order P and as it's a subset of G, that means G has an element of order P. So the result follows by induction in that situation. Well, what's the other possibility? Let's suppose that all of the indices are divisible by P, okay? So the order of G equals the order of Z, the center, and then you have all these centralizer indices, something like that. We could move these all to the other side of the equation. So we get the order of the center is equal to the order of G minus all of these centralizer indices. There's a bunch of those. So the point is this element is divisible by P, this element is divisible by P, all the other elements are divisible by P. So the entire sum is divisible by P, which means that the order of the center must also be divisible by P, okay? And so then we wanna focus here. So we know that P divides the order of the center. Now, we might just wanna say induction at this moment, but we actually don't know that the center is smaller than the group because what if it's an Abelian group? The center is the whole group itself. So the order of the center might not be smaller because the group could be Abelian, but whatever the case, we know that Z of G is an Abelian group. So the fundamental theorem of finite Abelian groups applies to the center. And by that structure theorem, an Abelian group always has an element of order P if, of course, its order is divisible by P that comes from the fundamental theorem. I won't go into the details of that right now. So therefore Z has an element of order P and as it's a subset of G, then G has an element for P and therefore we're done. We couldn't quite cite induction in that situation, but we got really close. If it was a non-Abelian group, then it would have followed immediately by induction, but we have to deal with Abelian groups themselves. So the fundamental theorem of finite Abelian groups takes care of it. And so that then proves Toshi's theorem. Every group whose order is divisible by a prime has an element whose order is a prime, that same prime. And so in particular, let G be a finite group whose prime P divides its order, then G has a subgroup of order P. So we switched from an element of order P to a subgroup of order P. But by Koshy's theorem, we're gonna have some element G inside of G such that the order of G is that prime. Well, then if you take H, then to be the subgroup generated by G, you're gonna get that the order of H is equal to the order of G, which is prime. So if you have an element of order P, then you'll have a subgroup of order P. So Koshy's theorem guarantees that every group has a subgroup order of the primes that divide it. So let's come back to A4, for example, like we saw. A4 is order is 12. Its prime divisors are two and three. So by Koshy's theorem, there is a subgroup of order two and there's a subgroup of order three in particular you're gonna have elements of order two and elements of order three, for which for A4, yeah, the two, two cycles, there's three of them, they have order two and then there's eight, three, three cycles in A4, they have order three. Those are the elements that were guaranteed by Koshy's theorem. I should also mention that Lagrange's theorem says the orders of an element can only divide the group and therefore you can't get any other prime divisors for the orders of elements than those that show up. So two and three are the only prime orders you can have in A4 and they're all realized. That's the limitations of Koshy's theorem though. We got these subgroups of order two and order three. We can't guarantee all subgroup orders like six is not present in A4 like we mentioned before. So six is problematic. Now A4 did have a subgroup of order four, which I said that's a prime power, it's two squared. Why is there a subgroup of order four? It turns out that to guarantee the existence of four for A4, we actually need something stronger than Koshy's theorem, which is gonna be the first Sealoff theorem, which we'll discuss in the next video.


What's up everybody Jason for Jasper's case brings calm we just sold out twenty twenty one Kenini obsidian basketball Asia team all addition Ten box half case break two random teams each number two and again guys This is an exclusive to team all in Asia guys You get two exclusive parallels per box one insert or autograph per box and again these colors Or you can only find in this edition of here of team all Asia Now we're gonna double up your names of course give you two random teams There's the teams there. Let's click the dice roller and let's do the randomizer It's just snake eyes two times. Here we go one Two Ben down to Nick. Let's get a little randomizer here Two times one two Boston down the San Antonio Spurs Two times All right Ben with the Celtics Simon with the Clippers Corey with the Rock is rich with the Bulls Allen with the 76ers John with the Kings Mike with the Pelicans been with the magic Allen with the Knicks Mike with the Mavs Allen with the Nuggets Raptors Pacers drew with the Hornets Allen with the Pistons drew with the Cavs Nick with the Wars Allen with the Jazz Simon with the T-Wolves Corey with the Blazers as well as the Suns Allen with the Hawks Nets Richard with the Bucks Allen with the Lakers heat Corey with the Wizards John with the Thunder Allen with the Grizzlies and Nick with the Spurs So if anybody does want to trade I'll give you guys a quick minute All right, Ben wants to trade the Celtics potentially Some mate dog sure that'll be my one request of the day Can't break I can't play it while we're breaking though because we'll get copyright in the video. We'll get blocked But I'm listening don't trade with a closing guys All right, so here we go To John Tim Murray for the Spurs little beef stew rookie Well the Marvin Bagleth We got a Jamal Murray little die-coat green Bojan Bogdanovich and we got a Michael Bridges for the Suns Let's quarry with the Suns Zignagi rookie Drummin sub onus. We got Trey Jones rookie For the Spurs San Antonio Spurs is a Nick and then we got a nice Pat Williams red color match for the Chicago Bulls, which is Richard All right next one guys Cure Louis St. Bama DeBio Tunnel vision Russell Westbrook for Washington Wizards going to quarry Yeah, Clint Capella and Yannis Valentunus. I think there's just to be at least two more past autographs hiding in here somewhere I feel like there's usually at least four this might have one of them No We got faked out And maybe there's an autograph who knows Man, you're quickly very nice Kelly Aubrey Josh Green You got a CP3 with a gold Bradley Beale and a green Marcus smarts Nice color match there And again, these are not numbered but they're exclusive colors in team all it's hiding a little bit right now, but we'll find them We'll find them All right fifth box Still more than halfway guys just have plenty and plenty of time plenty of boxes here and rookie Marcos folks Donovan Mitchell With a drew holiday And another Marcos folks CJ Elegy Chris Middleton Moises Brown rookie Tunnel vision Yannis onto de Compo With a very nice to deek Bay right there that Yannis is awesome That's for the Bucks and Richard the behinds to deek Bay is A lemon pepper Lou Lou Williams That's Detroit though. That's Alan with this one All right, four more to go. That's this is night in 2021 Mowgli would not be in this class. He'd be in next year's This is only Ant-Man, Tyrese Maxi Lamello quickly Like a 2021 rookies Jimmy Butler You got Donovan Mitchell with a Reggie Perry rookie a fattiest young RPA here, but I guess they're not guaranteed though Harden Hunter. Whoa. What is that? Missed this is probably like a super short print. I haven't seen this James Harden For the Brooklyn Nets going to Allen. That must be a super short-printing card or something because Not seeing that one Ben Simmons pitch block behind Kelly Obre is a Fred Van Blee for the Rafters All right, two more to go Sexton Rookie Jordan Noir for the Bucks Robert Covington Hitch black Zach Levine And a Moises Brown rookie with a Zion Williamson. Nice That's a Pelicans mic with that one. All righty guys come down to the last box. Good luck everybody This is definitely got a patch auto so this case only had like three Best for last all right the fact that we got this right in the last box Darren Fox and Ben Simmons. All right. Good luck everybody Here we go JJ Reddick Tyler hero Facundo compaso Theo Meledon RPA and that's gonna go to the Oklahoma City Thunder, which is John There you go guys one RPA with one of these super short-print mist die cuts and of course a lot of red green and gold Team all exclusive parallels fortunately, we didn't see any bigger bigger rookies on this side, but Thanks, we're still giving it a shot. We don't have any more guys But if we do we'll get some more Jaspie's case rates commons where it'll be posted coming up next is a filler For a hip rate multi-sport Jaspie edition and then we can break that next guys. Appreciate you guys


