Welcome to the last session, last technical session on just 2018. So this session will be on fault attacks, fault attacks too. So the first paper is on new Bleichenbacher records, fault attacks on QDSA signature. The work is a joint work between Kyoto University and Entity Labs. And the work will be presented by Akira Takahashi. Thank you for the introduction. So I'm Akira Takahashi, currently a master's student at Kyoto University in Japan. So today I'm going to present our recent results on Black-Hamburgers attack, and also our new fault attacks on QDSA signature scheme. This is a joint work with Meditibish and Masayuki Abe from Entity Secure Platform Laboratories. So here's the outline of the state stock. First, I'm going to introduce the problem we've addressed in this work. And then I'm going to present our three main contributions of the paper. So as a motivating example, let me briefly describe the Schnorr signature scheme, which is one of the simplest and most widely used digital signature scheme. So as everyone probably knows, the DSA and the ECDSA, those are the most notable variants of the Schnorr signature. As for the provable security, Schnorr signature is known to be secure in the random awkward model if the discrete log problem is hard. Also another important feature of the Schnorr is that it relies on an ephemeral random value known as nonce. So let's have a look at this nonce in the Schnorr signature. So here Alice is a signer holding a secret key D, chosen from integer mod n, where n is a large prime. And whenever Alice generates a new signature, she first samples this integer k from z over nz uniformly and at random. And this integer k is typically called nonce. And it usually satisfies this congruent relation. And since s and h, those are the public information as a part of the signature, of course, Alice should never disclose her nonce to anyone else. Otherwise, it would be very easy to recover the secret key. Also, Alice should never use the same exact k more than once. So these are well known facts. But what if this k is slightly biased? For example, as showing this figure, what if k's top bits are all c to 0? Or what if nonce k is partially leaked? Actually, even in such cases, by collecting sufficiently many signatures, adversely it could still bypass the discrete log problem and consequently steal the secret key by solving so-called hidden number problem. So we can see that nonce is a very sensitive part of the Schnorr signature, dsa and ecdsa as well. And it is very crucial to evaluate the risk of biased or leaked nonces. So in this context, our contribution can be summarized as follows. First, we optimize the statistical attack framework known as Blyhember's attack against noncedential signatures. Second, as a separate contribution, we propose new fault attacks against specific Schnorr like signatures called qdsa in order to obtain a few bits of nonces. And combining those two contributions together, we implemented a full secret key recovery attack against Schnorr like signatures, which was instantiated over a 252-bit group using only two or three bit nonce leaks. And actually, these targeted parameters have set new records in the implementation of Blyhember's attack. So here's the summary of the previous published records of attacks against nonces, including both Blyhember's attack and the lattice attacks as well. Basically, we solved this three-bit leak case and much harder two-bit leak case. And to the best of our knowledge, these targeted parameters have never been broken before. So now I'm going to present our first contribution, the optimization of Blyhember's attack. So what is Blyhember's nonce attack? It was originally already proposed 18 years ago by Daniel Blyhember, but it was only recently revisited by those two papers, the Maldau Tower at the chess 2013 and Araniel Tower at the Egecrete 2014. The main idea of the Blyhember's attack is to quantify the nonce bias by defining the so-called bias function, which is real valued between 0 and 1, and the final peak of it. Though I don't have much time to formally describe this bias function, essentially, the bias function becomes 0 if the nonce is uniformly distributed. And if the nonce is biased, then this bias function becomes close to 1. And essentially, the most important and costly phase of the Blyhember's attack is so-called range reduction of integers h. And this phase is necessary to detect the bias peak correctly and efficiently. So what do we have to do exactly during the range reduction? So we are first given s signatures, and we want to find the sufficiently many linear combinations of those h values. And these linear combinations have to satisfy two conditions. First, the resulting linear combination should be smaller than the certain threshold value. Second, the coefficients of the linear combination should be as sparse as possible. So those are the required conditions for the Blyhember's attack to work correctly. Also, please note that this range reduction problem looks like knapsack problem. But the main difference is that in the range reduction problem, we have to find many linear combinations instead of a single, exact knapsack solution. So the previous works I mentioned before have tried to address this range reduction problem in different ways. But unfortunately, their approaches were not very optimal, especially if the nonce biases is quite small. For example, at the Demalda et al's work, they used the BKZ lattice reduction algorithm. But with this approach, the resulting coefficients of the linear combinations are not sparse enough if the nonce bias is small. Also, another important approach is the certain difference conducted by Alanyato. But the certain difference requires many inputs and consequently consumes a huge memory space. So we took different approach. So here we apply the Schreppel-Schamier's knapsack algorithm. Actually, the use of Schreppel-Schamier was already mentioned by Blyhember, but interestingly, its merits have never been examined in the previous work. So what's good about Schreppel-Schamier? So first, the Schreppel-Schamier is highly space efficient. And second, it can be highly parallelizable by making use of how great Graham Jules variant of Schreppel-Schamier. So our range reduction algorithm was specifically inspired by this variant. So here's the brief overview of how the Schreppel-Schamier based range reduction works. So we first split the input integer list into four separate lists and sort them. Second, we search for the linear combinations of two whose top consecutive bits coincide with some certain fixed value and sort them and push them into two separate lists. And finally, we take the differences between those linear combinations of two. And since now it's guaranteed that the collision happened in the top bits, with good probability we can get very small linear combinations for each round. So that's the basic idea. We analyze this algorithm in detail and they observe a well-balanced time-space trade-offs. So here, as you can see, compared to the previous approach, sort of a difference, that our range reduction algorithm is much more space efficient. But the drawback is that the time complexity got a bit worse. But in practice, the Schreppel-Schamier-based range reduction still terminates within a reasonable time frame by making use of large-scale polarization technique, which I'm going to revisit in the implementation section. So now I will go over our second contribution, the fault attacks on QDSA signature scheme. The QDSA signature, which stands for the quotient digital signature algorithm, is a recent high-speed variant of Schnauer signature proposed by Renez and Smith at Egypt Crypt last year. The important feature of the QDSA is that it can be instantiated with a curve to 519-mongomeric curve. So this means that the signature generation algorithm can benefit from highly efficient mongomeric ladder, mongomeric ladder-based scalar multiplication, in which Y coordinate is not used at all. Also, please note that in QDSA, the non-scale is deterministically generated. Here, our technical idea is quite simple. So we basically exploit this group structure of curve to 519, which, as you can see, include non-prime and lower-order points. So by injecting an appropriate fault to the base point of the curve to 519, we can perturb it to non-prime or lower-order points. And using this 40 base point as an input to the ladder function, by looking at the resulting 40 output, we can obtain partial information of the secret scalar k, which is used as a non-synational QDSA signature. So here I give our fundamental observation on why our attack works against the QDSA signature. First, if we attack the EC-Synor or EC-DSA, since they use Y coordinate as well, if we perturb the point, the resulting point is not likely on the original curve anymore, that's why our fault attacks wouldn't make sense against such schemes. On the other hand, this QDSA makes use of X on the arithmetic, and therefore, it does not involve Y coordinate. The perturb point, the P-tuder, is necessary on the curve itself or its twist. So that's why, even after injecting a fault, we can still confidently make use of the structure of the underlying original curve. So following this observation, we proposed two types of fault attacks on curve 25 of 19. The first one is random, same ferment fault against the program memory. So using this fault attack, we can obtain the three least significant bits of the nodes. The second one is instruction skipping fault against the base point initialization. So this fault attacks only reveals two LSB of the nodes. It is much easier to achieve in practice. And we verify that this attack indeed works against the QDSA using the chip whisper light against the AVR microcontroller. The countermeasure is quite simple, just multiplying the noses by this common multiple of the curve cofactor and the twist cofactor. And this technique is commonly known as cofactor killing or clamping. So especially for curve 25 of 19's case, we can just multiply the noses K by this common multiple 8. So this clamping technique completely throws our proposed fault attacks. OK, so now I'm going to present our main result, the implementation of the noses attack against the signal like signature. So here's our main result. So we attacked QDSA using only two bit leaks. And first, as an input, we simulated 2 to the 40 QDSA signatures out of which we used only 2 to the 26 instances, having smaller H values. And we fed them into Blackhember's attack. And we highly paralyzed the implementation. So in this experiment, we used 256 threads in total. And after the first trial of the Blackhember's attack, we successfully recovered the top bits of the secret key. Also, as expected, the memory consumption was quite modest. So in this experiment, we only used 15 gigabytes of memory. Also, the CPU time was rather long. Thanks to the large-scale parallelization, the actual experiment finished only after 16 days. Also, once we know the top bits of the secret key, we can recover the remaining bits of the secret key by iteratively applying Blackhember's attack. And this procedure took only less than six hours. Also, please note that our estimation shows that a certain difference, the previous approach, would require much more signatures, which would be equivalent to the 2 terabytes of memory. So I would say that this attack would have been much more difficult to achieve in practice using the previous range reduction algorithm. Also, we attacked a bit easier case using 3-bit non-slicks. So here, of course, the woke up time was much shorter than the previous one. And since the CPU time was less than 10 days, also the memory consumption was less than 3 gigabytes, we can say that the Blackhember's attack would be even feasible using a small laptop.  And of course, as expected, that sort and difference range reduction was much faster than the Shreve-Peshami based one. However, the memory consumption was over 100 gigabytes. And it required billions of signatures, which would not fit in the commonly used laptop. So let's conclude. So in the first contribution, we optimized Blackhember's attack and successfully overcame the memory barrier of previous approach by applying Knapsack algorithm. In the second contribution, we proposed new fault attacks on QDSS signature instantiated over CUB25519. And here we discovered yet another situation where adversely could learn the partial information of non-slicks. Also, the important lesson here is that the cofactor killing is a crucial countermeasure when using X-only arithmetic of Montgomery's curve. And finally, we implemented a non-slick attack and presented the first large-scale polarization of the Blackhember's attack and successfully set the new records of non-slick attack. And of course, our parallel implementation of the Blackhember's attack is publicly available on GitHub page, so please check it out. Thank you very much. So we have some time for questions. Thank you for the nice talk. So I have a question. What kind of fault model you have assumed in this case? Can you explain the fault model? So in the first fault model, I mean, the fault that makes it possible to review the 3SB. So we basically assumed that the optical fault, in order to induce the bit flip. And in the second model, we basically used the clock glitch. OK, so clock glitch resulted in single bit flip only? Sorry? The clock glitch resulted in bit flips only? No, no. The clock glitch was just used to skip the one instruction to set the X coordinate of the base point. Oh, I see. OK. Thank you. So are there any other questions? So I have one question. So you mentioned about the Montgomery ladder being implemented using X coordinates. So what if I also have projective coordinates to implement? Like, will that actually work? So if you just use the X and Z, for example, as coordinates. So we basically assumed the X, Z in this experiment. So that's what exactly we need. It works. OK, so are there any questions? If not, then let's thank the speaker. Thank you.


