So this brings me to a tweet by Jeff Bezos now I know most people think Jeff Bezos is some kind of Some kind of leftist, but I don't I don't think Jeff is I Think he's probably more of a kind of a libertarian who doesn't have a political home And he probably voted Democratic because he hated Trump But probably he's not a Democrat and doesn't regularly vote Democratic Here's Jeff Bezos's tweet. This is in response to Joe Biden's tweets. So Joe Biden wrote this you want to bring down inflation Let's make sure the wealthiest corporations pay their fair taxes. They pay taxes Again Joe Biden wrote you want to bring down inflation. Let's make sure the wealthiest corporations pay their fair share of taxes How do would that bring down inflation? Where's the economic theory that suggests that corporate taxes are related to inflation? There's no relationship between the two, but that is the United States government official tweet coming out of Joe Biden Jeff Bezos response The newly the newly created Disinformation board should review this tweet or maybe they need to form a new Non-sequitur board instead. Whoops. Sorry my Jeff Bezos tweet tweets jumped Raising corporate taxes is fine to discuss Taming inflation is critical to discuss Mushing them together is just misdirection. I'll read you Jeff Bezos's tweet again. The newly created misinformation board should review this tweet or Maybe they need to form a new non-sequitur board instead raising corporate taxes is fine to discuss Taming inflation is critical to discuss Mushing them together is just misdirection. Other people comment on Deficits and other things and Jeff Bezos Later writes in fact administration tried hard to inject even more stimulus into an already overheated inflationary economy and only mansion Mentioned that senator from West Virginia saved them from themselves Inflation is a regressive tax that most hoods hoods the least affluent misdirection doesn't help the country This is exactly what the Democrats are doing. There's an inflation problem. We know what caused it primarily the Federal Reserve Monetizing the massive government spending that went into covert relief and spending three point five trillion dollars creating massive deficits and The Federal Reserve monetizing that that caused inflation Democrats Inflations caused by corporate greed Which didn't exist I guess when there was no inflation Inflations caused by corporations paying too few taxes Inflation is caused by anything except the thing that we Really advocate for which is lots of government spending Now I don't think anybody's buying the misdirection from the Democrats But it is interesting to watch This is the White House's response to Jeff Bezos Quote it doesn't require huge leap to figure out why one of the wealthiest individuals on earth Opposes an economic agenda for the middle class that cuts some of the biggest cost families face Fights inflation for the long haul and adds the historic deficit reduction The president is achieving by asking the richest taxpayers and corporations to pay their fair share It's also unsurprising that this tweet comes after the president met with labor organization Organizes including Amazon employees so again Yeah, I mean everything Biden is doing is gonna reduce inflation is gonna reduce the deficit Including spending additional three and a half trillion dollars Jeff responds Look a squirrel This is the White House's statement about my recent tweets They understandably we want to muddy the topic. They know inflation hurts the neediest the most But unions aren't causing inflation and neither are wealthy people Remember the administration tried their best to add another three and a half trillion dollars to federal spending they failed But if they had succeeded Inflation would be even higher than it is today and inflation today is at a 40-year high Jeff Bezos is not some nutty leftist He might own the Washington Post But does he have a tutorial say in the Washington Post as he hands off owner we don't know Everything I know about Jeff Bezos suggested he is a small L libertarian. There's a free marketer at the core It's not a socialist. He's not a redistributionist. He's not a You know big government type at all indeed. I think many people in Silicon Valley Who accused of being leftists are like this. They they probably support much of the social agenda of the left but an economic policy is they're free market people so What can I say? I always am happy to see a Business was standing up for himself talking back to politicians Defending himself and defending his ideas Yeah, he he none of these people are perfect. I get it. None of these people are perfect But the reason he lobbied for $15 minimum wage Do you know why he lobbied for $15 minimum wage because he felt like he was forced to provide it to his employees under threat of government action by Trump administration of all people and he said look if I am going to have to pay $15 minimum wage Then everybody should Now I'm not justifying. I'm not saying he's justified in doing it But if you ask Jeff Bezos, do you believe there should be a minimum wage and If we got rid of minimum wage across the board from Amazon and everybody else, would you be happier? I Bet you he would say yes Bet you he would say yes.  I'm sure is he somewhat of a compromise on other things. I'm sure Is he perfect? No God He's one of the most successful businessman in human history He's improved your life and my life dramatically and he has a voice that once in a while he uses to stand up to government I'm all with him I'll take a businessman like Bezos over any politician Particularly businessmen turn politicians Some of whom, you know Democratic party when it comes to inflation is aggressively trying to divert attention away From what causes inflation? Government spending out of control And a fed that bails out everybody left and right. Linda. Thank you for the support. Really appreciate it Who else did I miss here? Anonymous user. Thank you so It's They're afraid Inflation is going to really really hurt them Yeah, in in recent times alone muskets stood up to government more than jeff bezels says no question Still good to see jeff bezels do it I wish more did it There's no accident by the way that he's only standing up to government now that he's left amazon, right? He's left A ceo. He wouldn't be doing this if he was ceo of amazon Thank you for listening or watching the iran brook show If you'd like to support the show We make it as easy as possible for you to trade with me You get value from listening you get value from watching Show your appreciation you can do that by going to your own book show dot com slash support by going to patreon subscribe star locals And just making a appropriate contribution Uh on any one of those uh any one of those channels also if you'd like to see the iran book show grow Please consider sharing our content and of course subscribe Press that little bell button right down there on youtube so that you get announcement when we go live And for those of you who are already subscribers and those of you who already supporters of the show Thank you. I very much appreciate it


n nationals going at them these are so sweet very much like an impeccable style car with the the refractor kind of that silver on the five five tool phenom very clean two more boxes white should be our redemption and if redemption is the last card going to be ronnie doson cincinati red's edition of ronnie doson going to go to denis rookie auto and the redemption good look everyone let's get a good one here big hits congratulations you've won you are due to receive a five star autograph jumbo prime card green father we got a 25 of t liens of austin riley nice big bopper for the braves tanner said i was a bit above his budget still gets one knocks it out here austin riley jumbo prime patch and auto green out of 25 had in your way nice man congrats and the last box thanks again everyone for getting this done happy holidays last break before christmas without tuesday the 27th thanks so much for kicking it with me sticking around getting breaks done all damn day and all damn night so thank you thank you thank you let's get last two autos starting off for the milwaukee brewers jackson reitz how's going to go to chat looks like a horizontal should either be a silver or a gold has been fun i was i was hoping to be at one more bigger one at the patch auto besides the silvers one more for the philadelphia fillies bobby abray you first time we've seen this one all night 13 out of 30 for the potential future hallfamer philadelphia fillies last card in the break going to chad very nice nice fun way to end it tops five star baseball pyc number 10 is on the site right now i'm calling it a night though thanks again everyone jasky's case breaks dot com see you after the holidays later nice thanks y'all thanks for giving us to the goal line there thanks for converting in the red zone let's get this bad boy uploaded let's call it we can leave this it's fine that's already printed out wait will it done i i had nine of these no booklet i'm crazy it's it's annoying at this point really hey man have a great night too happy holidays thanks for sticking around with us thanks for breaking thanks for ripping i guess i'll put it here i'm still putting it this friday i think i will thank you again everyone i thank you chat happy holidays whatever you're celebrating be safe have fun happy holidays to all and to all to all a good night thanks again y'all we'll be back tuesday the 27 jasmine's case breaks dot com let's do it soon bye