Does he want to Yeah I Please let's join together in a few minutes of moments of centering silence Please remain seated as we sing our in-gathering hymn which is number 1023 in your teal hymnals However the words appear in your order of service Good morning Welcome to the first Unitarian Society of Madison We're so glad you came to be with us on this great drizzly day So gold stars for everyone This is a community where curious seekers gather to explore a spiritual, ethical and social issues in an accepting and nurturing environment Unitarian Universalism supports the freedom of conscience of each individual As together we seek to be a force for good in the world My name is Karen Rose Gredler and on behalf of the entire congregation I want to extend a special welcome to any visitors who are with us today We are a welcoming congregation so whomever you are and wherever you happen to be on your life journey We celebrate your presence among us Newcomers are encouraged to stay for our fellowship hour after the service and to visit the library which is directly across from the center doors of this auditorium Bring your drinks and your questions Members of our staff and lay ministry will be on hand to welcome you You may also look for people holding teal colored stoneware coffee mugs These are FUS members knowledgeable about our faith community who would welcome visiting with you Experienced guides are generally available to give building tours after each service If you would like to learn more about the sustainably designed addition or our national landmark meeting house across the parking lot there Please meet near the large glass windows on what is your left side of the auditorium after this service And I do show that we have a guide for this morning so please meet there We welcome children to stay for the duration of the service However because it is difficult for some in attendance to hear in this lively acoustical environment Our child Haven back in that corner and the commons behind the auditorium are excellent places to go If your child wants to talk, sing, dance, run around in circles, whatever normal childhood thing they want to do The service can still be seen and heard from those areas This would also be a good time to turn off all devices which might cause a disturbance during the hour Especially cell phone ringers I'd now like to acknowledge those individuals who are helping our service run smoothly We have Mary Manoring operating the sound system We have Tom Boykoff as a lay minister Upstairs you would have run into Carol Angel and Patty Whitty greeting today Our ushers are Marty Hollis, Paula Alt and Dan Bradley Sandy Plish is making coffee for us back in the kitchen We also want to thank Hannah Pinkerton for watering the palms and lilies this morning Thank you all The tour guide I'm showing is Michelle Mason So hopefully you'll see her after the service Now I'd like you to note the announcements in the red floors insert to our order of service Where you'll see descriptions of upcoming events at the society And see more information about what's going on today First I want to say I understand that it is 12 days to Oh sorry I didn't mean to do that Cabaret, Steve wants us to say that Also I have a couple of other special announcements The FUS Equity Ministry team invites you to an open conversation about today's reflections on white supremacy Folks will gather in the atrium auditorium at 12.30 following the 11 o'clock service Refreshments will be available and there will be a story time in the library for children Now for you folks it's great because you can just run out after this service Eat some brunch and then come back at 12.30 The second special announcement is this On Mother's Day weekend May 13 and 14 peace poles will once again be placed and decorated along the FUS property at University Bay Drive Between and after each of the two services these will be available to you If you would like to participate in this family friendly event Please sign up prior to the event so that we can make sure we have enough poles There will be a person with a sign up sheet in the commons after the service So hopefully many people will participate Those are really pretty out there every spring So now I'd like to say I hope you all feel welcome and I trust that today's service will stimulate your mind Touch your heart and stir your spirit Our opening words this morning come from the Harlem Renaissance poet Langston Hughes I dream a world where man no other man will scorn Where love will bless the earth and peace its paths adorn I dream a world where all will know sweet freedom's way Where greed no longer saps the soul nor avarice blights our day A world I dream where black or white or whatever race you be will share the bounties of the earth and everyone is free Where wretchedness will hang its head and joy like a pearl attend the needs of all humanity Our world I invite you to rise and body and spirit for the lighting of our chalice Our words of affirmation are responsive would you please join your voices in reading the bolded italicized sections For that which has been done to lift humankind to new levels of social order and health For that which we can do but have left undone For that which needs to be done but is as yet beyond our reach And now I do invite you on this chilly April morning to turn to your neighbor exchange with him a warm greeting Now I'd like to invite any children who are willing and eager to join me in the front for the message for all ages You got a shirt with all kinds of horses on it. You like horses? Yeah, I like horses too. And being loving. And we talk a lot about head and thinking. We talk about hands and what we can do with our hands and what we can do with our feet. But we don't spend a lot of time talking about the largest organ of our whole body. And what is that? Our skin. Our skin. That which covers us. That's right. And so this is a story about skin, okay? The skin you live in. We all live in skin, right? We don't? Ah, okay. Here we go. Some nice pictures up here too. Hey, look at your skin. The wonderful skin that you live in. The skin you're all day in. The skin that you play in. The skin you snuggle up and cuddle up and lay in. The skin that you beam in. The skin that you scream in. The skin that you dream about eating ice cream in. The skin you have fun in. The skin that you run in. The skin that you hop skip and jump in the sun in. The skin you laugh in. The skin you cry in. The skin you look up to the sky and ask, why in? It's baby born new skin and your family too skin and glows when it shows what it knows. We love you skin. It's face the rain. Bold skin. Snow angel cold skin and warm again. Let it in. Sunshine be hold skin. It's trembling fright skin. Get goosebumps and cringes at night skin. But turn off the lights to make birthday cake bright skin. It's whatever you do skin. Be happy it's you skin. You can't live without it. It's glad that it's me too skin. And look at all the shades that it comes in. The shades of your colorful skin. Look at the shades of the skin. Your coffee, your cream skin. Your warm cocoa dream skin. Your chocolate chip double dip Sunday Supreme skin. Your marshmallow treat skin. Your sponge sugar sweet skin. Your cherry top candy drop frosting complete skin. Your pumpkin pie slice skin. Your caramel corn nice skin. Your toffee wrapped ginger snap cinnamon spice skin. Your butterscotch gold skin. Your lemon tart bold skin. Your mountain high apple skin. Your cookie dough rolled skin. Bet you didn't know skin could come in so many different flavors. So look again at your skin and the skin of all the people. All the skin that they live in. It's not tall skin. It's not short skin. It's not best in the sport skin or fat or thin skin. Or you lose and I win skin. Not sad skin or mad skin. You're naturally bad skin. I'm rich and you are poor and you'll never have more skin. It's not any of those kinds of skin is it?  It's not weak skin or strong skin. I'm right and you're wrong skin. Nor she skin nor he skin or you're better than me skin. I'm lesser than your skin. It's me against you skin. It's not any of this because you are more than you see. All that you think and you hope and you dream.  