Hi everybody. So let me just remind you about the classical public key encryption setting. So let's say we have three players, Alice, Bob and Charlie. Let's say Bob has the secret key, Alice has the public key. So Alice can now encrypt messages posted on the bulletin board. However, but the problem with this is that only Bob can now decrypt the message and compute functions of the message. So the question that I asked in the stock is the following. How can we allow Charlie to learn functions of the message? So in particular, I'm going to say we're going to be dealing with circuits. I'm going to denote them by C. How can we ensure that Charlie doesn't learn anything but the output of the function? At the same time, we would like to make sure that Bob does as little work as possible and he can go offline. So let's look at some of the trivial solutions that you can think of. So you can say why not just ask Bob to send the secret key to Charlie? Of course this has an immediate problem because now Charlie can decrypt and learn everything about the message and not just the output of the function. On the other side, you can say why not say Alice and ask her to send the ciphertext to Bob? Bob can now decrypt, compute the function of the message and send the result to Charlie. Of course, this requires Bob to do the computation and that requires Bob to stay online at all times. So another solution you can think of why not use fully homomorphic encryption? So in particular, why not ask Charlie to compute encryption of the output of the function, send the ciphertext to Bob? Bob can now verify decrypt and send the result to Charlie. Now, the function encryption comes in to actually help us solve all of these problems. In particular, it says the following. How can we allow Charlie to learn a function of a message M? So I'm going to say let's see denote a family of circuits and let M be the message space. So now Bob can do the following. Now Bob can run a special key generation algorithm on the master secret key and a particular circuit from the family. He can send it to Charlie and now he can go offline without having to communicate to him ever again. So now when Alice sends the ciphertext, Charlie can run the special decryption algorithm on the secret key and the ciphertext and learn the output of the function. And the security says the following. Security intuitively says that Alice shouldn't learn anything about the message M except for the output of the function. And there are various ways of formalizing which I won't go into the details. Okay, so let me give you a simple example. Let's say we have three players. Let's say we have a doctor who wants to go on a vacation. At the same time, he has patients and he wants to make sure they're taken care of. So what he can do is he can define a special C as a circuit that opens special urgent emails defined by the doctor. He can send the secret key to the secretary, for example. The secretary can now get an email which is an encryption of an email, run the decryption algorithm and learn the email if and only if the email is urgent. So there are various special cases of function encryption, for example, identity-based encryption. In identity-based encryption, we view a family of circuits being indexed by an ID and the message consists of two parts, ID prime and a payload. And the property is the following, that the decryptor should be able to learn the payload if and only if ID matches to ID prime. And there are various other special cases of function encryption. Now in the stock I asked the following question, can we actually construct function encryption for all circuits? And the answer that I give you is yes, with a small catch. And what is the catch? Well, the catch is the following, is that the number of functions that Charlie can compute is now going to be bounded by some queue. And this queue is going to be fixed before the setup. In particular, I'm going to require that if Charlie gets secret key 1 up to secret key subq, he can compute and learn the output of all these functions, but he shouldn't be able to learn anything about the message m other than the output of these functions. A alternative way of looking at this is in the bounded collusion setting. In particular, let's say we have many charlots. Each one of them gets one or more secret keys for different functions. If all of them now try to collude and learn about something about the message m other than the output of the function, they shouldn't be able to do so. So let me just quickly mention that previously this relaxation had to be studied in a number of works, for example, CCA secure or bounded collusion IB. So our results as the following. We show how to construct a queue bounded, not adaptive simulation secure function encryption for all policy circuits. Assuming CPA secure public key encryption and PRG is computable in the low depth. Let me just say one thing. If you are interested in the bounded number of messages, this result actually extends to the adaptive setting. Now, if you're worried about the later assumption that is PRG is computable in low depth, you shouldn't be, because it's implied by the standard hardness assumptions, such as factoring discrete log and lattice problems. So let me make a quick note, in particular, if you're trying to construct public index predicate encryption, then you don't need the later assumption. So I already told you what queue bounded is, and you asked the question, why do we restrict ourselves with this bound queue? Why not consider an unbounded setting? We have a result that says for unbounded collisions, it's actually impossible to construct function encryption for all functions in the non-adaptive setting. On the other side, why do we restrict ourselves with a non-adaptive case? That is, the adversary can ask for the secret keys only before seeing the challenge ciphertext. Well, again, there is a result by BSW that shows that for adaptive simulation secure function encryption, it's impossible to achieve for all circuits. And finally, let me just mention that simulation security is the stronger of the two for function encryption, and particularly it implies in usability-based security. So let me give you a roadmap of our construction. So