The TA for this course is Yana Sotakova, and she'll be doing the problem sessions. So I wanna thank Yana to start with for her help with putting together this course. So I'm not gonna go in detail over the outline right now, but just so you know, the goal of my first lecture is to really explain all the words in the title. So super singular, isogenic graphs in cryptography. So let's just start with cryptography. I heard that Joe Silverman gave a nice lecture last week about cryptography as well, and some of you may have already some familiarity, but just from a very high level, very briefly, I'd like to say something about cryptography. So it's the science of keeping secrets, but more than that, so we often, most people probably know about encryption. People don't often think about it as authenticity. So cryptography is also used as a tool for guaranteeing authenticity. So one way to think about the importance of authenticity is that if you are connecting your web browser through a secure connection to buy something, let's say, from Amazon, and the secure connection is supposed to protect the confidentiality of your credit card number, that doesn't do you a lot of good if you're actually connected, instead of being connected to Amazon, if you're connected to some other party, and you're giving them your credit card number and your money instead of the party that you're actually trying to buy something from. So authenticity is very important as well, and the tools that we use for protecting authenticity are called digital signatures, and sorry, there was a little lag in advancing the slides, and the technology that we use for creating, for example, secure browser connections is key exchange. So I generally think of the three main building blocks in public key cryptography being key exchange signatures and encryption, and one way that you can see that this is roughly a good way to think about cryptography is that over the last five years, there's been an international competition called the Run by NIST, which is the National Institute of Standards and Technology, and they've been run for two parties to agree on a common secret using only publicly exchanged information, whereas signature schemes is some mechanism to allow parties to authenticate themselves. So one thing, and when we say public key cryptography, we're generally referring to the idea that there's an asymmetry, you also hear it called asymmetric cryptography, and that is that there could be some public key that everyone knows or that you publish, and then there's some secret key that corresponds to that public key, and so if you are the authentic user, for example, and you're trying to demonstrate your authenticity, you would use your own secret key to sign something, digital signature scheme, and then anyone could actually, using your public key, verify that signature. So one thing I think is interesting to point out, just from a societal point of view, is that this type of scheme requires reliable, right? It doesn't do you any good to verify somebody's signature with their public key if you have the wrong public key for them. So you can think of what we call PKI, public key infrastructure, as being a kind of a public phone book that actually is a trustworthy log of people's record of people's public keys. And so it's an interesting societal problem to even think about who and how to maintain a public key infrastructure. So I'm not all that up to date on everything that's going on in the world, but it used to be about five years or so ago that the only government that I knew of that actually maintained a public key infrastructure was South Korea. So in the US, we rely on private companies to maintain public key infrastructure to allow us to securely authenticate ourselves. So if any of you have had the experience of traveling with your laptop abroad and then all of a sudden you'll get these messages that say, hey, the certificate for this website or the certificate for this connection is not recognized. And that's because we don't have a global public key infrastructure. Okay, so now to get back a little bit to connect to the mathematics of public key cryptography, examples of public key cryptosystem that are deployed today are RSA, for example, which is based on the hardness of factoring large integers. Diffie-Hellman, or actually the discrete log problem, which is related to Diffie-Hellman. Unfortunately, I don't have time to go into all the details about these systems, but that's based on the hardness of the discrete logarithm problem, either in the multiplicative group of integers modulo p, some prime, or in the group of points on an elliptic curve. So ECDLP or ECDH are the discrete logarithm, or the elliptic curve discrete logarithm versions of the Diffie-Hellman and DLP problems. And then digital signatures, DSA also has an elliptic curve version of this, ECDSA. And typically, just as a little side note, the way things work is that, so how do we get to the point that we're all using RSA or elliptic curve cryptos? So there's actually a standardization process, which earlier in my career, I worked a lot on standardization of elliptic curve cryptography, where experts basically researchers like the people in this room get together and determine the security of the elliptic curve systems, like how hard they think these systems are to break. And then the government agencies, such as NIST and other agencies, and also professional societies, such as IEEE, create standards based on the researchers, what the researchers have determined. And then finally, the last stage is that there's industry-specific standards that are created for the use of those things. So for example, for elliptic curve cryptography, there's X962 and X963 are the ANSI standards, ANSI is American National Standards Institute, that created secure standards for using elliptic curve, both for digital signatures and for a key exchange, surprise, surprise. So that's how these systems actually come into widespread use. And over the last five years, I've been working intensely on standardization for homomorphic encryption, which is a different type of encryption technology based on lattices, which I'm not going to talk about today, but I'd be happy to talk with anyone who's interested on those things. So I've mentioned already a few of the important applications for deploying cryptography in our daily lives, so secure browser sessions based on the SSL or TLS standard, signed encrypted email, which almost no one uses, based on the SMIME standard, virtual private networking, which allows you to securely connect remotely, for example, to a corporate network using IPsec. And all of these things require authentication through certificates, for example, X509 certificates. And these are just examples of some of the standards. This is not an exhaustive list. There's a whole zoo of standards. Okay, so let me talk a little bit about elliptic curve cryptography, because it's kind of a precursor for talking about isogenic-based cryptography or a super singular isogenic graphs in cryptography, which is my title. So elliptic curve cryptography, you start from an elliptic curve, so I know many of you are probably very familiar with elliptic curves.  I'm just going to tell you what an elliptic curve is from a cryptographer's point of view, which is not my background, but it's the environment that I work in. So I'm very familiar with people that think of elliptic curves this way. So if you start with a very large prime, so when I say of cryptographic size for elliptic curve cryptography, that typically means at least 256 bits. So in cryptography, you just have to get used to all the numbers in sight being extremely large. Meaning, so if the prime is roughly size two to the 256, it takes you 256 bits to write it down. So that's the amount of space that it takes you to actually represent this number. So that's an important thing to think about how much space does it take. And then another thing you want to always be thinking about is how much time does it take you to compute on these numbers? And also even how do you compute? So on a computer, when I started working at Microsoft about 23 years ago, we were mostly using 32-bit architectures. Now it's 64-bit architectures. So you have to think about, you take a 256-bit number and you break it up into chunks of 64 and you have to implement multiplication, modular multiplication algorithms on these individual words. So the size of the prime makes a really big difference. So 256 bits is the minimum size for elliptic curve cryptography. And then you can think of an elliptic curve over a prime field that's so large because if P is that large, then in particular, it's not characteristic two or characteristic three. So if you were to, if you, there were standards actually for using elliptic curve cryptography over extension fields of prime fields, so you could have a field of size that is actually the field of two to the 256 elements. In other words, not a prime field. But there are various insecurities with having working over extension fields, especially when this extension degree is a composite instead of a prime. So in general, it's just easier and safer to stick with prime fields, which in particular means you're not in characteristic two or three. So that means that any elliptic curve over this field actually has a short wire stress form, which you can represent just as Y squared equals X cubed plus AX plus B, where A and B are just integers modulo P. So a really nice thing about elliptic curves is that there's such a thing as a J invariant, which it allows you to label the elliptic curve. Now this is not so important necessarily for usual elliptic curve cryptography, but it's gonna be very important for us for isogenic-based cryptography. So the J invariant is just, as you can see, it's just a rational function in the coefficient. So I've written, I'm sorry, PowerPoint isn't that great at displaying mathematical equations, so it might look a little messy, but it's just 1728 times 4A cubed over 4A cubed plus 27B squared. So you may recognize the discriminant in there in the bottom and this is the J invariant. So there's a lot we can say about the J invariant. You can think of it as modular form evaluated on an algebraic integer that corresponds to the elliptic curve through kind of the CM theory, but not gonna talk about that subject in this set of lectures either. So the important thing about elliptic curves in cryptography is they have both this algebraic, they have both the geometric structure, meaning it being defined by this equation being able to think of it as a curve in the plane, but they also have an algebraic structure which is defined by a kind of a chord and tangent method for computing the group law. So many of you will have probably seen the group law where if you wanna add two points on elliptic curve, you simply take a line and pass it through those two points and look for the third point of intersection. So all of this, as I said, is just kind of a pre-discussion for what we're gonna talk about today because this is how the usual elliptic curve cryptography works. In other words, the elliptic curve cryptosystems that I said are already deployed and in widespread use today. And the way those work is like I said, they're based on the discrete logarithm problem, which means that there's a public point on the elliptic curve that's known, it's published, and then when you want to use the elliptic curve in some protocol such as a key exchange or a digital signature, you will take some private secret number that you know, let's call it M, and you'll multiply that point, in this case, the group law on the elliptic curve is addition. So you'll add that point to itself, M times, but nobody else knows M, only you know M.  