You're a new day desire to reach even higher. Your feelings that start from deep down in your heart. You're the pride and the joy inside each girl and boy. So whenever you look at your beautiful skin from your wiggling toes to your giggling grin. Think of how lucky you are that the skin you live in so beautifully holds you within. They're like flowers in the fields that make wonderful views when we stand by side by side in our wonderful hues. We all make a beauty that is so wonderfully true. We are special and different and we are just the same too. Yes. Every page what? It does. There are a lot of skin words in there, aren't there? Yeah, that's right. It's kind of like a long, long poem that keeps repeating that word skin because the skin is really, really important to the story and it's really important to you. So having heard this little story, maybe you'll go back out and you'll think a little bit more about skin and what it means and what it doesn't mean as well. There's a ghost inside your body? Oh my goodness. I'm going to have to check that out. Okay, we're going to sing you all out with him number 120 and it's easy because it's the same tune as the one we sang when we came in here. Old 124th. Please be seated. One month ago today, on March 30th, the president of the Unitarian Universalist Association, Peter Morales, tendered his resignation. In so doing, he was but the second president of the UUA not to have completed his full term of office, the other person being Paul Carnes, who died of cancer after just more than a year of service. Morales, whose tenure would have ended on the 30th day of June, Morales jolted the denomination with his precipitous departure. Now the UUA presidency is a hard position to fill, and I know from conversations with colleagues and with Peter Morales himself that he has been no stranger to controversy. Nevertheless, he was elected to a second four-year term in 2013, and he subsequently presided over the relocation of our denomination's national offices to newer and more serviceable facilities. But clearly he was not able to weather the storm that was unleashed by a recent executive decision that was made at headquarters. In a letter to the Board of Trustees, Peter Morales conceded that I have obviously lost the trust of many people, and it is clear to me that I am not the right person to lead our association as we work together to address our shortcomings. We need space for healing. We need space for listening. Morales emphasized that the decision to step down was entirely his own, that no one even mentioned that he should resign. He simply felt that new developments had convinced him that it was time for someone else to assume the presidential mantle. Now the proximate cause for this leadership realignment was this. Denominational search committee, headed by Scott Taylor, the Director of Congregational Life. That search committee selected a white male to replace another retiring white male as leader of the UUA's southern region, a judicatory that stretches from Virginia in the east to Texas in the west. Now the UUA is made up of five regions. We belong to the mid-America region, and at present every one of the five is served either by a white man or a white woman. Although minority hires for positions in the UUA have increased, in the last decade, the decision to engage yet another white person for an upper level administrative position did not sit well with many of the people of color in our movement. And to make matters worse, another leading candidate to fill that vacancy was an accomplished woman of color, a Latina member of the UUA's Board of Trustees and Director of Administration and Finance for one of our southern region's more noteworthy congregations. In any case, reaction to the hire was swift. And in short order, the Director of Congregational Life and the UUA's Chief Operating Officer had also submitted their resignations. Presently three individuals, each with a different set of responsibilities are fulfilling the duties of the outgoing president. Now the issue at hand will not be resolved by a simple change of personnel or by revision of the association's hiring practices. The controversy has led persons of color and their white allies to call for some very serious soul searching, not just among the folks at headquarters in Boston, but throughout our movement. In effect, we are all being asked to confront directly a very challenging, very discomforting question. As a religious movement, are we consciously or unconsciously helping to perpetuate a culture of white supremacy? Over the past several years and in our own congregation here in Madison, we have forthrightly addressed such topics as the mass incarceration of African Americans, our nation's sordid history of discrimination which continues to handicap people of color, the problem of unacknowledged white privilege, the phenomena of implicit bias and how it affects white's perceptions of people of color and the importance of engaging with communities of color on their terms rather than our own. We have dealt with all those issues at considerable length. At the same time, First Unitarian Society has established ongoing working relationships with a predominantly African American congregation on the east side of Madison and with Latino organizations like Centro Hispano and Vosay's De La Frontera and with Young Gifted and Black. We have hosted community forums on Dane County's appalling equity gap. We have offered adult education classes on white privilege, sessions for parents on talking to children about race. There's another one to be held this afternoon. We've established vibrant ministry teams that are devoted to these very issues. But after all is said and done. Can we? Can we as one of Dane County's most progressive and socially active faith communities can we be justly accused of maintaining a culture of white supremacy?  And when this phrase was used initially to describe the UUA and its hiring practices, Peter Morales himself, the son of Mexican immigrant, said, if you call us that, then what do you call the Aryan nation? It's a loaded term. And it is one that sits uneasily on our tongues. But it's being invoked for a reason. First, because of its shock value, with the hope that it just might open our eyes to a deeply entrenched and unrecognized reality. And second, it's being invoked so that perhaps we can begin to appreciate how supremacy manifests itself in a wide variety of ways. And that's why we're interested to hate mongers. 25 years ago, Jake Lamar, a black Harvard graduate and a journalist who has worked for Time Magazine among other national publications, Jake Lamar called out an important liberal office holder in the state of New York who had exposed his racism with a subtle comment that he made to Lamar during the course of a political function. And so Lamar later wrote, liberals like us aren't racists, but this allows you to make easy calls against the David Dukes of the world while rarely questioning anything racial in your own enlightened sphere. You suffer from cognitive dissonance, Lamar said. You consider yourself astute on race issues, and yet you cannot acknowledge presumptions that you make solely on the basis of race. And so you steadfastly refuse to confront the mystery of your own manners. The mystery of your own manners. And that's what our brothers and sisters of color are alluding to when they use the expression white supremacy. As the current moderator of the UUA, Jim Key recently explained, it refers to a culture or a social narrative that places the needs, the desires, the stories, the well-being, the very lives of white people over and above those of people of color. And it is, he says, the water that we all swim in. We'll return to that definition and its relevance to Unitarian Universalism and to us here at FUS in part three of these reflections. The centuries, Unitarianism and Universalism have accumulated a fair amount of race-related baggage. And so if we are now challenged to examine some of our racially conditioned cultural assumptions, it is at least in part because we have not come fully to terms with our past. We remember with pride, on the one hand, notables such as Senator Charles Sumner, the Reverend's Theodore Parker, Charles Fohlen, Samuel May, and Thomas Wentworth Higginson, Lydia Maria Child, Mary Livermore. These early 19th century figures fought long and hard sometimes at great personal risk for the abolition of slavery. Both Parker and May harbored fugitive slaves and used a variety of means to prevent them from being returned to their southern masters. While serving in Congress, John Quincy Adams defended the mutineers who had taken over the slave ship Amistad, and ultimately he won their freedom. For their part, the Universalists adopted a strong anti-slavery platform at their very first national convention in 1790. And those Universalists who already held slaves were urged to free them and subsequently to educate their children. Dr. Benjamin Rush was a founding father. He was also a Universalist, and he assisted Richard Allen in organizing the first black congregation in America, which then grew into the African Methodist Episcopal or AME denomination. After the Civil War, Laura Matilda Town, a Philadelphia Unitarian, she carried her ideals and her skills to South Carolina, where she established the first school for freed slaves at Penn Center on St. Helena Island. Early in the 20th century, John Haynes Holmes, minister of New York City's Church of the Messiah, he co-founded the NAACP. He was a tireless opponent of Jim Crow and of racial inequality. The erection of a foolish prejudice into a condition of social organization, he wrote in 1923, that is bound to reduce the Negro to a level of inferiority where he simply does not