first of all, we're going to start with one query that is one bounded function encryption from SS10. We're going to combine it with some techniques for multi-party computation such as BJW, and we're going to obtain a bounded function encryption for bounded degree circuits only. And after that, I'm going to show you a bootstrap in theorem that converts a bounded function encryption for bounded circuits to unbounded circuits. So let me tell you a little bit what I mean by this bounded degree circuits. In particular, I'm going to say that for every circuit in the family, the circuit is going to be an L-variate polynomial over the field of degree D. And this D is going to come up throughout the construction. Okay. So let me just give you some intuition about one function encryption that is one query function encryption for all circuits from SS10. So in particular, you can view the ciphertext as being a universal garbage encoding of the message M, and the secret key for a specific circuit as simply being the set of input labels. So from the security of garbage circuits, we know that the user can now evaluate and earn the output of the function. However, he learns nothing about the message M. We also know that as soon as you give two secret key, two sets of input labels, the underlying schema can no longer be secure. Okay. So another property that I'm going to use in my construction is a property observed in BGW protocol about Shamir's secret chain. In particular, it says the following that you can actually perform computation over the shares. And what this means is that if I give you a share of a message M and you compute over the shares, what you get is a share of the output of the function. Now, this has a problem. And the problem is the following is that once you multiply two polynomials, the degree of the underlying polynomial increases. So in particular, this catch at this step of the construction is going to prevent us from achieving unbounded circuits. Okay. So let me actually go into the construction. So let's say I have a one query function encryption consistent of four algorithms. So our setup algorithm is going to do the following. So it's going to repeat setup for one query function encryption and number of times. And what is then, well, n is a simple parameter that I'm going to leave as unspecified for the rest of my talk. It's going to depend on both D and Q. Okay. Where D is this degree of the polynomial and Q is the number of collisions that you would like to achieve the construction for. So the second step, let's look at the key generation algorithm. Well, the key generation algorithm is given a circuit C. So the first thing it does, it actually chooses a random subset of the secret keys that it's going to use. Now, for each one of the secret keys, it's going to run the key generation algorithm for the same circuit for one query function encryption in order to get a secret key. And what it outputs at the end is a collection of all the secret keys. Now, the number of secret keys I'm going to set in a minute and you will see what it should be. Now, the encryption algorithm does the following. So it takes a message M. So the first thing it does, it secret shares M using degree T polynomial. Okay. And it obtains N shares and one up to M7. So now, for each one of the shares, it uses appropriate master public key to encrypt the share and get a ciphertext. Now, in the output ciphertext, it's simply going to consist of the collection of all the ciphertexts. Now, what does the decryption algorithm do? Well, the decryption algorithm is going to take the secret keys which it has and it's going to match it with corresponding ciphertexts. And it's going to run the one-query decryption algorithm in order to get C of M I for each I. Okay. Now, what is this? Well, I told you that by properties of Shamir's secret sharing, we can perform homomorphic computation over the shares. So in particular, in this case, C of M2 is going to be a share of C of M. So what this actually means is that what I get is a secret sharing of C of M. So given enough shares, what we can do is we can reconstruct C of M and learn the output of the function. Okay. Now, let's calculate how many secret keys we have to give to the decryptor. So in particular, let's say Mi is S of I, where S is a degree T polynomial. Okay. Now, I also assume that C is a degree D polynomial. In particular, this means that the composition of the two polynomials gives us a degree DT polynomial. So by Shamir's secret sharing, we know that as long as the decryptor has DT plus one shares, he can actually recover the output. Okay. Now, let's look at the second setting where a key generation algorithm is being run on a different circuit. Let's say C2. So again, remember that for each key generation call, we choose a random subset of the master secret keys that we're going to use in the construction. So let's say, for example, these are going to be the secret keys. So the view of the decryptor is the following. He learns C1 of M1, C1 of M2, and so on. He also learns some secret sharing of C2 of M. So however, what did I do here? Well, I did something illegal. In particular, I ran the one query function encryption on the same master secret key. And this could no longer be secure. So therefore, in the worst case, I'm going to assume that decryptor fully recovers the share M2. Okay. However, remember that we secret shared M using a degree T polynomial. So therefore, assuming that decryptor learns at most T shares, information theoretically, he learns nothing about the message M. So let me tell you about a few other technical problems that come into the proof. So first of all, I told you that we proved the construction to be secure under the simulation-based definition. So therefore, because the adversary learns C of M1, the simulator must also be able to simulate the same view. However, these are not random shares. Because how would we do that? Well, we computed the function over secret sharing of M. So therefore, it's unclear how to simulate it. The same problem has been observed in the BGW protocol. And the solution is actually quite simple. All the solution is to consider a new set of, a new family of circuits C prime, where the message consists