All right I'm here with Jack a big real estate mogul from Australia at the Sundance Film Festival and Jack was just telling me his story about how he quit drinking last year and all the amazing things that happen. Tell us your story Jack. I gave up drinking from February the first for five months before I followed the tour de France and I lost 13 kilos and felt absolutely amazing and so much energy but I felt great it's just tremendous. All right now what are you drinking right now? Maybe a gin and tonic. Before I get back to Australia I'm going to give up again from February the first until I follow the Giro d'Italia this year. Very nice.  Absolutely I mean you've got to lose the weight to do the bike riding to do the climbs you know the classic climbs like Alpe d'Huez you know all the fantastic climbs through Europe you've got it you've got a trim down you've got to be your best by not drinking it will be your best. All right there you go Jack from Australia. Thanks mate. Appreciate it.


This equation forces us to assume that epsilon is less or equal than a constant times n to the one negative one seventh. So we cannot push it all the way down to one, but at least we can get epsilon being a power of n. And there is no reason for this eight. It can be definitely reduced to one coordinate. This restriction is also an artifact of the proof. One can reduce it at the price of making the proof more involved, but we strived for the easiest case for the clearest result. So how would one approach such a problem? Of course, if the entries are bounded, if the tails are light, or even if the tails have a uniformly bounded fourth moment, this assumption, yes. But I wanted to separate the moment type assumption from the assumption on the entries. The tails enter the discussion in only one way via this event. They have to be on the sphere, but the sphere may be complex. So how would one prove this? First, let's get rid of the notion of eigenvector here. This is sensitive to eigenvectors related to eigenvalues, and if an ensemble is complicated, we don't have precise information on eigenvalues. Also, if the matrix is not normal, the eigenvectors are not orthogonal, and this strips us from using most of the spectral methods. So let's get rid of the eigenvectors and let's translate it to a different problem. So suppose that I consider a toy case where V on I is just 0. And then I have the equation of A minus lambda V is 0. But this means that I can reduce this equation to only the coordinates of I complement, and it means that this matrix has an untrivial kernel. But what is this matrix? This is the matrix with n rows and 1 minus epsilon n columns. This is rectangular, and if the matrix is random, it's improbable that the kernel would be non-trivial. We can play on this fact and reduce the problem to the invertibility of rectangular matrices. Of course, this is a toy case, but we can relax it, and if we instead assume that the norm of Vi is less than some delta, then we have again 0 is A minus lambda V, which is A minus lambda Vi complement plus Vi. And this means that A minus lambda Vi, I complement and here is also I complement. The L2 norm is less or equal than the norm of this matrix, A minus lambda times the norm of Vi2. Here I can use the triangle inequality, and if I have this event that the norm is small, I can bound it by m square root of n plus the absolute value of lambda times, and this way I assumed, was less than delta. And this lambda is a point in the spectrum, so its absolute values should not exceed the norm of the matrix, and we are getting down to m square root of n times delta. So this is small, and if it is delta small, I can also approximate this lambda by some deterministic parameter. So I'll assume that absolute value of lambda minus mu is less than delta, and let me keep the result, then we have that the norm of A minus lambda I complement, Vi complement will be less than 4m square root of n times delta. Sorry, yeah, minus lambda, minus mu. Mu now is deterministic. Of course, it's not completely deterministic, it should be close to lambda. Lambda is a random parameter, I don't know anything about its distribution, but we can run a delta net argument over all possible values of mu. We can take a disk of radius m square, and we can take a delta net in this disk so that any point, m square root of n, any point can be delta approximated by the point of a net, and then we can take a union bound over this net. These points belong to the net n delta, and the cardinality of n delta is bounded by, you can compare the area of the disk to the area of small disks around each point, it will be c square root of n over delta square, and this entropy cost, if we recall the lectures of Terry Tao, this entropy cost is negligible. Remember, we want to get something to the power epsilon n. If epsilon n is reasonably big, then these two will be suppressed. Then, after we have done it for a concrete set i, we can take the union bound over all possible sets i and get the following reduction, any mu complex with absolute value of mu less or equal than m square root of n, the probability for any i of cardinality epsilon n, the probability that the minimal singular value of a minus mu is less or equal than delta square root of n, is less than some number p0 less than 1, and then no gaps. Delocalization holds with probability greater or equal than 1 minus, and then we have to include the entropy cost, which is c square root of n over delta squared p0, and times the cost of choosing i, which is n, choose epsilon n, and this bound will be very important later. So, first how the minimal singular value got into picture? What we had here is that the image of this vector v reduced to the i complement falls into a small ball, but if I assume that the norm of v i is small and the norm of the whole vector v is 1, then the norm of v reduced to i complement should be large, it should be at least one half, and this means that this matrix should have a smallest singular value. So, if we manage to get a uniform bound on the fact that the small singular value is small, we are in a good shape. And there are methods to get such bound. The first one, the most straightforward one, is the epsilon net argument described in the lectures of Terry Tau, and we can run it after all. Sorry, not a minus mu, a minus mu, I complement this. Submatrix. If this submatrix has n rows and 1 minus epsilon n columns, this is a rectangular and we know to get the bounds on the smallest singular value of such matrices. So, if we run an epsilon net argument, we will get some bound. So, let's say if a tilde is an n by m matrix with nice entries centered unit variance and finite fourth moment, then the probability that the minimal singular value of this a tilde is greater or equal than c square root of n will be bounded by the exponential of the power of negative c prime. And if m is less than some beta n for a small absolute constant beta. So, we may try, this result was basically proved in the lectures of Terry Tau. So, let me save time on proving it again. We can look at this and try to apply it to our situation. It's very close. We only need to take some union bounds, but we know how to take them. So, what's the problems here? First, I want to keep the assumption about dependence. And this result require complete independence. Okay, maybe if I am not so ambitious, if I restrict myself to the generic type ensembles and I assume complete independence, I can easily finish the proof of this theorem by applying the epsilon net argument. And then we hit the second obstacle. And the second obstacle, this beta should be small and I am striving for a small epsilon, which means that the set i is small and i complement is almost the whole set 1, etc. And which means that the matrix is almost square. For almost square matrix, this is wrong. Sorry, less or equal, of course. For almost square matrix, this is wrong, but a more elaborate analysis, the composition of into compressed and incompressible vector, can yield some, less or equal and less or equal, can yield some bound on the probability that the smallest single value is small. The problem with it is that the bound will be not this time with the probability exponential in n, but with probability exponential in n minus m. If n minus m is small, in particular if it is epsilon n, we have the probability exponential in epsilon n. And we have to beat the union bound, which is super exponential in epsilon n. And then do not match. But maybe we can do something smarter than taking the union bound overall. Overall possible choices of i. So we have, so far we identified two obstacles, epsilon net argument. So the first is insufficient independence. The second obstacle is not sufficiently high, high probability. Although the epsilon net argument, at least the elaborate epsilon net argument with the splitting to compressible and incompressible vectors yields the exponential probability, we need a super exponential one. But then suppose that we can overcome even this, or suppose that our problem is more moderate instead of having it for all i's, we want to have it for a single i. Let's compromise on everything. But even then we would hit the third obstacle. And to describe it, let me recall the main feature of the epsilon net argument. We run the small, we play the small ball probability against the cardinality of the net. That means that we discretize the unit sphere, and if n, let's say n tau is the tau net in the unit sphere, then the cardinality of n real unit sphere, then the cardinality of n tau is bounded by a constant over tau to the power n. And if we are lucky, the probability that the norm of A tilde x2 is less or equal than say delta square root n is bounded by some constant c of delta, or c delta actually, to the power, this will be m, if it's in m sphere, and this will be n. So A tilde is the matrix, which has n rows and m columns. And then we multiply these two numbers, and if n is greater than m, and delta is of the same order as tau, we win. And this works perfectly in the real case. But suppose that our entries are real, then even in this case, the eigenvalues will be complex. So instead of the real sphere, we must consider the complex sphere. And for the complex sphere, the dimension, the real dimension jumps, and the size of the net also jumps from m to 2m. And then we multiply the exponential of 2m and the exponential of n. And if m and n are close, the entry cost is prohibitively high. We cannot get anything from the epsilon net argument. Of course, there is one case where we don't have to go to the complex sphere. The real would be enough. This is the case of the Hermitian matrices. But for Hermitian matrices, we don't have independence. So we cannot run the epsilon net argument for a different reason. And all of this looks like a hopeless case. It's the exercise in futility. So why did I discuss it here? First of all, it's not absolutely futile. We can, and we are going to use this later in the proof as a piece of the puzzle when we assemble the puzzle together. We wouldn't use it for the whole sphere, and we wouldn't use it for this matrix. But this will come handy in the critical moment. Second, this frontal attack, which failed, teaches us one thing. If we want to approach there are no gaps in the localization where this proposition, if we want to get a strong bound on the smallest singular values of rectangular matrices with very high probability, super exponential probability, we need a good, a very strong, small bound, a bold bound for such vectors. And we are going to spend the next lecture on developing some of these strong bounds, even if we don't know how to apply them yet. Let me stop here. Questions? So I had a quick question. So you made this assumption that the imaginary parts were deterministic, and then you said later you would take the expectation over that. So how do the imaginary parts come into the bounds? Is it just in terms of the M? The imaginary part, the only way the imaginary part comes into the bounds is that I want to keep this, the event BAM, the event that the normal of the matrix is more likely. Otherwise it can be anything. Any other questions? So if not, we'll start again in 10 minutes and let's thank Mark again.