belong. Unitarian Universalists were among some of the most fervent supporters of integration and civil rights during the turbulent 1960s. Hundreds of ministers and laypersons marched with Dr. King at Selma, Washington, D.C., Cicero, Illinois, and two white UU civil rights activists, James Reeb and Viola Liuso, joined dozens of African-Americans as martyrs for that cause. These are a few of the highlights that we proudly point to as proof of our progressive credentials. The good karma, as it were. But the other side of the ledger, that has to be considered as well. So a few brave abolitionists aside, the American Unitarian Association as a whole refused to take a stand on the issue of slavery right up to the Civil War. This was at least in part because some of Unitarianism's wealthiest and most influential members were mill owners and exporters and bankers who had a financial stake in the slave-produced cotton that came from the deep south. When the notorious Fugitive Slave Act was passed by Congress in 1850, a Unitarian president, Millard Fillmore, signed it into law. The Unitarian senator from Massachusetts, Daniel Webster, defended its enforcement in his home state. South Carolina Senator John C. Calhoun was one of the leading apologists for the institution of slavery. He was a member of the Unitarian Church of Charleston. In Boston, the Reverend Ezra Stiles Gannett, who served the prestigious Federal Street Church, he urged his parishioners to comply with the Fugitive Slave Act, causing Theodore Parker to comment, Gannett is urging members of his congregation to kidnap members of mine. Parker's church was one among the few that were integrated at the time. And despite his own dismay over slavery, Reverend Theodore Parker held deeply racist views. No doubt, he said, that the African race is greatly inferior to that of the Caucasian in general intellectual power, and also in that instinct for liberty, which is so strong among members of the Teutonic family. Such prejudice continued to inform Unitarian and Universalist culture once slavery had ended. Black aspirants to our ministry, to our professional ministry, were treated poorly, never really given the chance to serve established Unitarian congregations. In assessing the capabilities of one such minister, the Jamaican-born Ethel Red Brown, AUA President Lewis Cornish had this to say about Brown in 1932. I am told that this man shows the emotional temperament of his race and perhaps a weakness of judgment. I have lived much among Negroes, and I am inclined to be very sympathetic to their temperamental peculiarities. They are very lovable people and often very childlike. It would at once be unjust and misleading to judge Mr. Brown as you would an Anglo-Saxon. Finally, finally in the year 1961, and as the Unitarians and the Universalists merged to become one denomination, finally a black man, Lewis McGee, was called to serve as the lead minister for a predominantly white man. Now the minority presence in our movement grew during the 1960s, at least in part because of our involvement in the civil rights movement, only then to collapse in the face of a black empowerment controversy that erupted in the last years of that decade. By the end of the 1960s, many UU blacks wished to establish a separate stand-alone organization for themselves, for people of color, within the greater UU community. The integrationists in our midst, on the other hand, had other ideas, and they pushed back hard against the separatists. A clumsy and inconsistent response by the UUA led to a confusion over priorities, a sense of betrayal on the part of those who wished to create this separate black caucus and a subsequent exodus of many people of color from our movement. Much has been written about this turbulent era and its lasting impact on our efforts to foster greater diversity within the movement. Today, we do have more ministers of color serving congregations. We do have more administrators of color in leadership positions. Anti-oppression and anti-racism trainings are conducted at all levels and are freely available to all of our congregations. Defending black lives, protecting the interests of undocumented immigrants, these have become denominational priorities, and yet, as Leslie Mack, a black activist as she recently lamented, just when I think the UU faith has made some strides, I feel the pull of white supremacy in the faith and it snaps the back of my neck. It feels like whiplash. I now would invite you to participate in the giving and the receiving of our offering, and it will be shared this week with the Moses organization, which is doing incredible work statewide to end mass incarceration and to reform our criminal justice program. In responding to these complaints that the Unitarian Universalist Association has not done enough to address and to reform its inbred culture of white supremacy, in response to these complaints, the executive director of the Unitarian Universalist Ministers Association, a man by the name of the Reverend Don Southworth, he wrote, In a privileged and powerful position, I realized that for some, anything that I have to say must be filtered through the identities and the culture that I embody. And Don Southworth's comment holds equally true for me, and perhaps for all of those who operate out of what Joel Fegan has called the white racial frame that is our culture's long-standing default position. Honest self-scrutiny is hard to achieve, which makes it all the more important as Southworth acknowledges to seek counsel and wisdom from those who occupy a space outside of the prevailing frame. So let's try to bring the discussion home at this point. The First Unitarian Society of Madison is a substantial congregation. We are one of three or four of the largest in the association. The notoriety of our facility and our active involvement in the greater Madison community, this means that we do attract a fair amount of attention. And because of the distinctiveness of our non-doctrinal inclusive message, membership has remained stable in this congregation while most of the more conventional churches are suffering from significant membership attrition. The attributes that we possess have proven attractive to wide assortment of people, young and old, gay and straight, mystics and scientists, poets and philosophers. But when it comes to race and class, ours, like most UU congregations throughout the country, our congregation is pretty darn homogenous, isn't it? Blue and pink color folk are poorly represented as are persons of color. Mark Morrison Reid is an African-American minister who was raised as a UU in Chicago. He says that these two categories of race and class often intersect because many blacks, Latinos and even Asian-Americans are working class and they would not find our approach to church all that attractive or in the long run all that practical in terms of their own lives. Where does all this thinking you do lead, they might ask? What are you doing with all this on a practical level? It's not that we don't care about such things. Unitarian Universalists in Madison have been meaningfully engaged in the struggle for racial and economic justice for three quarters of a century. So for instance, in defiance of fraternization restrictions that were imposed by the armed services during World War II, Unitarians here in Madison held racially integrated USO parties in this facility for soldiers who were stationed at the air base out on the east side. After the war, the society fought against restrictive real estate covenants by opening a house on campus that welcomed students of all races. A few years later, FUS members helped to secure a passage of Madison's first fair housing ordinance. During the 1960s, our congregants enthusiastically supported the Civil Rights Movement both locally and in the deep south. Minister Emeritus Max Gabler writes that we established a close relationship with the civil rights effort in Sunflower County, Mississippi, a truck left Madison almost monthly with supplies for those who were active in the movement. And that remarkable lady, Fannie Lou Hamer, visited us twice in Madison and several of our young people participated in the Mississippi Freedom Summer Project. This is an admirable track record that we are certainly not allowed to rest on our laurels. White supremacy is, again, the brew that we are steeped in. And so it behooves us to understand how it affects us. And it's really not a matter, folks, of being guilty of white supremacy or not guilty, because it exists on a continuum with hate groups like the KKK at one end and well-meaning, conscientious organizations like the Sierra Club or the Summit Credit Union or FUS somewhere in between the extremes. As we can plainly see, FUS does suffer from a dearth of racial and ethnic diversity. Just look around. This is true of our staff. It is true of our congregation as a whole. One can cite several reasons for this. Our rather homogenous Near West Side location, a