What was the discrete log in this case? So that's how you do cryptography, the usual kind of classical cryptography on elliptic curves. And in that setting, you'd never use super singular elliptic curves. I mean, I shouldn't say never, my mom says never say never, but the point is that after super singular elliptic curves were proposed early on, probably late 80s or early 90s, Meneses Okamoto Van Stone found the MOV attack on super singular elliptic curves, which takes actually the V pairing, or you can use any other pairing on the elliptic curve, and it moves the discrete logarithm problem on the elliptic curve into a finite field, like if the elliptic curve was super singular over Fp, then it takes, because of the super singular property, the V pairing maps the discrete logarithm problem into the field Fp squared star, so a field which is relatively small compared to the size you would use for discrete log in a finite field, and so you end up with a very insecure elliptic curve. So in usual ECC, we do not use super singular elliptic curves. So I haven't even said what is super singular, so it's an important definition from the mathematical point of view, but it's not very important from the implementation point of view. So you could give the definition of super singular as just saying if your elliptic curve is defined in characteristic P over some finite field of characteristic P, there's no P torsion points over, even if you go all the way to the algebraic closure, so that's one way to define super singular. Something that is a little bit more important for our topic today, isogenic-based cryptography, is that the endomorphism ring of a super singular elliptic curve is actually isomorphic to a maximal order in a definite quaternion algebra. So I'm not gonna talk about quaternion algebras today, nor the maximal orders, which are the endomorphism rings, but I will talk about it in the third lecture tomorrow. But I like my favorite reference for reading about, this is actually Joe Silverman's book, section 4.1, I think, so if anybody wants to look a little bit more about that before tomorrow. So an important thing is that isomorphism classes of super singular elliptic curves have a representative over Fp or Fp squared. So when we talk about, if we fix the characteristic P, some large cryptographic size prime, and we talk about the set of super singular elliptic curves, isomorphism classes of super singular elliptic curves, you can think of them as all of them having a representative defined either over Fp or Fp squared. So this is a little interlude, I'm not sure if Joe covered this last week, but I think if there's one thing that's really important, when if you're a number theorist and you're thinking about working in cryptography, the main thing to really internalize is what do we mean by a hard math problem? And so I learned that I should include this slide after giving this talk on super singular isogenic graphs to countless number theory audiences. And having very prestigious esteemed number theorists ask me at the end of the talk, essentially like why the problem is hard, thinking about a very small prime and like the intuition of what you would do if P were very small. And the point is that you can have a cryptographic problem like discrete log, like inverting the vey pairing, like factoring numbers, which is very easy when the numbers are small. Like how do you factor the number of 15? You know, the intuition is that it's not whether the problem is hard or not, or like in the sense of a lot of mathematicians think some problem or an area is hard if the theorems are hard to prove. That's a typical thing that mathematicians think is hard. Like there's a conjecture or an unsolved problem means that there isn't, it's hard to find a proof. And that is not what we mean by hard in cryptography. So specifically what we mean is that if it takes you m bits to represent the problem. So like I said, an elliptic curve. In the beginning I said if the prime is 256 bits, then what we mean by hard is that the fastest algorithm that you have for solving the problem runs in roughly, so exponential time meaning like proportional to two to the m. So the m is the number of bits and the m is in the exponent. There's a little bit of an unfortunate difference between the math and the computer science community in this terminology, so I like to be very explicit. So exponential time in m means that the algorithm runs in time roughly two to the power m. Whereas sub-exponential time means, well let's go to polynomial time. Polynomial time is really easy. It means the best algorithm runs in a polynomial in m. And in between that we have this category which we call sub-exponential. I'm just giving you an example of a sub-exponential algorithm here. This is an L one-third algorithm. Which just means that the algorithm runs in time which is e to the power, so some constant c times m to the one-third times log m to the two-thirds. That's just an example. Like a one-half algorithm would be just replace the one-third and two-thirds just with one-half. And also the constant is very important in this, in terms of actually practically setting up crypto systems. So here's the motivation for the topic that I'm talking about today, the isogenic based cryptography. That is the quantum threat. So many of you probably know that many teams around the world, many physicists, are working on trying to build a quantum computer at scale. So there are already small quantum computers that have been built that can handle computations on like dozens of bits, let's say. But it's still an open question whether they'll be able to build quantum computers at scale that can handle computations on like hundreds or thousands of logical bits, q-bits. But if they are able to, the fact is we've known since the 90s, the Schor's algorithm, which gives you a polynomial time method for factoring large numbers like RSA module i. And it can also be extended to attack elliptic curve, crypto, the traditional ECC, which is deployed today. So the running time for if M is the number of bits of your RSA modulus, for example, the running time for factoring, using Schor's algorithm for factoring will be roughly four M cubed. So as you can see, that's a polynomial time algorithm. And you'll require a quantum computer that can process like two M q-bits. So a typical minimum size for RSA these days, M is usually 2048, 2048. So that means you'll need a quantum computer that can process around more than 4,000 logical q-bits. So once that is built, Schor's algorithm will give you a polynomial time algorithm for breaking RSA. So that's not good. So same thing for ECC, except for M is smaller. For ECC, like I said, M is like around 10 times smaller. It's 256 bits. And that's because of the difference in classical algorithms today. We do have sub-exponential factoring algorithms for RSA. We do not have sub-exponential algorithms for attacking ECC. So there's a very beautiful history of the sub-exponential algorithms for factoring, starting with like the quadratic sieve and moving to the number field sieve and the general number field sieve. And those are very beautiful number theoretic algorithms in case any of you are interested in that topic. But for ECC, we have just M is 256 because we have no sub-exponential algorithms. And so the Proust-Zalko's work already in 2004 estimated the constants to be around 360 m cubed time for attacking and that you would need a quantum computer that can process six m qubits. So six times 256 is roughly 1500. So you can see that you actually need a small or only a smaller quantum computer to attack the elliptic curve system, which is from the classical point of view of the same strength of as RSA 2048. So just as an interesting note, because of that difference, you may have noticed if you follow these things that in 2017, NSA released new guidelines for using ECC in practice and they increased their requirement from ECC 256 up to 384. So you can imagine that their motivation for that may have been progress on building a quantum computer. Actually, I'm quite familiar with the standards for elliptic curve cryptography. As I said, I worked on them early in my career and roughly my first five years at Microsoft from 1999 to about 2005, which is when we started deploying elliptic curve crypto systems in Microsoft operating systems and spread into all of our products worldwide at that time. So 2006 was when NSA and NIST released the sweet B requirements governing the use of elliptic curve cryptography for all government contractors, which was the thing that really helped push elliptic curve crypto into widespread use worldwide. And these 2017 guidelines released by NIST, the CNSA requirements in case you wanna look them up, really started to pull back those requirements. So they noted that the adoption of elliptic curve crypto was no longer required for government contractors. So again, you can imagine that motivation there being the advent of the quantum computer. So NIST launched its competition in 2017 saying it would be a five year competition to select new post quantum crypto candidates. So here we are, it's 2022, it's five years later. And so let me just first, before I tell you kind of the punchline, first of all, competition's not over yet, but they did make new announcements two weeks ago, which I'll tell you about in a minute. But from a high level, it's kind of interesting to see what mathematics was involved in this post quantum crypto competition.  