This paper reports on the synthesis of silver nanoparticles using an aqueous extract of seaweed entremorpha compressa as a reducing and stabilizing agent. The resulting particles were characterized by Fourier transform infrared spectroscopy, x-ray diffraction, scanning electron microscopy, energy dispersive x-ray spectroscopy, and transmission electron microscopy. The particles exhibited spherical shapes with sizes ranging from 4 to 24 nanometers and were found to be highly crystalline. The particles were tested for antibacterial and anti-cancer activities, and it was found that they had strong antimicrobial and anti-cancer properties. Additionally, the particles were shown to reduce the viability of cancer cells, suggesting potential use in cancer therapy. This article was authored by Vijayan Sri Ramkumar, Aravalgam Pugazendi, Kumar Gopalakrishnan, and others.


Karma will destroy the Narcissist. You may wonder if the Narcissist will ever experience the same kinds of things that they have caused you to experience. You may wonder if there are future consequences that will be shaped by their present actions. You may wonder if they are unknowingly planting seeds or do they just move on and find something else after they try to destroy you. The truth is that there is a universal law of cause and effect where things happen as a result of something else. Every action has a knock on effect. It causes other events and situations although not always directly. Some people call it karma, other people call it God or the universe but people understand that it exists and that is why we tend to do good by action virtuously because whatever you put out will always come back to you. It will return to where it belongs so it doesn't matter if they move on or if they try to begin a new life with someone else because karma will always come back to them but they will try to run away because they are trying to escape the results and effects of their actions because they don't want it to have an opposite and undesirable effect of what was intended. They believe that if they remove themselves physically then there won't be any unintended consequences of their actions. They will blame you to feel better about themselves because they are not considering that there is a spiritual side to this but they mistreated you and they know that so it is something that will play in their minds and you were not the only one because you are not the only person that they mistreated. They have already caused so much pain and suffering to people long before you came onto the scene and all of this will come back to them eventually because karma never loses an address and in my own experience I have seen narcissists experience their karma. I've seen them going through it and it gets them when they least expect it, when they think they're safe, when they think they've escaped but that's when they end up in a worse situation than when they were with you because they're always trying to gain an advantage, they're always looking for a way to get one over on you, they're always trying to find a better opportunity while they leave you hurt and confused but that pain helps you to heal and become stronger, it makes you a better person but narcissists are inadequate, they're damaged so they're flawed in character, efficiency and worth which is why all they can do is try to damage you, they will try to find someone to replace you but they're just going to end up damaging them as well, it might look good in the beginning but it won't last for long and it won't end well because they're damaged so whatever they get involved in they will always be disturbed some problems because not only do they have to deal with the new person's faults and failures but they also have to deal with what's playing on their mind from the past while karma is coming to hunt them down but you may not see them getting their karma because they're masters of perception management they control what you see to change the way you think about particular people situations and events they plan and coordinate the elements of a situation to produce the desired effect especially when they're hurt especially when they're going through hard times because they don't want you to see them when they're weak and vulnerable the way that they choose to live will cause them to receive the same destruction that they put out you may not see it but it is going on behind the scenes they will be struggling when they thought that they would get one over on you when it turned out to be a very unpleasant experience for them because karma is real they thought they were going to replace you but they didn't so they will constantly be looking for something else and they will be looking for someone who is just like you because you set the bar you established a level of standard and quality for other people to meet but they will never find something like that again so they will only experience disappointment and dissatisfaction thank you for watching if you found this video helpful please give it a thumbs up share your thoughts in the comment section hit the subscribe button to receive the notifications if you would like to support the channel you could donate at paypal.me slash narc survivor you could book a one-on-one with me on my website it's narc survivor code UK thank you for watching and I'll talk to you soon


This lecture is part of an online course about Lie groups and will be about some of the weird things that happen in characteristic P that's greater than zero So most of the time we've been talking about Lie groups over the real numbers So we can have things like the general linear group or the orthogonal group and lots of other groups like that and For much of this theory, there's no particular reason why you should work with the real numbers rather than some other field For example, if you've got any field whatsoever We can still define the general linear group over the field the orthogonal group over the field and so on So these are examples of things called algebraic groups so so Lie groups you define them to be a real manifold with a group structure algebraic groups you you can define them as Algebraic varieties with a group structure in other words, they're defined by some polynomial equations and A lot of the theory of Lie groups also applies to algebraic groups over any field for example for any group We can get a Lie algebra Which is of course going to be a Lie algebra over the field K And a homomorphism of groups gives rise to a homomorphism of Lie algebras and so on however, there are also some quite big differences between groups over the reals and groups over fields and and this lecture is going to be about some of the differences So so let's just start with an example Here I'm going to give examples of complex Lie groups so a complex Lie group is just like a real Lie group except you work with a complex manifold and Won't really matter because the ones we're going to look at are Very simple. So here are three examples first of all we can take the complex numbers under addition That's a one-dimensional complex Lie group Secondly, we can take the non-zero complex numbers under multiplication A third example would be an elliptic curve. This is a little bit trickier So an elliptic curve is something that looks like y squared equals x cubed plus b x plus c except We should add points at infinity which I not going to go into detail about and Elliptic curves together with a point of infinity all have a natural group structure on them and All of these have the same Lie algebra The Lie algebra is just the complex numbers with with the bracket being zero of course because it's one dimensional Now just as for real Lie groups for complex Lie groups if two things have the same Lie algebra They're locally isomorphic and we can write down local isomorphisms fairly easily So to go from the additive complex numbers to the multiplicative complex numbers, of course, we use the exponential function So z maps to e to the 2 pi i z and this identifies the non-zero complex numbers as the complex numbers modulo the subgroup of integers So these two are locally isomorphic as expected The local isomorphism between the complex numbers and elliptic curve is a little bit trickier and involves the viastrice p function So we take z to a point of the project of plain given by We need the derivative of the viastrice function and the Viastrice function and then I guess one So this would be in projective coordinates. I'm not going to cover the viastrice functional projective coordinates this lecture Because this is just a brief example anyway, this map from the Complex numbers to the elliptic curve identifies the elliptic curve as the complex numbers modulo a lattice So a lattice is isomorphic to z squared and a typical example might be all numbers m plus n i for m n in z So we see what's happening is we've got three locally isomorphic groups And if we take the simply connected one the others are just quotients of it by a discrete subgroup you can also find a map from This group to this group if you want So this looks pretty much like what happens for the reals Now let's see what happens over Over a field of positive characteristics, so we're going to take a field K of characteristic p greater than zero and Just as over the complex numbers we can find three algebraic groups, so we can find K plus which is the additive group of the field K or we can take the multiplicative group of none zero elements of K or we can take an elliptic curve over K Which will again be given by the points y squared equals x cubed plus bx plus c plus a pointed infinity Again, these all have the same Lee algebra And the Lee algebra is just the field K under addition. However as groups They're just totally different