remarkable facility that fairly shouts elitism, a Eurocentric style of worship, a departure from Christian norms and nomenclature, even our long-standing support for marriage equality in the LGBTQ community. Some of the foregoing does reflect white upper-class norms. And thus there are plenty of persons of color in Dane County who are more than willing to partner with us, but not necessarily to be a part of us. I wonder how we might go about changing that. Well, some have suggested that we could change our style of worship, reshape our Sunday celebrations. We could make them less programmatic, more spontaneous, change the tenor and the tone of the sermons, music that gets the congregation moving in their seats, pray more often. Because nothing requires that we maintain the liturgical traditions that have served us these past 140 years. As Hassan, the Indian chef that was featured in the film The Hundred-Foot Journey, as he said to his boss after he had altered a venerable French recipe, perhaps 200 years is long enough to prepare a dish in the same way. In any event, it is now a time of transition for this congregation. Perhaps it's a conversation worth continuing. But a faith community is more than its worship. The traditions on this subject of white supremacy say that many of the practices associated with it are much more covert. And thus they say that when an institution values quantifiable results over process values, when it's defensive about its norms and seeks to avoid controversy, when it opts for linear, logical problem solving while discounting other ways of thinking and knowing, when there is competition rather than collaboration, then it is manifesting in all these ways clear signs of white supremacy. So again, I'm a white guy and I'm still in the early stages of unpacking all of this stuff. And I have not gotten far enough along in my own discernment process to assess the validity of all of these critiques that I have been reading about. And yet I do think the questions that are being raised here are legitimate questions and they are questions that we ought to be dealing with. In her reflections of last week, my colleague Kelly Crocker spoke of the importance of belonging and of the need for us to try to widen our welcome in order to make FUS a more hospitable place for all people. An open discussion of white supremacy could very well help us to do that. And Kelly also noted that people always feel more at home in a place when you're trying to resolve the inconsistencies between your walk and your talk. And that she says is what we mean by striving for integrity, striving for wholeness to bring our walk and our talk closer together. And ultimately this is a question of what kind of faith community we want to be. Today is just the very beginning of this conversation. And I urge you to let it continue because to paraphrase the early 20th century Unitarian activist Fanny Barrier Williams, the hope of the dark races in America depends on how far whites can assimilate their own religion. Blessed be and amen. I invite you now to lift your voices in song once more. Our closing hymn is number 120, which is in your teal hymnals. Please be seated for the benediction and the postlude. If we have been unsettled by this time together then perhaps it is good for tranquility due to a lack of awareness brings only a false and precarious piece. If we have learned something new in our time together then this is good for new light must be ever breaking in upon us to dispel the clouds of darkness. And if we are leaving still unsure of where the truth lies this still is good as well because the truth is never all in. The quest for wisdom and understanding must be open-ended. If we are going out inspired to change our world or ourselves just a bit for the better then this too is good and a blessing for the brief time that we have spent together today.


Severino mentioned that there were some communication issues tonight. Why do you think it took you guys a while or why couldn't you really get on the same page? Yeah, tonight we were using a new set of signs. It was the first sign for us using the signs and yeah we definitely had some a little bit of trouble you know getting on the same page in that first inning you know I can definitely say there were some some pitches there that I should have done a better job but blocking them or protecting them from you know going to the back but yeah you know after that we were able to focus and make the adjustments and continue continue battling. Did you feel like things were kind of spiraling out of control on that first? Yeah definitely not anything in it right there you know at the same time you gotta find calmness and you gotta find a way to get through it you know once you do that you know you can't let what happened before rattle your your mind you know you still have a job to do and you're just trying to get the job done. Sign society it seemed like Severino was off overall what were you seeing from an advantage point in that regard. Yeah um yeah he was a little off tonight you know and like any ballplayer you know and any given any given day you know you're gonna have enough night he also executed some good pitches unfortunately the other team you know were able to put those hit those pitches very well and nothing now we just gotta look forward and you know see our way friend to bounce back.


So, we will be showing here a short presentation demonstrating various uses of open SSL toolkit, mainly it will be related to the presentation which we just saw. But before that we will take a look at a few Linux commands which are relevant to network security. So, the first command is the IF config, this command is used for viewing the settings of the interfaces of the machine. So, here you can see that the name of the interface is ETH3, there is a MAC address of the interface, it shows the IPv4 address, broadcast address of that network, network mask there is also IPv6 address over there and there are a few more counters below that which shows how many packets are transmitted and received and such things, below that there is also a loopback address, but which is just a self interface and after the IF config we will take the look at the ARP command. So, what ARP basically does is it shows you the ARP cache of the machine, ARP cache is basically mapping between IP addresses and MAC addresses in the local network. So, here we can see that in the ARP cache we have addresses of four other machines. So, I will just pick one of these machines and try to check whether it is alive or not, for that we will use the PIN command. So, what Swapnil is showing you is a bunch of Linux commands, the first one was IF config, IF stands for interface, it shows you what are the different IP addresses on the different interfaces and the MAC address and so on, subnet masks at each interface. The second command was the ARP, ARP stands for address resolution protocol, so with the ARP you can actually inspect the ARP cache at a machine and after that he is trying to use a PIN command. So, these are all basic Linux commands which you will try out in the lab today during the afternoon session. So, here we have PIN this machine <IP_ADDRESS>, so it is giving us replies right now. So, the PIN uses an ICMP echo protocol over here and the machine which is there at the destination replies us with ICMP reply. So, unless until we stop this PIN command it will keep on trying the echo equation reply. So, for that we can give it a minus C options which tells the PIN program to stop after a few PINs. For example, I have given here minus C3, so it will try it only three times. If in case the machine was not alive we will get here as 100 percent packet loss, but here we are getting that three packets have been received, so it means the machine is up and running. After this PIN command, we will have a look at the NSLOOKUP command. This is used to resolve the domain names of machines. For example, here I have used NSLOOKUP space IITB dot AC dot IN, it will try to resolve this into an IP address. So, here we have gotten reply 10 dot 200 dot 100 dot 1. There is another utility in Linux which is called host. It also does the same thing, here again we have got the same reply. And this is finally, we will see the trace root command over here. What it does it? It shows you all the hops in between our machine and the destination machine. Then here also it uses the ICMP echo protocol to find these intermediate routers, but there is a field called TTL in it in the IP which it varies, so that the intermediate routers give the reply. Now, we take a look at the secret key typography and for that we will take an example as the AES encryption standard. So, here we will again use the open SSL toolkit. Next I will show that I have a message file over here. This message file contains some text and each text I mean each line of this text has 15 characters and there is a new line character at the end of each line. It means that