of two things. It's secret sharing of M and it's secret sharing of zero. So what we're going to do is we're going to add the secret sharing that is this randomness to the output of C of M1. Now this creates another problem. In particular, because we add in random sharing of zero of the same polynomial between different outputs of the function, this relates some correlation. So again, it's unclear how to simulate this view. So the solution is to use a QY's independent random sharing of zero. So in particular, we're going to take a linear combination of some random secret sharing of zero and we're going to add it to the output of the function. And we're going to construct a function encryption for this circuit family. Okay. So what did we do? Well, we started with a one-query function encryption from SS10. We combined it with some multi-party techniques and we got a bounded function encryption for bounded degree circuits. However, I promised you more. I promised you for all policy circuits. And how would we do this? Well, the idea is to use randomized encodings. In particular, the observation that we use as the following is that the function that computes a randomized encoding for C is of low degree. So what this means is that we can actually now construct a function encryption that doesn't output C of M, but it outputs a randomized encoding for C of M. So given the output of that, we know that the decryptor can learn C of M, but from the security of randomized encodings and the security of underlying function encryption, he learns nothing but the output of the function. Okay. So let me just summarize what we have done. Well, we took a one-query function encryption. We'll combine it with some multi-party techniques and we got a function encryption for bounded degree circuits. And then we constructed unbounded decryptor for unbounded decryptor circuits using randomized encodings. So finally, let me just conclude with some open problems. Well, one of them you might ask is why not consider weaker definitions for function encryption, in particular, unusability-based definition. So you can ask whether or not you can construct function encryption for all policy circuits with unbounded collusions under unusability-based definition. Similarly, you can ask for new connections with multi-party computation, zero knowledge, and function encryption. That's it. Thank you.


Gweithio, ac rwy'n gweithio i'r cwestiynau coronavirus Cymru gyda'r Gweithio Welsh. Over the last month, coronavirus has grown significantly across Wales. As coronavirus has re-established a much larger foothold in our towns, cities and villages, we've introduced new measures to protect people's health and to help control the spread of this highly infectious virus. All of this has happened very quickly. There's naturally a lot of confusion about the virus and the action that we've taken. So today I want to try to answer some of the most common questions we've had from people about coronavirus and why we've taken the action that we have. I want to start by showing you a very short video which has been put together by the sale data bank team in Swansea University Medical School and is based on a model of the data from confirmed cases. It shows just how quickly coronavirus has spread. At the end of August, there was hardly any coronavirus in Wales, just a few patches in Wrexham and South Wales. As September starts, you can see the cases just beginning to spread in South Wales and some clusters in mid Wales. As the month goes on, you can see the spread in the South Wales Valleys, the lighter the colour, the more the cases. And coronavirus spreads quickly across South Wales with more and more people testing positive. And as September comes to an end, we see cases spreading across North Wales. The number of people living with the virus is continuing to increase. We will put this video on our social media channels. I want to thank the team at the sale data bank for all the work they have done to make this striking presentation. It shows just how quickly the situation can change and how coronavirus can spread right across Wales in a matter of weeks. As I said at the start, I want to answer some of the questions that people have about coronavirus and why we're taking the action that we have to control it. I'll begin with the general question some people still ask about, why are we making such a fuss about coronavirus? Is there really a problem? Coronavirus is a new disease. 35 million people have been infected worldwide and more than 1 million people have died. There are currently no drugs licensed to treat or prevent coronavirus anywhere in the world and there is no known cure. We do not have a vaccine but research continues. It is highly infectious and spreads quickly as the sale data bank video just showed. And we don't understand what the long-term impacts are. Many people who have recovered from coronavirus have ongoing health problems or low level have never been in a hospital, what's known as long COVID. Many people who asked the first question also say that more people die from the flu every year than coronavirus. Every year it's true, we have a flu season which sadly results in people dying, something that we call excess winter deaths. That's why we encourage people who are at risk of the flu and our LHS and care staff to get vaccinated every winter. Last year, about 1,900 people died from flu and pneumonia in Wales. Sadly, there have already been more than 2,500 deaths involving coronavirus. People continue to die from coronavirus in Wales. The next question is relatively similar and that's the one about the number of people in hospital dying are tiny and the threat has been blown up out of proportion. Over the last month, we've seen a rapid increase in cases across Wales. The virus returned as people came home from holiday abroad and has been spread also as people are socialised without social distancing. Crucially, most often in people's homes. The majority have had a mild illness so far but an increase in cases is followed within two to three weeks by the start of higher hospital admissions, higher critical care admissions and more deaths. Last week, an average of 73 people a day were admitted to hospital with coronavirus. The