They're not locally isomorphic at all. In fact, there are Any homomorphism between any two of these groups just has to be the trivial homomorphism and first of all These two here affine Algebraic groups, it means they're affine varieties and this one is projective and Affine and projective varieties are sort of almost opposite of each other And These two are also completely different for instance. This one is an example of a unipotent group What this means is if you take a matrix representation of it all elements of the group Have all eigenvalues equal to one for example We can represent this group as the following subgroup of the general linear group and you see all its elements have all eigenvalues one On the other hand, this one is an example of something called a reductive Algebraic group and this means well in this particular case. It means all the elements are diagonalizable and for non-Abelian groups that definition of reductive is a bit more complicated and and Reductive groups and unipotent groups are sort of implacable enemies of each other You can think of reductive groups as being good and unipotent groups as being evil When you when you study groups if anything is going wrong or causing complications It's because there is a unipotent group there somewhere causing trouble. So so these These groups of nothing to do with each other in characteristic p notice. There's no exponential function Because if you try and write down the exponential function You notice it's got denominators and these denominators don't always make sense in characteristic p So we we can't write down the functions between these groups that we had in the characteristic zero case because they're first They're not algebraic and second. They've got denominators and so on In particular, we see that Homomorphisms of lee algebras in general don't really say anything at all about homomorphisms of groups that there are simply no non-trivial homomorphisms between any of these groups even though the lee algebras are the same So so the the lee algebra and characteristic p greater than zero just tells you less about the group than in characteristic zero So now here's another example where characteristic p behaves oddly. Let's work over the reals We notice that high order differential operators Can be written in terms of First order Differential operators in other words These are more or less vector fields. So for example, if we take the lee algebra to be the left invariant differential operators of order One on a group G then we know that it's universal enveloping algebra is the left invariant differential operators of or of any order on the group G in characteristic P greater than zero this this just totally fails the high-order differential operators in General don't have all that much to do with the first order differential operators. In fact, this works even in the simplest examples Let's just take G to be the group of real numbers under addition So this is we're doing the characteristic zero case first then the differential operators Are easy to figure out the left invariant differential operators are just spanned by the operators D by dx to the n no problem at all. It's just a they just form a polynomial algebra Now let's try and work over Z so Here we can define an analog of differential operators acting on the ring Z of x and We can form differential operators D by dx to the n as before but we can there's a sort of extra wrinkle we can do because we can form the differential operator D by dx to the n and Divide it by n factorial and this gives you a perfectly good differential operator because it takes x to the k to K choose n X to the k minus n and you notice this is always an integer so so so we get more Differential operators and you might expect because of this extra possibility of dividing by n factorial You might wonder what is what actually is a differential operator in in these more general cases Well e is a differential operator of order less than n you can define that as saying that the commutator of e with multiplication by any function has order Less than we would n minus 1 so you can sort of define differential operators in general, but by this condition here Anyway, now let's look over what happens if you look over a field k of characteristic p And we want to know what are the differential operators on the polynomials in k? well here we can get the differential operators which are a sort of Divided power of D and these just take x to the k to k choose n x to the k minus n so it's just reduction of this differential operator mod p in some sense and Now you notice that these differential operators are not generated by the first order operator D one which is just D by DX in in fact D by DX to the power of p Is just equal to zero in characteristic p as you can easily check So this first order differential operator just generates a small finite dimensional subspace of the space of all higher order differential operators In particular in characteristic zero if you know the first order operators You know all of them in some sense, but in characteristic greater than zero Knowing the first order operators is only a small piece of what you need And so In the previous lecture We had this result that over the reels if we've got the Universal enveloping algebra, which over the reels is all high-order differential operators So in this algebra L is the subspace of Primitive elements So we recall the primitive elements are one such that delta of v is equal to v tensor one plus one tensor v and Delta is the map from u of L to u of L tensor u of L which for L in L takes L to L tensor one plus one tensor L and this is a homomorphism of algebra So What happens in characteristic greater than zero? well, this just fails and We gave an example of this last lecture. I'll just quickly recall it. So again, let's just take a one-dimensional additive group then the leap that then the Universal enveloping algebra of its leap algebra is just the polynomial ring in In D so we get 1d d squared d cubed and so on d to the p and so on and You can't really interpret these as differential operators because the piece power of The differential operator d by dx would actually be zero, but it's not zero in the universal enveloping algebra This is just the the the polynomial algebra in D On the other hand if we compare this with the differential operators The differential operators are spanned by 1d d squared d cubed and so on to d to the p and so on and These ones are just none zero multiples of each other, but these ones are definitely not zero none zero multiples of each other and the these two Things have nothing to do with each other. This one's a differential operation. This one just isn't or it's zero or whatever So in In this one What are the primitive elements? Well the primitive elements are easy to figure out They're spanned by d but d to the p is also primitive and d to the p squared is primitive and D to the p cubed is primitive and so on in general if you take a primitive element its p-th power is also primitive On the other hand in this one We can work out the primitive elements. We define you can define the Co-product delta by saying delta of d to the p K is the sum of d to the i tensor d to the k minus i and this time the primitive elements are Just spanned by d so So what this suggests is that in Positive characteristic the correct analog of the universal enveloping algebra is not the universal enveloping algebra Algebra of differential operators over the of the group and So when when you're working with groups of characteristic p you you don't really work with universal enveloping algebras What people often do in fact is they work with something called a formal group and the formal group turns out to be very closer related to the To the ring of differential operators over the field It's in some sense a sort of Something to do with the dual of this ring, but I won't go into that I should mention in characteristic p there are even stranger examples So we can get nought dimensional groups of order P so these are in some sense finite groups, but they are Connected Which doesn't seem to make sense if you're working over the reels. I mean a finite finite House door set can't be connected But in characteristic p the very weird things like this can happen an example of this is the group mu p Which is just the p th roots of One in some sense so its coordinate ring is K of x Over x to the p minus one and if you want to know more about this you can go to a Course on algebraic groups in characteristic p. I'm just mentioning this as a warning about just how strange things can get So The last example I'm going to present is that many lee algebras many lee algebras In characteristic p greater than zero Do not correspond to groups At least not in any easy way. So over the reels any real lee algebra. There's always a lee group associated to it So in positive characteristic this fails and let's start with an example in characteristic zero called the vit algebra So the vit algebra is more or less Polynomial vector fields on the line. So we're going to define l n minus one equals x to the n d by d x So we're going to take the lee algebra to have a basis of these and if you work out the common That the commutator l m l n Well, this is just x to the m d by d x x to the n d by d x minus x to the n d by d x x to the m d by d x Which if you work it out is equal to n minus m x to the m plus n minus 1 d by d x which is equal to n minus m l m plus n so So we have this nice infinite dimension lee algebra now in characteristic P greater than zero that we can define a variation of it We just define it to be spanned by operators l m with m only to find modulo P So this is now an algebra of dimension P and We use the same rule as before l m l n Equals n minus m l m plus n and This is a finite dimension lee algebra and it does not correspond to any Simple algebraic group And in fact there are quite a lot of finite dimension lee algebras in characteristic P, which just don't correspond to Algebraic groups and characteristic P in any obvious way and Sure Chevrolet classified the Algebraic groups and characteristic P the simple algebraic groups and over an algebraic you closed fields in characteristic P and Really amazed everyone by showing it's the same more or less the same as the classification of simple lee groups and Characteristic zero and this was a big surprise to everybody because they knew that the lee algebras in characteristic P We're completely different to the lee algebras and characteristic zero However, the lee groups and characteristic P at least the simple ones Are quite close to the ones in characteristic zero. So again, this is an example where the Lee algebras don't really seem to have all that much to do with the lee groups Okay, so next lecture we're going back to the case of algebras over the reels