there are 16 characters in each line and this is repeating sequence of three lines. So, we will encrypt this text using the AES. 256 is the key length of AES, minus E option stands for encryption, this is the input file which is the message and the output file will be named as ciphertextECB1.bin, it is a binary file. Note that here we have given the AES to use the ECB mode, it is the electronic code or book mode. So, here I will give some password over here to encrypt the file. So, these are the contents of the ciphertext. We can see here that the ciphertext is also getting repeated every 16 bytes. This is usually not a good thing because if somebody knows ciphertext before and then he can know the plaintext using that and maybe even the key. What do we notice out there? We had a bunch of zeroes, ones and twos, one full row of zeroes, one full row of ones, one full row of twos and the sync kept repeating. Now just look at the corresponding ciphertext. There is an M sort of thing on the first row, F and all that on the second row, six AES and the M thing that row keeps repeating with a periodicity of three. So, every three rows the thing repeats. Now, why does it repeat? This is like a known plaintext attack. If I know the plaintext on the first row, I can conclude what is the plaintext in the fourth row and so on and so forth. Now, suppose instead of having 15 characters, actually if you recall, in the original when you saw the plaintext, there were 15 zeroes and there was one new line character. The new line character was not visible on the screen but that is a total of 16 characters and 16 characters corresponds to each character's 8 bits. So, 16, 8, 128 bits is the block size. Now, suppose I make the plaintext just 14 bits, each of those lines, suppose I made just 14 bits, sorry, 14 characters, then what would happen? Would this thing repeat or not? So, that's a question I'll again state the question. The original plaintext that you saw with Swapnel had put inside a file, that plaintext was 15 zeroes followed by a new line character followed by the next line 111 repeated 15 times followed by 222 repeated and so on and then 00 all over again. Now, there were 15 zeroes, 15 ones, 15 twos. Now, suppose I had only 14 zeroes instead of 15 zeroes on the first line and similarly 14 ones and 14 twos and so on, would I see a repetition of the ciphertext? That's the question. Here, you see a repeat of the ciphertext but if I had only 14 instead of 15, would you see a repeat? That's the question and the rest of it is why or why not? Okay, so we will see now when the each line contains less than 16 characters, what happens? So, we have another file over here, it's named as, first we will see that if we encrypt the same file using the cipherblock chain mode, does the repetition occur in the ciphertext? So, here we have given the AES 256 CBC option, so we will encrypt this file and we will use the output. Here, you can see that there is no repetition anymore over here while before in the electron codebook mode, there was a repetition in the file. So, I hope all the participants can appreciate this. In the case of CBC mode, you don't see a repeat. Each line is 15 characters as before with the new line character which is another character, so 16 characters. 16 multiplied by 8 bits per character, 128 bits is the block size. But what you see is that those lines do not repeat as in the case of ECB mode, so you can verify this in the lab today. So, we saw that in CBC mode, there will not be any repetition in the ciphertext. So, now we will see that file in which there are less than 16 characters per line. This file is having 14 visible characters and so there will be a one new line character trend of each line, it means it has 15 characters per line and the sequence is repeating after each 45 characters. So, if I encrypt this file using the ECB mode again, we should not be able to see the repetition in the ciphertext. Again, I am using the same password over here. So, these are the contents of the encrypted file when I use the ECB mode on the message file which has 15 characters per line. There is no repetition over here. Note that this file contains 15 characters per line and a new line character and there is some modification in this file. Here I am having a 3 instead of 2 whereas in the original file it was all 2s. So, now we will see that if some byte in a message file is changed, then what are the changes we see in the cipher? I will encrypt both of these files using the ECB mode, electronic code book mode. We are seeing the repetition as before again, this is the same ciphertext as before. Now, we will encrypt the modified file. Here you will see that there is some repetition, but at one point the text changes. This is the point where we made the modification, this entire block has changed because of that. So, when we in ECB mode when we change one byte, the block in which that byte falls, only that block will change, rest of the things will remain the same. This is the original block and this is the modified block because of just one character. Now, we will perform the same test using the CBC mode. Again here there is the same thing one byte is modified. So, since this is CBC mode of course there will be no repetition in this file, but we will compare it with another ciphertext which has the modified plaintext. We will compare both this ciphertext now. You can see that these five blocks over here will look different in the another ciphertext. So, what happens is that just because of one byte the changes propagate forward in the ciphertext. This is because the chaining in the CBC mode. Next we will take a look at the public key cryptography using the RSA in open SSL. First we have to generate a public key private key pair to do that this is the command. This has generated a private key now. So, this private key is in the base 64 format. From this private key we will extract the public key using this command. So, this public key is also in base 64 format. So, but we cannot view whatever parameters are used in the RSA. To do that we have to use this minus text option in open SSL. So, I am opening the private key using the minus text option. It will show all the parameters it has used. So, the next set of experiments over here is to try to find the different parameters of RSA and to look at what the ciphertext is obtained when you do encryption. So, if we go back to the original project. So, these are the different things that you see. This is 512 bit RSA. You see the modulus over there in hexadecimal. So, you can actually compare and just check how many bits are there in the modulus. Are there really 512 bits or something else? Then the public exponent. Typically the exponents are small values. So, you might be surprised about this that the public exponent that is the encryption key is a small value. Around 65,537. And you can see it in binary just next to it. And then the private key. The private key is typically about as large as the modulus. Almost always. And then the two prime numbers. The two prime numbers P and Q which are about half the size, half the number of bits of the modulus. So, you can inspect all these parameters and try and make some sense of it. In particular, look at the number of bits. It says it is 512 bit RSA. Make sure that you have enough hex characters in those roles to give you actually 512 bits. Notice that the public exponent and don't be surprised that the public exponent is so small. There's no loss of security by making it small. Look at the private key. See that it is almost as big as the modulus. In fact, it is pretty much the same size. And then look at the two prime numbers and see if they are half the size of the modulus. Because the modulus is P times Q. So, this is some message file which we'll try to encrypt using the RSA. So, we have to give the public key as input here. So, this is the safer text. And we will do the reverse operation now. In reverse operation, we have to use the private key. We'll save it in another file and we will view the contents. It's the same. Using the RSA unit, we can encrypt and decrypt using the AES in the open SSL. Now, we'll take a look at various hashes and hashback, hash-based message authentication and signatures. So, again, I'm having the same text file over here. I will try to find the MD5 hash of this file. So, it is showing the hash in the hex format. We can output this in binary format also. We'll change something in the message file and see what happens to the hash. I will change just one character over here. I will change the i to j. You will notice that almost half the number of total bits will be changed in this. In hash, whenever we change, in the message, if we change even a small bit, the entire hash will change. And on an average, half the bits will be changed. Now, we will see the hash base. So, one of the important things that Swapnir just demoed was if you take some message and you compute the hash and you change even