But the ones that were really considered intensely in this last five year period in this, even though NIST is a US government agency, the participation in the NIST public workshops was international from teams from around the world, submitting candidates and attacking and evaluating candidates intensely in a series of public workshops run by NIST. So one of the things that was considered very favorably was code-based cryptography, which was proposed first in 1978 by McLeese. So there's, you can create crypto systems from error-correcting codes, and the security of those systems is based on the hardness of decoding random linear codes. So that's the hardness assumption there. So analogous to factoring, being hard, or like I mentioned, the hard problem for discrete logarithm problem for elliptic curves. So here the problem for code-based cryptography is decoding random linear codes. But there's an extra assumption there too, and that is that you can disguise codes, linear codes that have structure as random linear codes and in a way that nobody can figure out the structure because you need that structure for decoding your, for being able to decrypt your systems yourself with the secret key. So a second area for post-quantum cryptography was multivariate cryptographic systems proposed by Matsumoto NMI in 1988. So just think of many, many equations in many variables, so solving some system, multivariate system of equations, but obviously which is non-linear, right? So we don't want to do linear algebra here, but non-linear systems of equations in many, many variables. So a third area which you heard about last week is lattice-based cryptography which was introduced by Hovstein, Piper and Silverman in around 1996 with the advent of the NTRU system. And so 1996, that's already more than 25 years ago, so just kind of keeping track of time here. 1978, that's almost 50 years ago. And so the reason I note that is that so far our best way of understanding the security of public key crypto systems is to, once the system has been around for a long time, you see the best-known attacks on them and you can kind of plot how the attacks improve over time, like you could see the improvement in the sub-exponential algorithms for factoring over time and then you kind of predict the likely security of these systems in the absence of some disruptive, like new attack that is not predicted. So lattice-based cryptography has been around for more than 25 years. And so the last candidate that I wanted to talk about is the topic of my lecture series here is super-singular isogenic graphs. So this was introduced into cryptography by myself with my co-authors, Dennis Charles and Eyal Goran in 2005 at the NIST hash function competition.  It's more than 15 years old. And it's now an active area of research in isogenic-based cryptography. So just to give you the punchline, last week or week before last, NIST announced fourth round candidates. So super-singular isogenic graphs are still being considered for standardization in the fourth round. So Psyche is the super-singular isogenic key exchange, which I'll talk about in my second lecture here. But they actually announced four cryptosystems for standardization, which is pretty cool. And three of them are based on lattices. And so I'm not sure if Joe talked last week about NTRU. But that is pretty cool. And I'm a huge fan of lattice-based cryptography too. So I'm talking about, the rest of the talk today will be about super-singular isogenic graphs. So I've mentioned all the hard problems involved. Actually, I didn't explain what the hard problem is for lattices. It's a little involved, and you heard about that last week, so I'm not gonna go into it, but I'd be happy to talk about it offline with anyone. So the hard problem in super-singular isogenic graphs is finding a path between two given nodes. So imagine a very, very large graph, and you're given two nodes in the graph, and how do you find a path? So I should say, what do I mean by a graph? So a graph in this context is just V comma E, a set of vertices and a set of edges. So vertices are just nodes, and you could think of them in space, but that's just your imagination. They don't really exist in any particular configuration. But the edges are just specifying which nodes are connected to which other nodes. So that's what we mean by graph. K-regular just means that every node has K edges coming out of it. Undirected means the edges are not directed, so you don't have to say that this edge goes from this node to that one. It's just an edge between them. And then I'm gonna talk this afternoon in the second lecture about what we mean by expander graphs and expansion property. So the important thing for cryptography is if you wanna rely on the hardness of finding paths between two random nodes in the graph, that it should be really hard to find an efficient routing algorithm. So I always like to give the example of a really bad choice for a graph. So you could think of the hypercube as a graph, which has all of its nodes, so let's say a hypercube in n dimensions. So each node is labeled by a series of n bits, zeros and ones, and the edges are, two nodes are connected to each other if they differ by one bit. So you basically travel around the graph by flipping one bit at a time. Okay, so now give two sequences of length n of bits. And now it's easy to find gazillions of paths between these two nodes, right? You just flip every bit where they differ and you can flip them in any order and that gives you many, many different paths. So that's a really bad graph to choose for this cryptographic system. You don't wanna choose that. So I wanna tell you a little bit about the application here that caused us to come up with this proposal for using super singular isogenic graphs in cryptography. And that is cryptographic hash functions. So cryptographic hash function is very fundamental for cryptography. They're used as a building block in almost every cryptographic protocol to maintain security. But they are, in fact, it's somewhat difficult to actually define the security property that you want for them. And we often use what's called the random oracle model. But this random oracle model is actually very hard to describe what it means. So in practice, what you actually have to do when NIST was running its hash function competition in 2005, you have to look at a bunch of candidates for hash functions and just try to see whether you can reverse engineer them or whether they have the properties, at least the minimum set of properties that you want them to have. And so first let me say what is a hash function? It just is a function that goes from some sequence of bits of some length, which I'm calling length N right now in here, and it maps it to some other sequence of bits which have length M. So for example, a common use of a hash function is as like a message digest where you have a very large object like a movie or something that takes as many, many bits and you have some kind of very short representation of it by mapping it into a sequence of bits where M is much, much smaller, like 128 or 256 bits. But an important property of a hash function is that it should be very fast and easy to compute, efficient to compute. When we say hash function, we usually mean it's unkeyed so it does not require a secret key to compute the output as opposed to like a Mac where you are assuming that the person who computes the Mac has a key to compute it. And the cryptographic properties that you want are that it is actually collision resistant and well, what we call pre-image resistant and that there isn't a bias in the output so it's essentially uniformly distributed output. So a hash function is collision resistant if it's computationally infeasible to find two distinct inputs that match, map to the same output. So in the message digest example, definitely there's gonna be a lot of things that do go to the same output if N is way, way bigger than M but it should be very hard to find those two inputs that hash to the same. And we say a hash function is pre-image resistant if given a certain output, it's computationally infeasible to find an input which hashes to that. So clearly if you can find pre-images then you can definitely create collisions so that's kind of the harder problem to solve. Okay so here's kind of the key thing that I need to describe is how do you create a hash function from a, well I'll just call it an expander graph for now but it's gonna be the super singular isogenic graph. So given a K regular graph, each vertex, it's very important that each vertex in the graph has a label. That's why I mentioned at the beginning it was very important that elliptic curves can be labeled with the J invariant. So what we're gonna do is we're gonna have a starting point for a hash function which is specified publicly and then when you get some kind of input, so the input is some long bit string, we're gonna use this bit string as directions for walking around the graph and the way we do that is to divide the bit string up into chunks and we process one chunk at a time and each chunk tells you where to go in the next step of your walk around the graph. So you start somewhere, you process the first chunk and you follow one of the edges. So another very key point is that you're not allowed to go backwards.  