Well, I want to thank the panelists for coming, and Bernie, I really appreciate you joining us today. But it's a somber day. It's an astonishing thing that happened in Texas, because it effectively bans abortion and completely interferes with the right of a woman to choose. It's completely in conflict with Roe v. Wade. It appears to be absolutely and clearly unconstitutional. But you have the political authority of a state essentially passing a law that completely compromises the constitutional rights of women throughout this country. The second thing it does is, you know, is staggeringly dangerous. It essentially creates a vigilante mob enforcement mechanism where there's a financial incentive provided to people who oppose the right of a woman to choose to not only sue that woman, but anybody who, quote, aides that woman. It could be a mom taking a daughter to the clinic. It could be a sister. It could be a good friend that is just essentially an Uber driver providing transportation. And we don't need imagination to see how that sets citizen against citizen in a state of enormous conflict as opposed to citizen debating citizen, which is always fair in a democracy. In the House, I believe this week we're going to take up legislation that all the Democrats I believe in the House are cosponsoring, Representative Judy choose legislation to codify Roe v. Wade and protect the right of women throughout the states to continue to have the protections of Roe v. Wade. Of course that's something that our Vermont legislature, thank you, led on. So we're here to talk about that today and hear from the folks here who have been on the forefront for years protecting the right of women to choose and I want to thank you and just for the run of show so you know we're going to hear from Senator Sanders in a minute and then Lucy, my former colleague in the State House from Planned Parenthood, Alain Gregory Davis, thank you, UVM medical students for choice, Falco Shilling from the ACLU and Senator Lyons, my colleague, Representative Ampute, my former colleague, House is Senate and House, Josh Diamond from the Attorney General's Office in our House, and really I think the godmother of advocacy, Melinda Moulton, we're all grateful to you for your years. So I want to turn it over now to my colleague, Senator Sanders, Bernie, you've been there from day one on this and I really want to express on behalf of all of us our gratitude for your leadership. Bernie? Well Peter, thank you very much and let me thank all of the individuals and organizations not only for being here today but for the years of long struggle to make sure that at the end of the day it is women and non-politicians who are able to control their lives and I want to say a word to the men out there and I want you all to be thinking about how you would feel if the government, state government, federal government told you what you had to do with your own body. You would say this is outrageous, this is unacceptable, this is a denial of my basic rights and it is. We are in this together. This is not a women's issue, it's not a men's issue. This is a human rights issue. Anybody who believes in freedom understands you cannot force and coerce somebody to do something with their own body except for whatever reason they choose not to do. So I want to applaud what the Vermont State Legislature is attempting to do and certainly I think we have in Washington got to do our best to codify where we wait. The other point that I would make is in this rather unprecedented moment in American history, one might think that there would be an effort on the part of political leaders in Texas and all of them to try to bring us together around issues of common interest and yet here we have a nation today, the richest nation on earth where we continue to have the highest rate of childhood poverty of almost any other nation say here we have these folks talking about children and the rights of children and yet you got millions of children in this country who are struggling with poverty and I might say that just recently and I want to thank the president and colleagues Peter and the house and said to the American rescue planet we have cut childhood poverty in America by over half. That's a pretty good thing. You know how many Republicans we have supporting that in Congress? Zero. All right. We have over 90 million people who are uninsured or underinsured and yet we have our Republican colleagues trying to end repeal the Affordable Care Act and not expand healthcare in the way that has to be done. So this is not only an attack on women's rights only an attack on human rights. What it is is another effort to try to divide us up by the same people who refuse to recognize the crisis of climate who we fuse in some cases to urge fellow Americans to get the vaccines they need to combat this terrible pandemic that we face. So I again just want to we're very grateful for the leadership here in the state of Vermont. I hope Vermont leads the nation in this direction and just want to thank all of you for the great work that you're doing. Thank you. Thank you. Lucy. Thank you for the leadership here turn the floor over to you. Yeah. Well, thank you so much. I really appreciate you doing this representative Walsh and thank you for joining.  Thank you for having us and thank you for convening this group. Representative Walsh and thank you for joining us, Senator Sanders. We are and I want to also thank the press. I see there's quite a few members of the press here. I just want to thank you for your good coverage and for talking about the Texas law. I think it's really important and really relevant for us in Vermont and all across the nation. And here's why. In Texas, we have a law that's clearly unconstitutional. Our U.S. Supreme Court has decided to in a midnight decision to let it stand and go into effect. It effectively bans abortion after six weeks. So let's put that into perspective for a second. Six weeks is before many people know they're pregnant. Most people won't even have a positive pregnancy pregnancy test until four or five weeks. That means to get an abortion. They they have a they have a week, maybe two at best to suspect that they may be pregnant, confirm that they are a confirm an appointment with a provider, have an ultrasound, wait 24 hours, and then have a second appointment with that same provider to have a procedure. So or to to go through the process. So this is clearly a it effectively bans abortion. So it really matters to us because as Representative Welch had mentioned earlier in his remarks, it creates a vigilante justice kind of system, a Sue thy neighbor system where people get financial rewards for turning in their neighbor. A person who's subjected to harassment could be further harassed and encouraged to be further harassed by an abusive partner or disgruntled neighbor. And the attorney's fees aren't even recoverable for people who prove that they are innocent. So we have a really unjust law that's gone into effect and already right now there are states that are jumping to pass copycat laws. And guess what? You might think Florida, yes, but right next door in New Hampshire we have a lawmaker who is vowed to jump on the bandwagon and pass a similar law in New Hampshire. So that's the state of affairs with regards to the Texas law. And then just for the larger context nationally, we've seen over 600, 600 separate restrictions or attempts at restrictions introduced in the United States. That is the worst year in terms of restrictions introduced nationally since the road decision was decided in 1973. So now our next large hurdle or threat here is that we have a Supreme Court case that the Supreme Court has decided to take up a case. It's a Mississippi 15-week abortion ban, DOPS versus Jackson Women's Help. And this is the case where oral arguments are going to be coming up very soon. I'm sure maybe our attorney general, our assistant attorney general, Josh Diamond, may want to talk about this a little more. But this is a very concerning moment for us because many believe that with the new composition of the Supreme Court we have three Trump appointees, and this will be the first case, abortion case, that the court will have heard since Amy Coney Barrett joined the court. So we know what the composition of the court, we can count the vote already. We know that there is not support for Roe v. Wade to continue on the current Supreme Court. And that's why it's so important that these legislative heroes, who are here to my right, Representative Ann Pugh and Senator Ginny Lyons, and also the heroes at the attorney general's office, TJ Donovan and team, have worked together to move forward and have the foresight and vision to move forward, anticipating this day a constitutional amendment called Proposition 5 or the Reproductive Liberty Amendment. So it would enshrine reproductive rights in our constitution. So I will leave more discussion of that to those speakers, and I will now yield the floor. Thank you, Lucy, very much. Kailin Gregory Davis, thank you for being here. I'm the leader of the UVM Medical Students for Choice. Kailin. Thank you. Hello, everyone. Thank you for inviting me to speak today. So my name is Kailin Gregory Davis, and I'm a fourth year medical student at the UVM Learner College of Medicine, and I'm currently serving as the president of the Board of Directors of Medical Students for Choice. So this is an international organization dedicated to ensuring abortion access and training for medical students worldwide. And so at Medical Students for Choice, our mission is to ensure that medical students have the tools and skills to become the abortion providers that this world needs. This feels precarious when legislation closes down clinics and overwhelms training sites that remain. So what's happening right now in Texas feels particularly close to home to me as I grapple with what this will mean for me and my training in the next couple of months. So I start my reproductive health externship in two weeks in New Mexico, and I'll be spending three months training with the abortion providers there to supplement my medical education with firsthand abortion training. So I've been in touch with the providers at this clinic, and they're seeing inflex in patients from Texas to the point where they've had to hire additional providers during this time to handle the increase in volume. And this for me has an eerie feeling of history repeating itself, because I actually studied midwifery in Texas, and I was living in El Paso in the aftermath of House Bill 2 that shut down the vast majority of clinics in the state, so it was in 2014. And as a provider, I saw firsthand the devastation that comes when a huge state bans abortion. So I saw people traveling upwards of 15 hours to get the care they needed, and I actually worked for an abortion fund that had to send people to other states, such as New Mexico, to get an abortion. We knew people were crossing into Mexico to get misoprostal, which they could get over the counter there, and then out of desperation, self-administering at home. And so we held self-managed abortion workshops for harm reduction. And I then worked as a counselor and a midwife at an abortion clinic in New Mexico, where over 50% of our patient population was coming from Texas. So I've had countless patients whose stories I've witnessed to share just a few.  