number of people in hospital with coronavirus has almost doubled in the last fortnight. And sadly, the number of people who are dying is increasing week on week. It's easy to forget what things were like just six months ago when around 150 people were admitted to hospital each day with coronavirus. And our intensive care units, including the additional capacity that the NHS created, were very close to being overrun. If we can't control the spread of the virus locally, we will see this happening again. And the next question is why other countries like Sweden, which is commonly mentioned, which didn't introduce lockdowns have lower cases than us. I understand why attempting to look at other countries which didn't go through the difficult months of a nationwide lockdown. But it's just not as simple as comparing the experience in Wales or indeed the UK with other countries which didn't lockdown. If we look at Sweden, it's home to just 10 million people and it's almost twice the size of the UK. There have been more than 5,000 coronavirus deaths in Sweden compared to hundreds in its Scandinavian neighbours which introduced stricter measures. And the final two questions are about our regulations here in Wales. First, the question about why can't I see my family but I can sit next to strangers in the pub? It's a question I often hear and it's been put to me before. I do know from my own experience that it's very difficult for families who have been separated by the pandemic. But if you're going to a pub, you cannot sit with a stranger. You can only go to a pub with someone you live with or in the same exclusive bubble with. Gathering with friends, sitting on another table is not an exception to that rule. The law is very clear about what the owner and manager of a business must do to keep everyone who uses it safe. These cover everything for maintaining social distancing and collecting people's details and booking a table to the cleaning regime which needs to be followed. We do not and would not put similar laws in place to regulate people's private homes in this way. Because so many cases of coronavirus had been linked to people gathering in each other's homes, we put limits on who and how many people we can meet socially indoors at home, in the pub and elsewhere. In areas under local restrictions, we have had to suspend bubbles except for people who live alone or our single parents. The final question, why are children still going to school when the virus is on the rise? We've been very clear that we want children back in school learning alongside their classmates. Our schools, teachers and local authorities have worked really hard over the summer to make that happen. More than eight out of ten schools in Wales have not had any cases of coronavirus since the start of the autumn term. We'll continue to do all that we can to make sure it's safe to keep children in school because it is undoubtedly better for their long-term health and well-being to be in school with other pupils and their friends than to be missing out on another term of lessons. I know there will be many more questions about coronavirus and our regulations and the First Minister will be holding and ask the FM Facebook live session tomorrow afternoon. If we are to make a difference in Wales, we need everyone to follow the rules and to do all the small things which make such a big difference. Keep your distance, wash your hands often, work from home wherever possible, wear a mask in indoor public places and think carefully about where you go and who you meet. The more places you go, the more people you see, the more contacts you have, the greater the chance of catching or passing on coronavirus. I will now take journalists' questions and we will broadcast all the answers to these questions on our social media channels. The first question I have today is from Teleri Glyn Jones from BBC Wales. Good afternoon. We've had reports from schools across Wales of pupils and teachers having to self-isolate. 456 formers at Ulchfa Comprehensive in Swansea all of Year 11 at Allyn School in Moldon. 24 membys of staff at Plas Mawr School in Cardiff self-isolating. That obviously has a massive impact on pupils' education. Are you still comfortable that this policy is working? Recently, last week, the Welsh Government said there's no evidence of transmission in schools. Is that still the case, Minister? As I've just said in my opening, more than eight out of ten schools in Wales haven't had a case of coronavirus. The evidence is that the school community that is affected, the spread is taking place outside of the school, or it's taking place within a school between adults working in the school. The figures you give about the larger groups of people who are now self-isolating, that's a challenge about how successful each school is in getting people to maintain social distance or to have in-school bubbles. So if a whole-year group is mixing, then a positive case means that a whole-year group needs to isolate. With that larger number of staff, that is because that larger number of staff are being close contacts with each other in the school environment. That's why the efforts of school leaders and local authorities have gone through to try to create an environment where people don't mix across and within year groups where possible, and for staff to follow the rules on social distancing at work are really important. But yes, I am comfortable. It's been the right thing to do. We know that if we took an alternative approach, if people were at home, we know that many of our children and young people did not gain the same benefit from home learning, both their education, but also the impact on health and well-being. So it remains the commitment and priority of this government to keep our schools open for as long as possible, and we will take measures to end other areas of activity before we close our schools. Thank you. This morning, Frank Atherton, the chief medical officer, said there wasn't an introduction of the virus by tourists over the summer coming to Wales from England. What evidence therefore supports the First Minister's request that Boris Johnson should stop English tourists in Covid hotspots travelling to Wales? That was a point about the