a single bit in that message, roughly half, I'm not saying exactly, but roughly half the bits in the hash value will change. That will be diffused all through the hash value. It's not like if I change only the first part of the message, the first part of the hash will change. That the change will be all over the hash value. So, Swapnir, can we just see the two hash values and where the bits are changed? Like over here, in the modified hash, all ones are here in this byte, but here only three ones are there. Similarly, over here, we can see that the bits are changed. If we count these ones over here and the number of bits toggled over here, it will be roughly is equal to 64. So, roughly half the bits are flipped in the hash value and where exactly they are flipped, we don't know. It could be diffused over the entire hash value. Now, we will take a look at the hash-based message authentication. The hash-based message authentication also takes input key along with the message. The HMAC option over here lets us input a key. This is some random password I am putting here. So, we will be using this HMAC in binary. Now, we will see that if we change the key little bit, what happens to the hash? Here also on an average, half the number of bits will change. So, this is a property of hash. If in any input, if you change a little bit, the hash will change entirely and roughly half the bits will change. This HMAC is used for a message authentication. In this, the board sender and receiver will be having the same key and to verify the hash and to create the hash. There is another way to do this using the signature, but it has different advantage in hash. In HMAC, if someone uses HMAC and sends a message and later he can say that I did not send the message. But in signature, there is a property of non-repedition. So, here I am using the same public key, private key pair which I created in RSA. This is the message file of which we will create a signature. This is a signature in the hex format. And now we will verify whether this signature is correct or not. Notice that when we created the signature, we use the private key which is known only to the sender of the message. And the signature can be verified by anyone using the public key. So, I have given as input the signature over here and the message file. So, it is saying that verified OK. And what if the message file was changed a little bit while transmitting? If I change something in the message file now and verify it using the same signature. It has to show that the signature is incorrect. It says verification failure. So, this is what I was talking about. The demos that will be done in conjunction with the lectures. It is this demo that will be part of the lab today. So, whatever commands you have seen in open SSL that Swapnel has typed and demonstrated to you, the same thing you have to type in the lab. So, 70 percent of the lab will be basically typing those commands and appreciating the different things that we are trying to illustrate by those commands. So, for example, if you change just one bit in the message, then the hash value approximately half the bits change and there is no locality in this change. So, if you change one bit in the start of the message, any bits in the hash value could possibly change. The goal was also to tell you about RSA. What does the, what do the two primes look like p and q? How many bits are there in the two primes? What does the modulus look like? What does the phi of n function look like? What does the encryption key look like? That it is a very small encryption key of just a few digits, while the decryption key is almost as large as the size of the modulus. It was also to show you what is the effect of using CBC rather than ECB. That the known plaintext attack will be thwarted if you use CBC rather than ECB. Also the repeating plaintext, which was a deliberate choice of plaintext. If you, if we showed you a repeating plaintext, which had a periodicity of 128 bits and a periodicity of less than 128 bits, say 120 bits and then the plaintext does not repeat or you cannot see it repeating because the size of the block is precisely 128 bits. So, these are the things that we are trying to illustrate with this open SSL lab, which you will do be doing today. Now, we understand at this end that you might have problems viewing the video. So, when you came here, when the coordinators came over here, the screen was pretty clear and so on, but when you are doing it at the remote end, there might be problems. We are going to try and address these problems in the coming days. For the time being, however, you can take a look at the video. The video, hopefully you have already downloaded. The organizers will tell you how to download the video. Download the video as soon as you can and during the afternoon session, maybe the coordinators can replay part of the video so that the participants feel comfortable in these very important experiments with Linux where we looked at IP config, where we looked at Ping and ARP and so on and so forth and now in this crypto lab, we did not have the crypto lab in the 5-day workshop. I thought it would be very enlightening for the participants to not just derive equations, but also to understand what these keys are and what is the effect of encryption and what does it actually look like, what does the plaintext, ciphertext look like. And you can choose a key and change a bit of the key and change a bit of the plaintext, etc. and see what is the effect on the ciphertext. So, I hope that these problems with viewing will be sorted out. We would like your feedback as soon as possible. You can tell us, especially there are problems that linger on with the viewing because there are many demos, every session will have a demo and we want to make sure that you can see the demo as clearly as possible. So, please tell us what is going on and let us, if you have any suggestions also please inform us. There are some questions that have come which I would like to address. So, we have still about 10 minutes to go. So, one question was regarding the values of E and D. How did you obtain E and D in the case of RSA? So, in the case of RSA you choose these two primes, the software that is actually generating the key. Say it is a Java API to generate an RSA key pair. They will choose a large prime number of say for example, 128 bits. If you are talking about 512 bit RSA, they will choose P and Q to be about 1 and 28 bits. So, that is the first thing that will be done. The software will choose prime numbers. Now, how do you know that something you have chosen is prime? There are well known algorithms to do this actually that I mentioned in various books. I think I have a little mention about how to generate prime numbers.  The product is the modulus N. Now, the question that came up from one of the remote sites is how did you obtain E and D? So, the first thing is you just choose an E and D that is relatively prime to phi of N. Now, how do I know that something is relatively prime to something else? I will perform the GCD test. So, there is an algorithm called the Euclidean algorithm. Again that is explained in the text. The Euclidean algorithm to find out the GCD of two arbitrary integers, positive integers. If I want GCD of M and N, the Euclidean algorithm will enable me to find GCD of M and N. So, I choose an E and I make sure that I do this algorithm for E, E and phi of N and I make sure that this thing is equal to 1. So, that is the way I choose E and I try to choose a small number. So, obviously, I cannot choose an even number because phi of N is going to be even. It is an odd number, this is an odd number, this is an even number, even number, this is a number that will be divisible by 4. So, phi of N is going to be an even number. So, E will be typically an odd number. So, typical choices are something like 3 and I think what we had in the demo 65537 I believe. So, you choose an E and then there is the extended Euclidean algorithm which will enable you to find D as the inverse of E modulo phi of N. So, these are standard algorithms and in the textbook this is actually shown the different steps of the algorithm. So, you can program this or you get the program from somewhere and just verify that the D that you choose is relatively prime to phi of N. So, that was the question about how did you get E and D? I need an explanation for the calculation of E and D. So, how did I obtain E? I just chose an E which is and then I verified say I choose an odd number and I verified that that E was relatively prime to phi of N. I choose it small because I want encryption to be fast as fast as possible. There is no harm in choosing a small value for E, there is no security issue we are choosing a small value of E. So, choose a random number which is odd and make sure that this test is satisfied. Then go ahead and compute D as the inverse of E modulo phi of N. The inverse is computed using the extended Euclidean algorithm. To