So these are all just a few, but all of these patients traveling hours and hours piecing together what little funds they have for food, lodging, and traveling. And this was all for a procedure that could have been less than 10 minutes and could have happened in their hometown. So it's financial, emotional, and physical stress that simply doesn't need to be. And I feel really saddened by this hardship. And I'm fearful of other states following suit, as Lucy was mentioning.  So as a future abortion provider and a student whose training will be affected by this situation in Texas, it brings to light these broad ripples that such bans have on the lives of everyone involved. So for me, this brings to light the importance of Proposition 5. And I'm currently a member of the Vermont Speaker's Bureau. And I feel absolutely committed to doing what is needed to support the reproductive liberty amendment here in Vermont. Because I've seen a devastation of what can happen when legislation heads in an anti-choice direction. And I am so ready to see the growth and the safety that comes when a state does the opposite and declares reproductive freedom a human right. And then on a personal level, I'm a 31-year-old Vermonter with a uterus. So I'm considering options of birth control, pregnancy, parenting, adoption. So Proposition 5 also feels personal to me. And I need my home constitution to support my own reproductive freedom. And for me, that means the ability to pursue whatever methods of conception work best for me and my partner. Means I have an option to end a pregnancy if need be, to access birth control if desired, to have fertility assistance if need be. And ultimately, to give birth where I feel the most supported and safe. These are basic human rights. And on a professional level, as the president of medical students for choice and as a student at the Larner College of Medicine, I need my home constitution to support my medical practice so that I can safely provide people the options that they deserve. So I'm studying medicine in Vermont because I believe in this community. This is my home. And I ask in return that state legislation believe in me and my fellow students and providers and community members and patients by entrusting us with reproductive freedom. That's the fundamental right to safe, legal and accessible reproductive options. I've witnessed that the alternative is too dangerous and substandard to accept. And so I've witnessed what you all know to be true. That legislation matters. So thank you for being here today. Thank you very much. Excellent. And now let's hear from Falco Schilling from the Vermont ACLU. Thank you. Thank you so much. Thank you so much, one for everyone for being here. Representative Welch for convening this. Senator Sanders for joining us today. And just for the great legislative leaders we have here, both on the federal and on the state level, we put Vermont on a very different path from many other places in the country. Which is one reason that I know many of us feel so proud to call this state home. This is a deeply personal issue for so many people, both here in Vermont and around the country. Because we know we're not really free unless we have the freedom to make our own decisions about our own healthcare and our own life paths. As someone who is expecting a child and is planning a family, this is deeply personal for me and so many other people I know. And when you hear the stories coming out of Texas, it's deeply troubling and saddening. You hear the stories on the radio just this morning of people queuing up in line, hoping that they might be able to get the medication they need before this ban goes into effect. People fearing how they're going to travel hundreds of miles out of the state to get the care they need. Especially when these are some of the people that have the least access to resources in the first place. Laws like these only further exacerbate the health disparities that already exist in our country for so many people. And we've seen late bear during the pandemic. And that's why the ACLU joined by Planned Parenthood, ACLU of Texas, Center for Reproductive Rights, sued to stop this law from going into effect. We brought suit in the federal court in Texas because we knew the way this law was designed, it was made to evade judicial review. And it's doing exactly what it was intended to do right now, which is an incredible problem. This law targets providers and those who would aid people in getting lawful and constitutionally protected abortions with vigilante bounty hunter suits, which are completely unconstitutional yet because of the weird the loopholes written to this law that other states are now eyeing and hoping to put in effect the day the Supreme Court might uphold the law like this. We know we need to push back and we know this is something that we cannot let stand. So the ACLU and our allies across the country are continuing to fight this case in the courts in Texas as well as at the Supreme Court. And that begs the question, all right, we're doing that. What can we do here? What can we do here in Vermont? And I often say, when I'm talking to my colleagues, I know my job isn't easy working for the ACLU. That is never easy. But it's not as hard as some of my colleagues' jobs across the country. I see some of the things they're having to push back again since the things they're having to battle. And I feel quite grateful to work in the state that I work in and with the leaders that I get to work with who have the vision to ensure not only that right row versus way is protected, but the reproductive liberty of all Vermonters is protected now and going forward. That's why this is reproductive liberty has not only been enshrined in our legislation, but now we're moving to enshrine in our Constitution because we need to have as much protection as possible to these fundamental rights for all Vermonters for as long as possible. And that's what we do. When we want to enshrine our rights, we put them in our Constitution. And that's why Vermont voters will hopefully have the chance to go to the ballot box this November and say, these are the rights we believe in. These are the rights that we need to enshrine going forward. So I just want to thank everyone who's been here today, everyone who's been part of that effort, and everyone who's going to be part of that effort going forward. Thank you, Falco. Thank you very much. And now, Senator Jenny Lyons, my colleague from years we served together in the state senate and she's the original senate sponsor of the reproductive liberty amendment, Senator Lyons. Thank you very much, Congressman Welch. And thank you, Senator Sanders, as well as Lucy Lyrich and all the others who are here. We couldn't be proposing a constitutional amendment for the fundamental right of reproductive liberty if we didn't have the leadership that we have at the federal level. There's just no doubt about it. We're so fortunate in this state to have you there and we really greatly appreciate the work that you do. This constitutional amendment, Proposition 5, began in 2018 when I talked with some folks about my concerns around reproductive liberty and abortion rights and had a bill drafted so we could put this proposition in. And then, fortunately, we had good support from our pro tem, Tim Ash, at that time, and then also Becca Ballant, whose current pro tem. So the proposition has gone through one whole biennium and been presented to the senate and voted on 28 to 2, almost unanimously. And then this past, during this current session, the senate again voted on the proposition and has sent it to the House for confirmation. So the role of the senate is to draft the bill and the language so that is consistent with the constitution. We've done that. As you know, Article 1 of our constitution states that all persons are born equally free and independent and have certain rights. That is a key part of the purpose for our constitutional amendment. The other article in the constitution that we feel supports what we're doing is Article 7, the Common Benefits Clause. That affirms the public health of Vermont citizens. So the constitutional amendment gives us reproductive liberty as a fundamental right. Currently, we have that fundamental right in this state and we've had it for nearly 50 years as a result of Roe v. Wade. But it's only a right as a result of court cases. So now we can move forward and move this Article 22, Article 22, into our constitution to build a fundamental right for reproductive liberty. It is time to put the question to the voters. The lack of definitive enumeration of reproductive liberty in Vermont's constitution, the threat of Roe v. Wade being overturned by a very conservative US Supreme Court, and the cloud of a multi-state initiative to pass restrictive, punitive laws all build a strong claim for Article 22 to go to the voters. Voters in the state of Vermont support what we're doing. A majority of voters will support what we're doing. Our job now is to complete our work in the House, Representative Pugh's leadership there, and then take this to the voters in November of 2022. One cannot emphasize enough the role that family planning and decision-making have in one's life course. One cannot emphasize enough the significance and the importance of economic security, social security, and other important aspects of one's personal life that relate to fundamental reproductive liberty, right of fundamental reproductive liberty. So I will close there, and thank you all again. The testimony that we took in our committee came from many of the people sitting around this table. As well as others, it was a broad range of testimony. I want to emphasize that. It wasn't one-sided. We heard pros and cons. And now we're going to hear the pros and cons from the citizens of Vermont, mostly pro. Thank you. Thank you. Thank you, Senator Lyons. In your counterpart in the House, a leader on this constitutional amendment, Representative Van Pugh. Well, thank you, Congressman Welch, and thank you, Senator Sanders. I could not feel luckier than to be a representative in the state of Vermont to work with and my colleagues in the Senate to have leadership from longtime advocates and citizens such as Melinda Moulton and to see have new doctors come on to the field soon to be.  