summer, and as we saw in the summer, we did have foreign travel with people who came back, and that definitely introduced coronavirus into areas of Wales. You'll remember the flights from Zanti in particular with high numbers of people testing positive. The summer was a time where coronavirus was effectively suppressed all through the summer in all four nations of the UK. The picture we now see in all four nations of the UK is a much higher prevalence with some areas with very high concentrations. The reason why the Welsh Government has asked the UK Government to introduce travel restrictions in areas of England of high prevalence is because we do know that travel brings with it additional risks. We know that if people from Liverpool come and mix and they're in the same pub, the same environment as other people, there's a risk of there being spreading events. It's why we talk about the rules in different places and about people following those rules. We're still considering, given that the Prime Minister announced in an interview as opposed to in response to the correspondence from the First Minister, that he is not minded to introduce any travel restrictions of people from high prevalence areas in England who can travel anywhere. We're actively considering what we should do, and I've discussed it this morning with the First Minister. We have quarantine regulations for international travel, so for some of the hotspot areas in the north of England, the north-east, the north-west and the west midlands, if they were other countries or territories, we would have quarantine regulations for them to return to the UK. So we're having to consider how we use our power to protect lower prevalence areas of Wales, but at the same time we don't want to take a whole nation approach because a low prevalence area like the south-west of England, there's no good reason to prevent someone from Devon at this point in time coming to visit a pre-booked holiday or trip to Pembrokeshire. So we are thinking about how we use something that is proportionate and deals with the reality of the threat that we face. Thank you to Lerrie. We've now got James Crite-Smith from ITV Wales. Health Minister, thank you. There's a lot of talk today about 16,000 cases and even not being reported. Now, last week, Public Health Wales also tweeted that there were around 2,000 cases not included in the Welsh data. Now, is the 16,000 cases, is Wales linked to that in any way? And if so, how many Welsh cases are involved and how many potential contacts might not have been traced as a result? Well, there's part of that question that I can't answer because I wouldn't know how many contacts haven't been traced without seeing the actual figures and the number of cases and the number of contacts that they would generate. So that would be an entirely hypothetical guessing game. We are in a position where we have had a delay in a number of Welsh cases from the Lighthouse Lab system. We understand that most of the 16,000 cases that have not been reported accurately and in a timely manner into, I think, not just accurately, but that haven't been reported into contact tracing systems are largely about cities in England, but I've asked my team here to check what the position is so I know accurately where that is. I've yet to have a full report back in that because my team haven't had all the answers. As we go through the day, I do want to make clear what the actual position is with whether we're still awaiting data and what that then means, but we are seeing all those cases that are being reported are going into our own contact tracing system. So all the information we have is being followed up, but naturally I'm concerned about the potential for people who have tested positive, not knowing about that result, and crucially in contact tracing terms, us not being able to track down their contacts in the way we have successfully managed for a number of months now. I'll come to my second question shortly. So just to clarify then, the 2,000 cases that were mentioned... This is your second question James, you don't get to ask three questions, so if you want this is your second question opportunity. I was just getting some clarity minister, that's all. I will move on then. Are we heading for life of rolling lockdowns, basically turning the taps off, turning the taps on in order to contain coronavirus, and if so, how long might that go off? Well, it's the reality that we may have areas go in and out of restrictions, as the chief medical officer said. As indeed my colleague Ken Skates said last Monday, he said last Monday, it's likely that over the next few weeks and months we'll see a rolling programme of restrictions to help us respond. We need to be dynamic in our approach to responding, so actually we've already contemplated that areas may go in or out. That depends on the pattern of behaviour we see, on the pattern of the cases that we see, and our understanding of them. So today we know that on the figures that we're seeing over the weekend, that we've seen a rise in cases in both Aberystwyth and Gwynedd, but we think there's a link to a small number of student cases that has a big impact on the number of cases per 100,000 in those areas. So there may be targeted lockdowns that then get released, there may be county-wide restrictions where that's the right thing to do that may get released, but if figures do go up, then it's entirely possible that we'll have to take further measures to suppress the virus again. And we're really in this position until we have either effective treatment, which we don't yet have on a population level, or if we do have a vaccine and we get good population coverage that allows us to make different choices. So this will be a difficult autumn and winter ahead for all of us, whether we run a business, whether we're in work, whether in school, or whether we're working in our frontline health and care services. But I'm really grateful to all those people across Wales who are diligently following the rules and doing their part to help keep their local community safe and to help keep Wales safe. Thank you James and I'll get Andy Davies from Channel 4. Thank you Minister. Can I just clarify what you said to Larry earlier? Is the