So now you only have K minus one choices and so that's called backtracking, no backtracking allowed and then when you get to the end of your walk, the label for that node, that's the output for the hash function. So this was a general kind of hash function that we described in our paper with Gorin and Charles in 2005 and so this is kind of just a picture of the representation if you had a three regular graph. So after the first step, you only have two choices for each next step. So those two choices are just determined by one bit. So you just have a zero, you process the bits in block sizes of one. So one bit at a time, you just determine how to walk around the graph. So that's for a three regular graph, for example. And so this is kind of a very simple idea which we propose in general that you could create cryptographic hash functions from K regular expander graphs. The expander part, I'll talk a little bit more about later this afternoon. It basically is good for assuring that you have a uniform distribution of your output. It's not all that important necessarily for the collision resistance and pre-image resistance. Okay, so now we have the job of finding graphs where routing is hard. So I've already mentioned routing is easy in the hypercube and so that's not a good idea. And I'm gonna talk like I said later today about the expansion property in Ramanujan graphs. And what we're trying to do is avoid that it's easy to find paths or that it's easy to find collisions. So collisions would be like when you have an actual cycle in the graph as you can see here. These are like I said, undirected graphs. I'm just putting the arrows in there to show when you're walking, what your direction of your walk. So we actually proposed in our original paper two different graphs that you could use for this purpose. But the one that I'm talking about now is the super singular isogenic graph. So what is this graph? That's kind of the whole point of this lecture is to actually explain what is this graph. So the nodes are going to be isomorphism classes of super singular isogenic, super singular elliptic curves, modulo p. So you fix some p of large cryptographic size in our original paper and still today for this application of hash functions, p can be 256 bits. And then the isomorphism classes of super singular elliptic curves are going to be, each of them as I mentioned will have a representative over fp or fp squared. So you can label them, each of them, so the J invariant is an isomorphism invariant. So any elliptic curve in that class has the same J invariant. So you take the J invariant of your representative, which is over fp or fp squared, and that's the label for that node. And I mentioned earlier, it's very easy to compute the label given the equation of the curve. It's just this rational function in the coefficients. Okay, so that tells you what are the vertices of this graph, and now let me tell you what are the edges. So given an elliptic curve, one way you can think of that elliptic curve is as an abelian group. So I haven't told you a lot of the nice theorems that about elliptic curves that many of you probably know, but if you look at the group of points on an elliptic curve over like a finite field like fp or fp squared, it's actually an abelian group. And the nice thing is that what can you do with an abelian group? Well, you can actually quotient it by subgroups and you get another group. And so actually there's a nice kind of geometric theory behind this that allows us to extend from just thinking of these as abelian groups to actually thinking of them as geometric abelian groups. So I'm blanking on the paper, but I think it's geometric invariant theory by Mumford. Maybe somebody can remind me or correct me if I got that wrong. But for your purposes, and for the purpose of cryptographers, you can forget about a lot of that theory and you can just think of these objects as being abelian groups. So how are we gonna start from one elliptic curve and get to another one? We're gonna quotient by some subgroup and then we're gonna get some other elliptic curve which is gonna be an abelian group. And these are, so the size of the group that you quotient by gives you the actual degree of the map that takes you from the elliptic curve you started with to the elliptic curve that you end up with. So in this, it's very easy to assure that everything works because we're gonna let P be very, very large of cryptographic size and then we're gonna quotient only by subgroups of very, very small size. So for example, subgroups of size two. So the main application of super, the main instantiation of super singular isogenic graphs is when the degree is either two or three. So that's pretty easy. So what that means is you start with a starting point in your graph which is an elliptic curve and you just wanna quotient by subgroups of size two. Well, those are really easy to find. Those are generated by the torsion points of the two torsion points which are just the roots of the polynomial f of x if the curve is y squared equals f of x. So I'll show you the formulas in just a minute for computing the isogenes of degree two. But now that we have said what the vertices and the edges are for this graph, we have some very, very nice properties of the graph. So first of all, if you take degree L isogenes where like I said L is gonna be two or three in this setting, then these graphs are actually K regular where K is just equal to L plus one. So if you think of L equals two, this is very nice because you just have three two torsion points for each elliptic curve defined over the algebraic closure. They're not necessarily defined over the base field. That's kind of an important issue. But you have three two torsion points and so those are the three edges that allow you to get a new elliptic curve. And actually you can end up getting the self isogeny. It can end up being the same elliptic curve but we still count that as an edge. So we have a three regular graph if we take L equals two. We also have some very nice theorems. We can make this into an undirected graph if we assume that P is congruent to one mod 12. These were actually the first proposal for Ramanuj and graphs by Pizer were described in terms of quaternion algebras and under the assumption that P is congruent to one mod 12. But in cryptography, we're actually ending up using the graphs even when P is not congruent to one mod 12. I'm leaving aside a little bit of a technical issue here which is that we're, because L is small, L is two or three, these isogenies are going to be separable because they are co-prime to the characteristic which is very, very large, as I mentioned. So the formulas for computing isogenies were actually, and the equation of the new curve were actually given by Vellu in the 1970s. So as you can see here actually, again, these are very, very easy to compute. So if you have a two torsion point Q, which is represented by R comma zero, R is a root of f of x where the elliptic curve is y squared equals f of x. Then you have a nice equation for your new curve E2 which is just in terms of these coefficients A and B of the original curve and R, the two torsion point, and you have a nice representation for the map between E1 and E2. So now I think in the interest of time, I'm going to just end up by showing you a picture of these super singular isogenic graphs. So maybe I should have done this before I defined them, but this is, I think, something that is very nice to remember from this lecture is kind of a picture of what do these super singular isogenic graphs look like. So in roughly 2000, like I said, 2005, we proposed super singular isogenic graphs at the NIST hash function competition. And I spoke about this proposal at a lot of different kind of number theory crypto conferences, including ECC and things like that. And I spoke about it at JMM, the joint math meetings in 2008, and Dana McKenzie, who's a science writer, was at my talk and the hash function competition was kind of finishing up around that time. And so he wrote an article in Science Magazine about hash functions and in particular our proposal for super singular isogenic graphs for hash functions. So we actually created this picture for him. We took a prime, which was about 2500. So it's a very small prime from cryptographic point of view. I think it was P equals 2521. And so one thing that I maybe didn't include, I'm not sure if I put it on a previous slide, no. So we have what we call the Eichler class number, which tells us how many isomorphism classes of super singular elliptic curves we have. And it's roughly P over 12. So again, that same section of Joe Silverman's book on elliptic curves gives you the exact formula, greatest integer part of P over 12 plus, either zero or one or two. And so if I have a prime of roughly 2500, not 2500 bits, but 2500, 2521, then the number of nodes in the graph is gonna be roughly P over 12. So somewhere around 200 nodes. So there's about 200 nodes in this graph, all smushed together and it looks really kind of ugly and it's hard to even figure out how to represent such a graph. And the point being that if you look at this graph, and like I said, this is very small from a cryptographic point of view. And you take two nodes in some random parts of the graph, it's very hard to see how to get from that one point in the graph to another. And that's the whole point of the hardness of this underlying problem. So just to illustrate that, we took two points that are the end points of the blue path that you see and we showed the path that you find there. And you can see there's no sense of orientation. There's no sense that if you go in one particular direction, like one particular step that you'll be coming closer to the endpoint that you want. We also showed this red path is a cycle in the graph which as I mentioned is important for collision resistance of the hash function. And cycles, again, are very hard to find. So the point of, I guess the point of this picture is to try to give you an idea that these are very strange graphs which don't have a natural kind of orientation or routing in them. So this afternoon what I'm gonna do is I'm gonna talk a little bit more about the actual expansion property of these graphs which is important for being able to get kind of a uniformly distributed output for your hash function. And I'm also gonna talk about the key exchange that you can define from these graphs. So I think, so I've just left about five minutes in case anyone wants to ask questions. So I'd be happy to take questions but otherwise you can also save questions for the problem session. So thank you very much. Questions I see.