Almost, OK. And to realize that Planned Parenthood has been a strong advocate as well as the Attorney General's Office and ACLU. We are not in this alone. Article 22, Proposition 5, that an individual's right to personal reproductive autonomy is central to the liberty and dignity to determine one's own life course and shall not be denied or infringed unless justified by a compelling state interest achieved by the least restrictive means. It sounds simple. It's one sentence that we would like to add to the Vermont's Constitution. We have added phrases to Vermont's Constitution on several occasions. It is a living document. As stated, it would amend the Constitution to enshrine personal reproductive liberty as a fundamental right. It would be the umbrella that all other state policies and state laws, even the ones that we passed last year, about reproductive liberty, be it umbrella under which they all stand. It restricts, but it does not eliminate the state's ability to regulate. Personal reproductive liberty, it's about more than just abortion. It's about the freedom for those of us who call Vermont home to decide when and whether to have children, to become pregnant, to carry a pregnancy to term, to choose or refuse sterilization, and to choose or refuse contraception. The importance of Proposition 5 to Vermonters cannot be understated. It was important before, and it's important even more now, given what is happening nationally. The house is poised. The house under the leadership of Speaker Krowinski is poised to continue the amendment process in the house soon, as soon as we get back to the state house in January. And I am committed to bringing it. I chair the House Human Services Committee, where Prop 5 currently resides. We will have a public hearing, as we did last time, where the public is able to speak. And I have full confidence that the committee will vote out Proposition 5, as it did in the last biennium. And we will take it to the House floor. The proposition will sit on the House calendar, on the House notice calendar for four days so that people can read that simple sentence and know what they are voting on. And after the fourth day, the House will take a vote. And I am confident, as it did two years ago, that the House will pass this, and it will continue its path towards having the citizens of Vermont, giving the citizens of Vermont the ability, and I might say the right, to make the vote for whether or not, and would make the vote, and I believe, to support reproductive freedom. Vermont's policy has long recognized that decisions related to reproductive health and abortion are deeply personal and private. And they're best left to the individual and their health care provider. Prop 5 keeps things the way they are now and the way they've been for half a century. And I believe that the voters of Vermont will agree. Thank you very much. And now from the Attorney General's office, Vermont Deputy Attorney General Josh Diamond. I would love, Senator Sanders, thank you so much for this opportunity today to come and talk about the fundamental right of reproductive freedom or reproductive autonomy. Prop 5 and the Attorney General's efforts on these matters. And before speaking about Prop 5, as some of the speakers here today have discussed, I'd like to frame this within the national framework, because I think that really bears on the importance of why Prop 5 has been proposed. And really for the last almost five years, the fundamental right to reproductive freedom and reproductive autonomy has been under assault nationally. And I just want to highlight a few of the areas that our office, the Attorney General's office, in coalition with many other states have been leading the fight against these efforts to roll back this important fundamental freedom. Back in 2018, there was a decision by the Trump administration to cut funding under Title 10 to plan parenthood, which is really, as we know here in Vermont, an essential service in the delivery of healthcare to Vermonters. And the Attorney General's office filed comments in opposition of that and led efforts to gather comments from our citizenry to oppose those efforts. There was a decision by the Department of Homeland Security to deny access to abortion services for detained immigrant minors. We filed an amicus brief in that case to oppose such efforts. In 2019, there was once again a change in Title 10 rules that would require family planning clinics to withhold information about abortion and referrals on abortion. We joined our sister states to oppose those efforts as well. In 2020, Vermont with a number of other states filed a challenge to a rule, restricting access to contraception under the ACA, creating an exception for those employers who had a religious or moral objection to providing such healthcare opportunities. And then, as well in 2020, Health and Human Services rule requiring separate billing for abortion services on insurance programs that were provided, once again, Vermont, along with their sister states challenged those efforts as well. And then that brings us to these two landmark cases that are before or had been before the Supreme Court or will be coming before the Supreme Court. Whole Women's Health versus Jackson, which was the challenge to SB 8, as we've heard about today, which is really, again, an assault onto this fundamental freedom that we have recognized for almost 50 years, where it pits neighbor against neighbor, where it tries to undermine the norm, the legal norm, that states do not have the ability to prohibit access to abortion pre-viability. And in this case, in a shadow docket decision, the Supreme Court essentially let this law go into effect. We commend the U.S. Department of Justice's efforts to challenge that law that was filed last week and Vermont in the Vermont Attorney General's Office with a number of other states have filed a friend of the court brief in support of those actions. Coming up in this fall term, in the case of DOBS versus Jackson Women's Health Organization, there is a Mississippi law that is coming before the Supreme Court, which again also challenges the fundamental freedom of reproductive autonomy in banning most abortions after 15 weeks, which again is a direct challenge to the legal precedent that pre-viability abortions should not be restricted. So it's within this context that the Vermont Legislative Leadership, which the Attorney General's Office supported, did two very important things back in 2019. It passed the Freedom of Choice Act, which was a statute that recognized the fundamental liberty interest in reproductive freedom. But because statutes can be changed and no one should be at the subject of the whim of a different legislature, although we are very fortunate to have a wise legislature that recognizes the important freedoms that could change over time. And so hence Proposition 5, a constitutional amendment that does two very important things. One is it expressly recognized as the fundamental freedom to reproductive autonomy. And does that so that there's no ambiguity, that there's no reliance upon a judicial decision that could change over time? But secondly, it states clearly and unequivocally that the level of judicial scrutiny on any ever to limit that fundamental right must be strict scrutiny. So that there is a presumption that government action to limit that right will be prohibited. And that only when there is a compelling governmental interest and that interest is furthered by the most narrow means, the least burdensome means may that fundamental right be impacted. And this is very, very important for Vermonters, not just today, but going forward for our children, our grandchildren and the Attorney General, Attorney General Donovan is very much pleased to support the passage of Proposition 5. Thank you very much. Thank you. And our last speaker is Melinda Moulton who has been the Vermont champion. Thank you Congressman Wills and Senator Sanders for pulling together these remarkable people. My name is Melinda Moulton and I'm 71 years old. The legal right to an abortion did not exist in this country until I was 23 years old and that was back in 1973. I am here today to lend historical context to this discussion and to tell you why the Texas abortion ban should scare all of us. The year is 2021. Imagine that you live in a place where your neighbor could sue your family and friends, healthcare provider or even your pastor and therapist for helping you secure an abortion after six weeks. Well, welcome to Texas and this is real time. Texas citizens can sue anyone who assists a woman seeking an abortion after six weeks. The Lone Star State has taken its long horn and stabbed it into the road versus way constitutional right for all women to obtain a safe and legal abortion. On anti-abortion sites in Texas, you will find language that says in quotes, the law authorizes citizens to sue and seek financial damages from the support networks who help women get an abortion, such as families, friends, counselors, pastors, healthcare providers and more. And there is a designated area on these anti-abortion sites where you can leave in quotes, information anonymously about aiding or abetting a woman from securing a safe abortion after six weeks. Incredulously, the Lone Star State received 5-4 support from our Supreme Court. So what does this mean for American citizens of reproductive age as anti-abortion legislation supported by our now conserved Supremes heats up across the country? Well, let me tell you a most personal and deeply painful story about my mother. When I was just eight years old, my father and mother divorced and my mother became a single parent with four children to raise. I remember the night my mother was rushed to the hospital because she was found bleeding on the floor of our bathroom. Years later, I learned from my aunt that my mother attempted and succeeded to abort her pregnancy. Eventually, my mother fell in love again, but when this man got her pregnant, he abandoned her. What I remember is that my mother left for what she said was an operation. In fact, she was going to a secret hospital where women went to give birth in secret without anyone ever knowing. After she gave birth, the doctor performed a hysterectomy, which went wrong, and my mother developed peritonitis and over a five month period, lying alone in that hard so-called hospital. She withered away, got pneumonia and died three days before Christmas. My mother died in 1962, 11 years before abortion was made safe and legal in this country. I was 12 when she died and she was just 40 years old. Her four young children were left motherless. In closing, I wanna explain to you that women will always, always choose their own reproductive destiny and the reality is it will either be at the dangerous end of a coat hanger or it will remain safe and the capable and trained hands of a professional. We must ensure that it remains the latter. Vermont is taking strong bold steps to protect women in their reproductive health choices. We passed age 57 a couple of years ago and there is the Constitutional Reproductive Liberty Amendment moving through the legislature that will enshrine these rights. Women cannot realize full equality if we are not allowed control over our own bodies. Reproductive freedom for women is a human right and most Vermonters agree. Passing Prop 5 is an extremely crucial step to ensure we never, ever allow anyone to force us to