Welsh Government actively considering imposing some sort of quarantine regulations on people coming from high incidence areas of England into Wales? From high incidence areas across the UK, yes we're actively considering it. And what might be the timetable for that just as a follow-up? Well the timetable for that will be, we'll have to consider the matter today. We'll need to take some advice from our scientific and medical advisor, public health advisors here, and we'll then need to consider whether this is the right course of action because the measures we've introduced in Wales are about isolating areas with a higher prevalence of coronavirus and protecting lower prevalence areas too. So that underpins the rationale we've taken. It's consistent with the approach that all four UK nations have taken to international travel and quarantine restrictions where we recognise that higher prevalence areas in other parts of the world represent a risk to coronavirus being re-imported or having an opportunity to spread further within the UK. Now it's disappointing that we haven't had a response to First Minister's letter but we've seen the interview with the Prime Minister where he indicated he's not prepared to do that at this point in time. So we then have to consider our own responsibility, our own powers, and how we'll draw that in a way that is proportionate to the risk that we face. Thank you Andy, we've now got Dan Bevan from LBC. Sorry, can I ask a second question? You've had your two questions, thanks Andy. Now you've got Dan Bevan from LBC. I was trying to clarify your answer on the first one. No, you've asked two questions. Dan Bevan from LBC. Thank you Minister, good afternoon. We've seen more reports this weekend from police being called to student populated areas like Cataise and Cardiff. With those areas, house parties, as we've already seen, can lead to mass spreading of the virus. Do you think the message is getting through to students? And also do you think they understand the implications that having house parties can have? Well, I don't think it's just an issue for the student population. If you consider the cases that we've seen and the animation we showed from the rise in actual confirmed cases at the end of August, that spread was taking place well before students started moving around the UK in larger numbers. It's an issue for all of us, whether young adults, whether students, whether older adults. It's really about understanding, it's our indoor contacts and having that close contact, not following social distancing that is leading to the significant spread of coronavirus over the last five weeks. And that is causing real harm. You've seen the numbers of hospital admissions going up. We are reporting more deaths now each week than we were through August, where we had many days where no deaths were reported at all. So the harm is there. And the message is for all of us, not just for students, for all of us. Younger adults may be less susceptible, but there is a risk to them. There is an obvious risk of spread to other people that they will come in contact with as well who may not be so fortunate. So that's the challenge for all of us. It's a significant change the way we are used to live our lives and times before coronavirus. But we all need to learn from the last six months and reduce our risk and to all play our part in helping to keep well safe. Thank you. And picking up on something that you spoke to James about, the rolling lockdowns that the chief medical officer has discussed. At the moment, only one place in the UK has come out of a local lockdown nowhere in Wales. So what is the criteria at the moment being looked at for somewhere coming out of local lockdown here in Wales? And is there any area that's close? Well, the criteria we're using is partly about the data and it's also then about the broadband talents we get from our public health officials and our test trace protect teams. So we're looking at, wanting to see a significant and sustained reduction below 50 cases per 100,000. We're looking to see the positivity rate, the number of positive test cases per 100. We want to see that significantly reduced below 5 at very least. And if we see those two data figures coming down, it will allow us to consider that broader community intelligence of where the spread is or isn't taking place. And to give you an example of where we didn't introduce restrictions again in the key pack incident in Murther where actually the rates at Murther was the highest reporting in local authority per 100,000 cases. We knew where that was. We also knew that it wasn't a matter of community transmission. There was a particular workforce community that was affected. We isolated that group of people successfully and avoided community spread at the time. If we see the figures go in the right direction and if that community intelligence is there from our test trace protect teams, that we understand the spread that is and isn't taking place, it will allow us to consider taking different measures. And Caerphilly over just a period of a few weeks has seen a sustained reduction. We want to see that reduced further to allow us to seriously consider the opportunity to ease some of the restrictions in the Caerphilly County Borough area. So, we're course optimistic about what might be possible, but it does so rely on all of us doing our part as well. Thank you, Dan. I've got Mark Smith from Wales Online. Thank you very much indeed, Health Minister. Firstly, I was wondering if you could explain why there appears to be a growing number of positive cases being reported by public health Wales from people resident outside Wales, and would you be able to tell us the importance of including them in the Welsh daily stats? We think this is largely driven by new student tests, and so where people are travelling into Wales and asking for their address off and giving their home address and where they've moved from. But we're looking to bottom that out to understand all of that within the figures that we have. So, it's something that I'm aware of, and I'm keen to understand in more detail what that means. But that's my understanding at present that it's largely about a population that has moved into Wales