See how Akra, Ghana is becoming the fastest developing city in Africa. Akra is the economic and administrative hub of the greater Akra region, as well as the anchor of the greater Akra metropolitan area, Gama, which has a population of around 4 million people, and is Africa's 13th largest metropolitan area. Akra was designated a Gamma-level global metropolis by the Globalization and World Cities Research Network in 2020, indicating an increasing level of global impact and connectedness. It is home to important government and non-government ministries, insurance companies, bank headquarters, and large markets that import the vast majority of the world's food supply. Akra has a land area of 225.67 square kilometers, or 87.13 square miles, and a population of 4.2 million people. In 2020, it is divided into 12 local government districts, 11 municipal districts, and the Akra Metropolitan District, the only district with city status in the capital. Akra was formed by the merger of various towns centered on British Fort James, Usher Fort, and Danish Fort Christiansborg as Jamestown, Usher Town, and Christiansborg, respectively, and served as the capital of the British Gold Coast between 1877 and 1957. The architecture of the capital reflects its history, from 19th century colonial architecture to modern skyscrapers and residential towers. In today's video, we'll look at how Akra is quickly becoming Africa's most developed city. We will base this development on factors such as economic growth, infrastructure development, industrial growth, cost of living, tourism, and so on. Please subscribe as we ride along. Geography of Akra Because of its location in the Dahomey Gap, where the Gulf of Guinea and the Atlantic Ocean coast run parallel to the prevailing voice Monsunnel winds, Akra has a very marginal hot, semi-ary climate that borders on a tropical wet and dry climate. In Ghana, the average annual rainfall is around 730 millimeters, with the majority of it falling during the two rainy seasons. The primary rainy season lasts from April to mid-July, with a second weaker rainy season beginning in October. Rain often falls in short, intense storms, causing local flooding and clogging drainage canals. The temperature varies very little throughout the year. The average monthly temperature ranges from 25.9 degrees Celsius in August to 29.6 degrees Celsius in March, with an annual average of 27.6 degrees Celsius. The humidity level is higher in the colder months than in the warmer months. As a result, during the summer months, the city experiences a breezy, dry heat, particularly during the windy-harmitin season, which feels cooler than the cooler but more humid, rainy season. Because Akra is so close to the equator, daylight hours are nearly constant year-round. The relative humidity is high, ranging from 65 percent during the day to 95 percent at night. The main wind direction in Akra is from the WSW to the NNE. The average wind speed ranges from 8 to 16 kilometers per hour. Thunderstorms generate strong wind gusts that travel along the coast in squalls. Akra has the highest recorded wind speed of 107.4 kilometers per hour, or 58 knots. Economy of Akra In 2008, the World Bank estimated that Akra's economy accounted for only about US$3 billion of Ghana's total GDP. The economically active population of Akra is expected to be 823,327 people. Akra's major industries include manufacturing, marketing, banking, insurance, and transportation. Akra's economy is made up of primary, secondary, manufacturing, power, gas, water, and construction, and tertiary sectors. Supermarkets, shopping malls, hotel, restaurant, transportation, storage, communication, financial intermediation, real estate service, public administration, education, health, and other social services. Tertiary services, which employs approximately 531,670 people, is the city's main industry. The secondary industry is the second largest, employing 22.34% of the workforce, or approximately 183,934 people. Unemployment in the city is estimated to be 12.2% of the workforce, or 114,198 people. Industrial Growth Ghana was the world's seventh largest goal producer in 2019, producing 140 tons. Ghana's output surpassed South Africa's for the first time, establishing Ghana as Africa's leading goal producer. In addition to gold, Ghana exports silver, lumber, diamonds, bauxite, and manganese. And it has numerous untapped mineral reserves. In terms of diamond exports and reserves, Ghana ranks ninth in the world. Ghana's government has proposed nationalizing the country's mining industry in order to increase government revenue, because Akra is Ghana's economic and administrative center. All of the country's industrial successes can be attributed to her, as she plays a significant role in the development of the country's industries. Infrastructural Development Ghana's economy has expanded steadily over the last two decades, with improvements in nearly every sector. Poverty has fallen to 24.3%, a 4.2 percentage point decrease from the previous figure. Despite these achievements, the economy is still be set by gaps in both economic and social infrastructure, which has slowed the economy's rate of progress. Despite this, Akra's infrastructure development is outpacing that of other major African cities, such as Addis, Ababa, and many others. It is safe to say that Ghana is currently carrying out development projects in Akra that will propel the country's economy to new heights by 2003. Tourism in Akra Akra is a bustling modern metropolis where the streets smell delicious foods, and the ocean invites you to relax. You'll find everything you need from a bustling local cultural scene with traditional song and dance to sophisticated clubs with the newest live music. There are traditional crafts, as well as cutting-edge contemporary fashions. The Dutch, British, and Danish influences of the city blend with its West African heritage to create an unforgettable experience. Visit the Gannon National Museum, the Akra Arts Center, and the Kwame and Krumah Memorial Park and Mausoleum for Artisan Crafts. Bojo Beach and Lombardi Beach are just two of the many beaches near Akra that offer clear waters and activities like horseback riding and surfing. Cost of living in Akra As in most countries, the cost of living varies greatly depending on location and lifestyle choices. Akra has the most expensive lodging. Some development agencies provide a $3,000 monthly rent allowance, which is insufficient. You can live for less if you're young, healthy, and don't need the Western way of life. If you don't mind living without air conditioning or electricity for long periods of the day, in a neighborhood with some noise and dust, and with no official security, you can live quite cheaply. Another option is to share housing. Dividing the rent of a three-bedroom apartment in three equals approximately $1,200 per month per person. Regrettably, landlords now demand a one-year rent deposit. And certain living expenses are exorbitant, for example. Electricity recently increased by 75%, and you may have to pay to have water delivered to your home during the dry season. Cuisine and Nightlife in Akra Akra Ghana has a lot to offer visitors, but the nightlife scene is especially appealing to party goers. The vibrant nightlife scene combines African and European cultures. Akra has something for everyone, whether you want a quiet lounge with an intimate atmosphere or a wild night out on the town. To name a few, the Zen Garden, Inside Carbon Nightclub, and Champ Sports Bar and Grill are all located here. Gendian cuisine has mastered the art of transforming everyday ingredients into sublime delicacies that have become staples in the Gendian diet. Chefs all over the world are praising their brilliant Gendian brothers and sisters for upping their culinary gain and serving up dishes that are as appealing as they are delectable. Culture Akra is known for its many historical and cultural sites. The Du Bois Center, for example, contains a research library and a manuscript gallery, as well as the graves of scholar W.E.B. Du Bois and his wife Shirley Graham Du Bois. The Kwame Nkrumah Mausoleum is the final resting place of Ghana's first president, Kwame Nkrumah, who advocated Pan-Africanism and oversaw the Gold Coast's independence from Britain. Ghana's National Symphony Orchestra performs at Akra's National Theater. Akra also has the National Museum and the National Theater, both of which feature eye-catching modern Chinese architecture. The Usher and James Fords, as well as Osu Castle, also known as Christian's Borg, were built by Danish settlers in the 17th century in the Jamestown area. The Ghana Parliament House, the Ghana India Kofi Anin Center of Excellence in ICT, Black Star Square, the Akra Sports Stadium, and the Akra Center for National Culture are all worth seeing. The African Fashion Foundation, based in Akra, provides assistance to the African fashion industry, particularly in Ghana. Akra has an Atlantic beachfront, with the most popular being Lombardi Beach, as well as Cockerbite Beach, which is 25 kilometers, 16 miles, west of Akra. The Academy of African Music and Arts is also near the beach. It is important to note that these historical sites play an important role in the growth of the nation because tourists flock to the city to visit these sites. Ghana has taken in upon herself in recent years to carry out development projects that will bring the country to the same level of development as other countries such as Nigeria and Kenya. Because Akra is the country's capital, it is only natural that these initiatives begin there. Stay tuned to Africa Reloaded for more exciting and thrilling videos.