return to the days where women like my mother must endure humiliation and potentially life-threatening procedures that often cost them their lives. Thank you. Thank you. Melinda, thank you very much. So that's the real world. Any questions to any of the folks who are here in the press? I have a question for both of you and Senator Sanders. There is the expectation that there is going to be anti-abortion groups, funding anti-abortion campaigns with Prop 5 does come up to vote in Vermont. What is your message to those groups? You want to go burn? I'll go first. Vermonters are to support the right of women make this choice. By the way, that is going to happen. There is an enormous campaign organized and financed as we saw in Texas to limit the reproductive rights of women and that's why this action that the State of Vermont is taking in the efforts that Senator Sanders and I are making to codify the rights of Roe v. Wade is so important. This is a fight that's engaged. This is real. The threat is real. And the consequences of Texas style law becoming, spreading out throughout the country is very threatening. So we all have to mobilize. And that's a major part of the reason we're all here together, renewing our commitment to protect reproductive rights for women. Bernie? People can do whatever they want in terms of advertising. But I think what makes Vermont an extraordinary state is our belief in freedom and human liberty goes way, way back to the inception of the state. And I think at the end of the day, I think all of the speakers have acknowledged we have absolute confidence that the voters in the state are going to overwhelmingly support the right of a woman to control our own money. Able to briefly go through the timeline of what the launchers can expect, just like simplified of what the next step is for this proposition. You go first. I'll try. Prop five currently sits in the House Human Services Committee. And we will take testimony. We will have a public hearing. We will do that between January and February before the time being recessed. The committee will take a vote on whether or not to move Proposition five to the House floor. I have full confidence that the committee will be voting yes. It will sit on the House calendar. For four days so everyone can see it and read it. And after four days, the full House will take a vote. It'll be a roll call vote by majority of the members present in the House. And it is an up or down vote. We cannot change any of the words at all. Once we vote, and I am confident that the House will vote yes, as we did two years ago. It will go to the governor. And the governor has to issue a proclamation where I'm putting it on the ballot in November of 2022. When the voters in Vermont will have the ability to vote to support something that has based reflects Vermont's values and beliefs and what has been the policy for half a century. So just one last comment on that. When it once it leaves the House and it goes out to be put on the ballot, the Secretary of State will take out Article 22 and Article 22 alone is what will be voted on. So that is the second part of Proposition Five. All the other language will be gone. So the purpose will be gone. It'll simply be that one sentence that Representative Pugh read earlier. The Secretary of State will put it, then put it on the ballot for the voters. There's another set of small rules regarding how the proclamation is issued, the timing and so on. But that'll happen. If the voters approve the amendment, Prop Five, when would it be effective? When would the Constitution actually change? Once Article 22 is voted favorably, it will go directly into the Constitution and I believe it becomes effective immediately, does it, Josh? That's my understanding, Senator. So as soon as, so Article 22 is added. 22. So right now there are 21 articles in the Constitution. 21 changes. 21 changes. Additions. So if you look at Vermont's Constitution and you look at the articles that are listed, it ends with the 21st and the next publication would have Article 22. And that's the one sentence that you read. Yes. Could you read that again for me? Certainly. That an individual's right to personal reproductive autonomy is central to the liberty and dignity to determine one's own life course and shall not be denied or infringed unless justified by a compelling state interest achieved by the least restrictive means. Is that the same as H.R. 3755 that was in the press release for today? Is that Article 22? Is that the same as H.R. 3755? Or is that different? I didn't hear what you said. I'm sorry, I'm not hearing what you're saying. I couldn't hear what you said. You might be referring to the congressional. Yeah, this is the Vermont legislation and the federal legislation that I believe we're gonna be voting on in the House protects and codifies Roe v. Wade rights to women throughout the country. And it would, in effect, it would prohibit a law like the Texas law from interfering with a woman's right to choose. That might be coming up for a vote in the House, is it starting in the House? It's in the House, and I think it's gonna come up this week. We spoke to Speaker Pelosi and her intention is to put that on as soon as possible. So we may well be voting on that this week. Do you have any indication if it's gonna pass easily through the House? It won't be easily, but I believe it'll pass in the House. I think it probably faces more challenges in the Senate, but Senator Sanders can speak to that. But I have confidence that we will pass that in the House. Passed what? The Article 22 or something different now? No, this is... For Vermont, Air Washington. Right. Article 22. What? I'm sorry. I'm sorry. Article 22 is only dealing with Vermont. The Vermont Constitution. And then S1975, the Women's Health Protection Act is going to be voted on by Congress, by the House of Representatives in D.C. And the Bards to Vermont? No, this is towards the whole nation. Okay. Vermont is amending its constitution. That's what the leaders here have described. We propose a statute, not a constitutional amendment, but that statute would provide legal protection for women throughout the country. Senator Sanders, do you have any indication if the Congressional Reproductive Act will be able to pass through the Senate smoothly? It's gonna be a struggle. There are some Democrats who may not support it. There are some few Republicans who will. Whether we get the votes or not is not clear at this point. If it passes, what would the implications be for the Texas law and other laws that are being considered that would be similar to the Texas law? Well, that's a lawyer's answer, but... What we're talking about is codifying Roe vs. Wade and taking it out of being a court decision to being the law of the length. But would it be retroactive in some way and therefore negate the Texas law? Well, it would in fact negate the Texas law. As of the day it became law. So every woman in Texas would be afforded the protection of that legislation immediately. We can't undo what Texas has done retroactively. I mean, it's doing damage each day as the calendar page flips, but we can't stop it with legislation. So maybe one last, I'm sorry, one last question because Senator Sanderson, I don't have to head out to the airport pretty soon. I think you have a timeline with this. If I have that one or not, can we have a little bit? I'm calling it HR 3755 because that's what it was in the release, but would you clarify the name for me again in the timeline of it? Well, let me just clarify for you. The HR refers to house resolution and that's the Congressional Act. And Senator Sanderson and I are here in our federal roles where we're promoting legislation that would essentially protect reproductive freedom. The Vermont legislature has taken the lead really in the nation by putting those protections in the constitution of our own state. And that's a tremendous act of leadership by the Vermont General Assembly. So there are two separate issues. The Vermont legislators are working on the constitutional amendment. Senator Sanderson and I are working on congressional statute. So the congressional statute, does that have the timeline for that? Well, the timeline is the House will vote on it, I believe this week. No, later than next week. And then Senator Sanders. I don't think there's clarity yet when the Senate will vote on it. One more time. Okay. Senator Sanders, it looks like the social policy bill will not be able to include a pathway to citizenship for undocumented foreigners. Do you have a next step in mind or a next plan about how that pathway could be secured? Well, it's gonna be more difficult now, but we do. Look, I think you've got 11 million people in this country who are undocumented, who in many cases have worked here for decades, who in the midst of the pandemic have put their lives on the line to keep the economy going. We got young people who came to this country at the age of one or two, some of whom are now serving the US military, the teachers, the doctors or whatever. So I am disappointed and frankly not surprised by the parliamentarian's decision that we're gonna do everything that we can to continue the fight to make sure that there is a path towards citizenship for the undocumented. Do you think Democrat leadership will go forward despite the Senate problems very abstract conditions? Well, that's the political decisions that the leadership has to make based on votes. Thank you. The other thing you guys need to deal with as you go back is the debt ceiling. Do you think that you'll be able to raise the debt ceiling, or are we running him only into the government shutdown? Well, you're running not only into the possibility of a government shutdown, you're running into the possibility of the most powerful economy on earth and not paying its bills, which will lead to economic chaos throughout the entire world. And it is really rather astounding that you have a United Republican Party, supposedly the conservative and responsible party prepared to not pay its bills and bring the economic disasters that will follow. So we will see what happens. But obviously, the United States of America has got to pay its bills, especially the bills that were accrued under the Trump administration. You know, the debt ceiling will be raised.  Yeah. Yeah. Thank you. Yep. President Sanders, do you have a statement on the Department of Defense waste and fraud and adequate statement you released today? Do I have a statement on the statement? Yes, you got it. I do have a statement on the statement. Let's try it out. Look, the Department of Defense is the only major government agency that has not undergone an independent audit. Right. It is a huge budget, $740 billion. I think, I think most observers think it's full of waste and fraud, cost overruns, et cetera. So I think the time is long overdue for the Department of Defense, like every other agency of the government to be held accountable. Okay, thank you all very much. Thank you.