and still considers their home address to be somewhere else rather than where they're living now. But what we do understand is that the figures, the data that we get is still going into our system for contact tracing to take place. So, it isn't hampering our contact tracing service, but I do accept this point to clarify in the way we present our data to the public. Thank you very much indeed, and my second question was briefly mentioned by you earlier. I've spoken to patients and NHS staff who are suffering with the effects of COVID-19 many months on from their initial symptoms. What is the Welsh NHS and the Welsh Government doing to support people with so-called long COVID, and what help and rehabilitation will be given to those who were clinically diagnosed with the condition, but were never given a positive test to ensure that they don't slip through the net for treatment? Well, we're still learning about long COVID. We're still learning about the long-term impacts for people who have had COVID, but haven't necessarily been in hospital or still suffering post-effects. And it's part of the challenge that I was mentioning earlier in the opening of today's conference, and that is that this is a new disease that we're still learning about now. The Welsh Government is involved with other governments in the UK in research about long COVID. That's both about understanding how and why it affects some people in different ways, but crucially to understand how we help support people in their rehabilitation and recuperation and what we'll need to do. And as ever, if people are concerned their conditions haven't been picked up and haven't been recognised by treating healthcare professionals, they should have that conversation with them. Because I recognise that for people with long COVID, it is a particularly debilitating condition, and there's real concern about the understanding that doesn't exist about how to help them through. Now, the puzzle that new coronavirus has brought into to us is not one we have all the answers to. That's why the research we're undertaking is hugely important. And again, I'm tremendously grateful to people in Wales who have volunteered to take part in that research. Whether as those are a professional research end or indeed members of the public who are keen for their condition to be better understood. So I hope we'll have more in the coming weeks and months. And certainly when we do have more information about long COVID, I'll be happy to make that clear in public and crucially to make clear what that means in terms of our treatment, rehabilitation and recovery options. Thank you, Mark. Now I've got Rupert Evelin from ITN. Thank you very much. Can I just ask about local lockdowns in Caerphilly? It's almost a month now since Caerphilly was put into a local lockdown. The stats appear and have appeared for some time to be going the right direction then. Why is Caerphilly not out of local lockdown measures yet? And when is that likely to happen? Well, Caerphilly is about 50 per 100,000, so still fairly high. I know it was an authority move in the other direction. We'd be considering intervention measures. Now we still also see a positivity figure in Caerphilly that is above four. So it's still somewhere on our concern list. And what we want to do is to see a further reduction in new coronavirus cases so that we can have much greater level of confidence that the virus is being effectively suppressed together with, as I said in answer to earlier questions, that community intelligence about where new cases are. Because if we understand that those cases are linked in the clusters we already understand, that gives us a level of confidence. We can have a smart hyper lockdown around families and cluster groups. The challenge that led to a number of areas going into local restrictions is that we've seen new cases popping up that are not linked to other clusters. And those unexplained events are a real cause for concern. They give us evidence about community transmission taking place. So it isn't a question of there is a number of days when we can be confident that, say, in three days' time we'll be able to take restrictions off in order to understand the figures. And we're in a period of reviewing those figures, not just daily when the figures come in, but a formal review every week. And we'll consider that together with the Health Service, Public Health Wales, and, of course, the local authority and other partners. And my second question. It's my understanding that you and your officials believe that your approach to local lockdowns is fundamentally working, which is a good thing. Why do you think it's better than the way they're doing it in England? Can you explain that to us? Well, I'm not in the decision-making tree in England, but you've heard the well-advertised concerns about local authorities hearing a decision that's been made rather than being part of it. We deliberately take an approach that draws all of those partners together. So in each of the local restriction decisions that we've had, before ministers make a decision, we have a conversation with the police, with the local authority, with the leader and the chief executive involved. We have a conversation with Public Health Wales and the local health board. And we also have a conversation with our chief medical officer's department and our chief scientific advisers on health. That is a joint conversation all at the same time to discuss what happens. And we normally get a series of suggested recommendations from the local incident management team within that area. That means we're all pooling and sharing the intelligence and resource we have. It's also the case that local authorities have also got a clear line of sight in every part of Wales about the intelligence we're getting from our Test Trace Protect service because it's housed in local authorities, it's local authority staff doing the contact tracing and it's a real partnership with our National Health Service here in Wales. So I think the partners come to the table for a start. They come to the table with lots of the same information. We share what's there and we then have to go away and make choices upon that. Now, that isn't the picture that I hear described