compute the GCD you use the Euclidean algorithm, to compute the inverse use the extended Euclidean algorithm. Both of those are shown in the text. Ok, so I hope that answers the question about an explanation for the computation of E and D. There was another question about the S box. What is the functionality of the S box? So, it turns out S box is the one non-linear component in the secret key cipher and it has to be chosen very correctly. Now, what does it mean to choose it correctly? It means that it should be immune to things like a crypt analysis various crypt analytic attacks. And one attack is linear crypt analysis, another is differential crypt analysis. You can study the details of linear crypt analysis in the text. So, this non-linear text box is necessary so that you do not succumb to a linear crypt analytic attack. And what do I mean by that? You are unable to find a linear combination of bits in the plain text, in the cipher text and in the key. So, it should not be possible to say for example, your plain text is p 0 to p 127. I should not be able to find a linear combination say p 1 0 5 exclusive or with p 98 exclusive or with p 31 exclusive or with cipher text bit number 25. Suppose, I can find expressions of the following that hold all the time. This is a linear expression which is these variables are operated upon by this exclusive or gate. So, if I can find expressions of this type then I can get clues as to what could be bits of the certain bits of the key. And then the this cipher is vulnerable to these kinds of attacks linear crypt analysis for example. That is why the design of this S S box has to be it has to be very very careful when you are designing this. And again details of this are in the chapter on secret key cryptography in the text. The last part of that chapter deals with linear crypt analysis with a very detailed example of an attack on the S box in a secret key cipher. So, ciphers such as DES and AS will make sure that these kinds of things will not occur. So, basically what it is is that if I have got this as the S box and there are inputs to this. Let us say 3 inputs 3 outputs and a simple toy S box in the case of AS there are 8 inputs and 8 outputs. Then if I take any combination of inputs and outputs say this bit this bit this bit this bit and this this this and something like that. I cannot find a linear relationship between them. I should make sure that regardless of what are the inputs and what are the outputs. Something like say these inputs are I 0 I 1 I 2 I write up to I 7 the case of AS there are 8 inputs and 8 outputs. O 0 write up to O 7 then I should not be able to find expressions of the form. For example, I 1 exclusive or I 5 exclusive or O 0 exclusive or O 6 for example, such things these things will never be true all the time. But it should be the case that even 80 percent of the time I should not find this thing to be true regardless of what kind of input I give over here. Suppose I give some in 7 bit input to this thing I get a 7 bit 7 bit output and then if I look at the different values of the input I gave and the corresponding outputs. It should not be true that this thing holds all the time. So, 50 percent of the time if it holds that means this is 0 and 50 percent of the time it is 1 then I am very happy. There should be no bias towards 0 or 1 it is called a bias. So, it is these kinds of things that I will use to design my S box I will try to make sure there is no bias towards 0 or towards 1 for any combination of inputs and outputs. And once again some detailed examples of these are given in the textbook. So, I guess that this is an attempt to answer the question about the S box and the design of the S box.


Hi, this is Dr. Ruscio and let's discuss how vitamin D can help you heal your gut and more importantly, how that healing of your gut can lead to healthier bacteria in your gut and most importantly, how all these changes can lead to an improvement in your digestive symptoms and your symptoms of irritable bowel syndrome, IVS. So I'd like to start with this first study here. I'm going to go through a few studies and tie together a few thoughts for you. So the first study, bolus, weakly vitamin D3 supplementation impacts gut and airway microbiota in adults with cystic fibrosis, a double-blind, randomized, especially low-control trial. Their conclusion, this exploratory study suggests that vitamin D insufficiency is associated with alterations in microbiota composition that may promote inflammation and that supplementation with vitamin D has the potential to impact microbiota composition. Now remember, if you have inflammation in the gut, that inflammation will create an environment that allows unhealthy bacteria to grow and is poisonous to healthy bacteria. So let's continue along this theme and look at another similar study. The role of vitamin D on gut microbiota in cystic fibrosis. There seems to be a fair amount of research for whatever reason on vitamin D, the microbiota, and in the condition of cystic fibrosis, but I will connect this to regular non-cystic fibrosis populations in just a moment. So to quote, recent evidence shows that vitamin D acts to maintain the integrity of the gut mucosal barrier by enhancing intracellular junctions that control mucosal permeability aka leaky guts and reduction of pro-inflammatory cytokines or pro-inflammatory molecules. Due to effects on the intestinal mucosa or the mucous membrane in the gut, maintenance of sufficient vitamin D status may be essential for the development of a healthy gut microbiota, particularly in conditions defined by chronic mucosal inflammation. So again, seeing this theme that vitamin D helps with essentially leaky gut function, the gut barrier, and that leads to healthier gut bacteria. One more study, gut microbiota interactions with immunomontoratory role of vitamin D in normal individuals. The role of vitamin D in the maintenance of immune homeostasis or balance seems to occur in part by interacting with the gut microbiota. And just a couple of specifics from this study, lipopolysaccharides increase with reduction of vitamin D or lower your vitamin D. The higher lipopolysaccharides are in lipopolysaccharides are just one measure essentially of leaky gut. Continuing, after adjusting for age, sex, season, the season of the year will affect vitamin D levels of course, sunny months higher vitamin D, and BMI or body mass index, there will still maintain an inverse association between vitamin D and Bifidobacterium. So after we adjust for people of different age, people of different sex, different times of the year and body weight, all of which can affect vitamin D, we see that as vitamin D goes down, you have a decrease of Bifidobacterium. So there seems to be this positive impact of vitamin D on Bifidobacterium levels. So all this is very interesting. We see that vitamin D can help with inflammation, leaky gut, and the bacteria in the gut, but it's very important to be able to tie these observations to an actual measurable, meaningful outcome in humans. Do these people feel any better? Because if all this stuff was happening underneath the hood and patients felt worse or the same, then it wouldn't really be that meaningful. So it's very important we tie these mechanistic changes to an actual measurable positive outcome in humans. So continuing, effective vitamin D on gastrointestinal symptoms and health-related quality of life in irritable bowel syndrome patients, a randomized double-blind clinical trial. Summarizing, vitamin D improves IBS symptoms compared to placebo, key, key finding. And in this study, 50,000 IUs were administered every other week, which if you broke that down to a per-day dosing would leave you with about 3,500 units, which is pretty common daily dosing for vitamin D, usually between anywhere from 2,000 IUs per day to 6,000 IUs per day is what's used. And I'll also put a link in here to our previous episode where we reviewed this particular study in more detail. So that's very exciting. It's also important to keep in mind that there are health benefits that you can only obtain from sun exposure and you cannot obtain solely by supplementing with vitamin D. So it's important to practice safe sun exposure in addition to supplementing with vitamin D. And in my book, Healthy Gut, Healthy You, I lay out specific guidelines for how to use safe sun exposure to optimize your vitamin D levels. So in summary, we show very exciting evidence that vitamin D can essentially heal the gut. That reduction of gut healing, reduction of leaky guts, and reduction of inflammation in the gut leads to healthier bacteria in the gut. And then most importantly, all these changes summate to an improvement in people's health, their well-being, and their digestive symptoms. So vitamin D, very important for one's gut health, one factor of many to include in your gut healing regimen. This is Dr. Ruscio, and I hope this information helps you get healthy and get back to your life.