Thanks. Starting? Okay. Let me know if I win. All right. As Jason said, I've been doing social media for a few years. I started off at Linux Pro Magazine, and so this is a summary of what I've learned. This summer I heard at an event, I don't believe in social media, and I have really good news. I've seen it. It's totally real. And so I'm going to tell you you should believe in social media, regardless of what kind of open source project you have. You might be overthinking it if you don't believe in it.  Even if you prefer to hang out in IRC, you can be directing people to your IRC channels by using social media and letting them know that they exist. So first of all, you want to share relevant, interesting, and accurate information. Consider your social media streams to be your own little publications, and so make sure you're curating what content you have in there. You don't want to just be off on whatever topic.  Also, you want to make sure you stay on message, even with your retweets. So if you're retweeting something relevant to your project, be sure that you check the link and make sure it's actually a relevant link and a current link because often people are sending out tweets that are outdated to articles that are no longer current or to a previous release of a project or software. Also craft the text. The title of whatever somebody wrote as their blog post is not necessarily the best content for you to be putting on an 140-character tweet. Make sure that all they have to do is hit retweet and they can send out the exact thing that you wanted them to send out whenever possible. I personally won't hit retweet if I have to go in and reword the whole tweet. So use hashtags and refer to other people's accounts but you also don't want to overdo it. Anytime you see a whole bunch of hashtags, it's like screaming, it's like all caps, and it's usually from somebody who's in marketing and not necessarily in community on a project. So be very selective in the hashtags you're using. Avoid a PR verbiage. People will send that to you all the time and marketing speak. Talk like you would normally talk to somebody who's working on your project. Please don't do any of the upwardly stuff.  Numbers do well and how to tweets do well. You'll see that on opensource.com. These were number one and two the week I was pulling this example. Note about Facebook, unless you're marketing and have a budget to spend on marketing, I would avoid it. I mean, oh, when I avoid it, I would go ahead and send stuff out but I wouldn't dedicate a lot of time to it. For every 1,500 stories a person might see when they log on to Facebook, the news feed only displays about 300 because of the algorithms that Facebook is using. Part of being on social media is social, which means you don't just send stuff out all the time. You need to retweet, reshare, respond, and reply. So like I said, make sure that resharing your message is very easy. Make sure you reply to people who have questions or commenting. Make sure that you are getting credit for people who have actually sent you the original information and let followers know where you heard about it first. Show your reach. Ask your network to share particularly important content. If you go to the Linux underscore pro Twitter account today, for example, you'll see that they shared a tweet that I wanted them to share about the new women in open source award here at Red Hat and they did that because I just sent them a note and said, hey, we used to send this out. Consider the schedule. If you're trying to talk about an event that you're hosting in Paris and all of your tweets are going out in East Coast time, you're missing everybody who's actually going to go to your meetup in Paris. So make sure you are scheduling all of your tweets to hit the time zone that is most relevant if you're talking about an event, for example. Avoid scandal. I use two computers when I work every day. If I'm doing something personal, it's on the computer over on my right. And if I'm doing something work-related, it's on the computer on my desktop. And that's another good reason to recheck those links because occasionally people will send out really naughty links in well-meaning tweets. And measure success. These are examples of how I've been measuring success of the Red Hat open account, which is one of the accounts I manage. I keep an eye on about 34 accounts in the open source and standards team at Red Hat. And then I have a spreadsheet where I kind of do an update every week on, and just keep an eye on how successful we are. And then make sure you promote all of your accounts. And this is an example of promoting all your accounts. Do it at a conference if you're giving a talk or make sure it's on your business card or on your blog. Do as I say, not as I do, because these aren't actually listed on our blog right now, but they will be. So, and that's it. Do I win? Yes. Yes!


The war in Vietnam is a combined and joint effort, whatever you are. Soldier, sailor, marine or airman, American, Korean, Arvin, Australian or New Zealander, you already know this well. A message read to this unit illustrates the challenges faced and met by units of all services. I have a letter addressed to all personnel of this command from the commanding general. During our tour in the Republic of Vietnam, we have all seen what communism can do to a struggling nation in our world community. It is therefore most important that we conduct our affairs with the people of the Republic of Vietnam in a manner that will show the Vietnamese citizen that we are his friends and that we are here to assist him in the improvement as well as the security of his country. It is then with great pride that I note the fine work done by the soldiers of this command during the recent operation. It is with pleasure that I say, a job well done. These men have just emerged from many long days of combat, during which they faced and fought the Viet Cong, drove him from areas which he had heretofore controlled, opened roads for travel which he had dominated and thereby contributed substantially to the turning of the tide in Vietnam. Yet it is not for their military accomplishments alone, important though they are, that they are commanded by their commander. It is for other activities, more subtle, less easy to grasp and define. Yet every bit is indispensable to the final decision in the war that is being fought in Vietnam today, a war unique in the long history of armed conflict between men and nations. This is the story of the fighting man and his unit in Vietnam. Defending against aggression, but more than that this is the story of their special task, helping to build a nation at the grassroots level among the people in the villages. This is the battleground in Vietnam, the varied and changing face of the land, but this too is the battlefield, the villages and hamlets which dot the countryside, the communities across the land where four-fifths of the people live. And it is in this battlefield that the issue of victory or defeat will ultimately be decided. The reason is rooted in the special nature of the war that is being fought here, for this is a war whose objective is to win the hearts and the minds of the people. To understand why this is so, it is necessary to contrast this war with those we have fought in the past. Take Korea, for example. As in all the wars before it, it had a battle line. When the war was going against us, the battle line pushed in on us. When we were on the offensive, the movement of the battle line described the progress of our victory. In this normal kind of military situation, the other side of that line was known to be enemy territory. And tragic though it was, it was a fact to be accepted that cities and towns lying in areas controlled by the enemy. Often had to suffer the cost of victory's advance. War in Vietnam is different. Although we do engage in conventional combat with well-organized North Vietnamese military units who support the Viet Cong guerrillas, our enemy here is active over most of the countryside. And rarely is there a conventional battle line. Our job here is to search him out wherever he hides, however he travels and supplies his forces. To seek him along all the avenues through which he infiltrates the Republic of Vietnam carrying the weapons of his aggression. Our job is to find him, but because the enemy over here is fighting basically an unconventional war, he does not rely on holding terrain, but more importantly, the people. The foundation of unconventional war rests on the need for their support. In this kind of struggle, whoever has the people wins the war. So we must deny the enemy the popular support he has to have in order to survive. And more than that, we must enlist that popular support behind our own effort and behind the government of the Republic of Vietnam, whose cause we represent to the people through whose villages we travel and fight. So the issue turns on the people in their villages and hamlets. And these become the ultimate battlefields of the war, as we and the enemy both endeavor to win their support. As the war thus ebbs and flows through their lives and villages, the people themselves have little control over that constant movement. Nor ironically, is the grave issue of the war, the threat to freedom which communist aggression poses, always clear to the people whose support will determine the ultimate decision. Their lives are lived at the edge of fundamental experiences. And it is the fundamentals with which they are concerned, poverty, hunger, disease, and a deep yearning to live in peace without fear. For they have known the trials of war now for many long years. Their allegiance will go inevitably to the side which most clearly demonstrates a sympathetic interest in those fundamental problems. The winning of their support then will take other weapons beyond those in our arsenal of firepower. It will take the intangible weapons of compassion and goodwill. In the early phase of the war, the Viet Cong proved themselves adept at using those weapons. Simply by persuading the people that they were their friends and protectors would help to bring them a better life. The real objectives of communism were always underplayed or obscured entirely. They're still doing that as much as they can, but one factors changed the direction of the war significantly. The big build-up of American forces on the countryside started to loosen. And as that has happened, the enemy has turned more and more to the use of terror and murder as a way of maintaining its grip on the people, hoping to intimidate them into support as the voluntary support weakens. It has been his greatest mistake. For although it can have immediate effect, its long-term result can only work against him. The enemy's mistakes, of course, work to our advantage, but there is an obvious and urgent lesson. We must never duplicate those mistakes, not to be sure that there is any danger about ever using terrorism as a weapon. But to the people in the villages, there might well be little difference between the calculated cruelty of communist forces and fire which rains on them and their families from an American rifle or an American plane. These special demands of this special war place a particular responsibility on every American over here. Commanders at all levels are obliged to consider the psychological as well as the military objectives of each operation. And because the circumstances call for an exercise of restraint beyond that normally required on the battlefield, they must strike a balance between the force essential to accomplish their mission and the necessity of inflicting as few casualties as possible on the civilian population. This does not mean that the rules of safety and security can or should ever be relaxed. Every commander must always be able to accomplish his mission with the least loss to his own troops and the fewest possible restrictions on his planning and his freedom to fire and maneuver. Nor does it mean that men going into battle should ever place their own lives or those of their buddies in jeopardy through excessive caution. Every man must take with him into combat a certain knowledge that he has the inherent right always to defend himself against hostile action wherever and whenever it occurs. Special responsibility does mean that firepower of all kinds has to be employed with care taken in so far as humanly possible to avoid unnecessary damage to the lives and property of the civilian population. For the blunt truth is that death or injury to helpless civilians, even when it is unavoidable, works against us. This knowledge is simply one of the conditions of war which we must accept and work with as best we can. This special responsibility extends all the way along the chain of command on every level and in every service. And it very specifically includes the man on the ground who must face his personal responsibility through the sights of his personal weapon. This man's job carries him into the very heart of this unique battlefield. Precautions to eliminate needless casualties should be taken whenever possible. One example of how this can effectively be accomplished is the Air Force's practice of carefully marking targets so they can be hit with pinpoint accuracy. Born civilians of the military vulnerability of their area or of impending airstrikes. And still another is the use of broadcasts to warn villagers of ground operations. All such warnings, of course, must always be made with due regard given to the security and success of the mission. Leges have to be entered. They have to be searched and made safe. This remains an inescapable duty of war. But with careful plans made and executed, with every precaution taken that can be within the bounds of safety, and with every man aware of his own increased responsibility, it can be done with the least possible damage and danger. Operations planned in coordination with the country's province or district chiefs help smooth the path the rest of the way. For they can explain better than anyone else, at least in the beginning, that the Americans are here as friends. As friends, that's how we come to the villages of Vietnam. How we must come if we're going to win this war. Friendship is one of the most basic gifts a man can offer. It is built on the desire really to know people. Look around you and look deeply to understand what Vietnam and its people are like. Vietnam is an old country with a recorded history that goes back more than 2,000 years. Its traditions and its customs are ancient and deeply cherished. The amese himself, humble peasant though he may be, is the inheritor of this old tradition. And anyone who would claim his friendship must understand that about him. He must know and respect the fact that he looks on the world in a very set way, defined and shaped by long tradition. It is a world that lies always close to the soil and the elements. His village is more than his home. It is his universe and he seldom if ever leaves it. The values which shape his coat of behavior are his most precious possessions and they are honored ones. Dignity, self-control, respect for elders. His family is all important to him. All his family. Not only those immediately close to him, but distant relatives as well. The living, those dead and those in generations yet to come. Regard for the family is the center of his religion. Although many villages in Vietnam like this one are Christian and the majority are Buddhist. Whichever it is, the place of worship in the village is the focal point of community life. Knowing these things about the Vietnamese people and understanding the importance they have in their lives will help in establishing the friendship that is so crucial to our mission. Since knowledge is the foundation of friendship, endeavoring to know and understand the Vietnamese, there is much more we can do. The American Fighting Man establishes rapport with the children in the lands through which he travels. He is an enduring part of his legend. There is no mystery about it. For the hungry language of childhood is universal. And men at war have known for generations that candy and chewing gum bring smiles and warm the hearts of children in any land. There is more to this legend than candy and color. And every man knows it who has watched happiness brighten the eyes of a child as the result of his own effort. Even a child's friendship is sometimes slow to come. And how much more true this is of people who have known bitterly only the brutality of war and warriors and who can escape only behind thick walls of silence and suspicion. Their friendship is not obtained easily, but with patience and effort it can be won. A civic action plan supports every military operation. Not because we believe that friendship has a price which we can pay with giveaways and handouts, but because we thus have the opportunity to show in a definite, tangible way that our interests really lie with the people and their problems. And sometimes clothing supplied by the various aid agencies are often as important to take on combat operations as any other kind of cargo. They are not likely given, nor are they likely received. To people who live always on the edge of want, whose days blend into a continuing challenge to rest enough from the earth to still their children's hunger, such gifts are deeply important. And the gesture of giving becomes deeply meaningful. To people whose lives are shadowed and haunted by disease, who must suffer pain and illness because no help for suffering is available to them, whose health and well-being are doomed to the harsh conditions of a harsh life. To such a people, the attentions of a military medical team are powerful symbols indeed of a desire for friendship. Even soap, in a community which has none, and therefore is vulnerable to the ravages that such a simple item can prevent, becomes a weapon with almost magic powers. Every military unit has the capability of helping to improve a community in some way. Simple construction material, for instance, and a little paint, and a few hours' work can pay rich and lasting dividends. Particularly work done side by side with the people themselves, for the Vietnamese people are both proud and industrious. They welcome our assistance, and together we can accomplish much. A new community meeting center, place for the children to go to school, a new face for the house of worship. A well, a bridge. These can work for us with wonderful effect. Helping the people to gather in their precious rice crop and protect it from the Viet Cong, as these Marines are doing, can be as important as any military engagement. For all this is evidence which will stand as lasting reminders that the Americans who came this way as allies of their government came as friends. Establishing friendships in the villages of Vietnam can also produce tangible results. Intelligence information, for instance, which could save your life. The war we wage in Vietnam today is slow and hard and grinding. From the storehouse of our nation's power, we are bringing to it the might with which we can eventually destroy the enemy's strength. From reservoirs of compassion and generosity, we bring a spirit and a demonstration of friendship, and these two become weapons. As powerful as any other in this war, which is a contest for the hearts and the minds of the people of Vietnam. They are weapons that will touch and stir the hopes of those through whose villages we travel, and when we pass on, we'll leave a deposit of warm feelings behind. A demonstration of the people's new appreciation for us in any one village cannot perhaps be taken too seriously. It may not mean that the village has been secured forever. The war is not that simple, but we must not falter. We must continue until the enemy is convinced his aggression will not pay. And hand in hand with the fighting in Vietnam, we must press ahead with the crucial task of helping to build a nation. All the way to the heart and soul of the smallest hamlet.


This year's Human Rights Day begins the countdown to the 75th anniversary of the Universal Declaration of Human Rights, which is a marked by UN wide call for the dignity, freedom, and justice for all. The enjoyment of the all human rights, including the rights to the adequate food has been under pressure of the past years. As the world face a new and the ongoing straights, including the climate crisis, the pandemic conflict, growing equities and gender-based violence. Global food security challenges are more daunting than ever before, around 2.3 billion people in the world where the moderate or severely food insecurity in 2021 that is nearly 30% of the global population. Food and life are interlinked. The Universal Declaration of Human Rights and the Food and Culture Organizations of the United Nations confirmed that without food, there is no life. Ever cemented the secure adequate food for all and ensure the freedom from hunger for has been the driver of our actions. Supporting the accountability of a member's nations and building up a capacity to ensure that the food is not a privilege, but a right for everyone, everywhere, every day. Ever continues to be firmly committed to the declaration of the human rights to its recognition of the human dignity and the inequality in rights. Its values are the foundations for our collective actions in ensuring the four betters, better production, better nutrition, a better environment, a better life for all, for the bad world and the bad future where no one is left behind.