in England but of course I'm not responsible for the way they do that and other people in England may claim the system runs differently but I think the way that I describe the Welsh system it really does mean that people have confidence in the data, in the information and then ultimately the choices that are being made. Thank you, Rupert. I've got Lydia Morris in the Daily Post. Good afternoon, Minister. How likely is it that Gwynedd and Anglesey will follow suit on local lockdown? Well, it isn't clear that we need to take any action in those areas. Anglesey isn't at the point where we think that we would need to take action based upon the figures. With Gwynedd, we're seeing a rise in cases but we think that is linked to a handful of positive cases within the student population. Now, we're looking to separate that out in exactly the same way we have done with contained communities in the past so think about two sisters in Anglesey, think about Rowan Foods and Wrexham Keypack in Merthyr. If we understand that's what's happening and that's my understanding at present then what we need to do is to isolate part of the community where the infection is actually there and where there are new cases and not take collective action across the whole community. That may mean that some people in part of the student community may need to isolate but not the whole student community so we're treating our students exactly as we would do with any other part of the population if that's the case then we may not need to take action in Gwynedd or Anglesey any other parts of Wales that don't currently have restrictions so it'll be that open process in discussing it with local partners and understanding the intelligence and the information as I've described earlier. Has there been a notable drop in cases since the four North Wales counties were locked down last week? No, there's been a rise in cases since we introduced those local restriction measures and you'll recall that we introduced those restrictions when we could see that a rise was taking place and I appreciate that some local actors doubted the need for the action that we'd taken and that was disappointing. Some of those elected representatives in particular would have expected to be a deal more supportive and responsible. We have those seen that despite us isolating those four authorities with the travel restrictions as we thought would be the case there would be an increase in positive cases that came from that because we had early evidence of community transmission. As we've seen in Caerphilly for example it's likely that we'll see an increase for at least a period of a fortnight or so potentially two or three weeks before we start to see if those cases are going to start falling back down again but as you'll have seen all four local authorities where we've introduced restrictions in North Wales have seen an increase over the past week I expect we may see more cases over the next week we'll then need to look at the end of the first two weeks whether there is evidence that cases are falling and that's crucially about the community response and as I say I really am tremendously grateful to all those people in every part of Wales who are doing their very best to follow the rules and to find a way to live their lives within the rules rather than looking for a reason to get around the rules or to find a loophole. Thank you Lydia. We've now got Rob Taylor from rexham.com Good afternoon Minister you mentioned isolating North Wales counties there since lockdown was enacted with other range of reports of cross border issues be it rexham to Flintshire or even England and to use one specific example an off-roading event attracting a large number of people from the north west of England to place in rexham yesterday due to local concerns the police did attend and the event appeared to continue as planned presumably because no rules were broken is that cross border travel and such events the regulations operating as you planned or do the regulations need improvement? People can travel within county but they need to follow the rules when they do so on social distancing so it may be possible for events particularly outdoor events to go ahead within that county but there should not be cross border travel and I would certainly not consider it to be a reasonable excuse to travel from outside a county with local restrictions for an off-road event people need to think seriously about how they follow the rules not try to distinguish out what is plainly not considered to be within them and we're at a point of real maximum seriousness and risk for the country I said previously I think we're in an equivalent position to February I think we're much more now like the end of February in the start of March we're seeing a rising tide of cases we're taking earlier action because our line of sight our local intelligence about what is happening our national intelligence picture is much clearer than it was in February and March but it is not inevitable that the tide of coronavirus cases recedes the cases will rise if people don't look again at how we avoid contacts minimise the groups of people we're indoors with especially not to follow those rules on regular washing of hands maintaining social distancing wearing a mask where you should do in accordance with the laws of Wales Thank you Last week RCT council tweeted a heat map style of graphic for COVID spread per lower super output areas and you appear to have used similar data for the animation earlier in this presentation as it appears that information is around and it's localised will similar information be made public for around Wales or is it just localised lockdown areas and how regular will it be updated? The animation that we've provided today will be on our social media channels for people to be able to view it and to share it I'm absolutely looking at how we provide a regular and reliable update in data we need to make sure that in areas of lower preface we're not identifying individuals in the way we present our data and the maps but I'm confident we can find a way through to present that and then to be able to announce how it's going to be presented and then announce the regular pattern for doing so as well Thanks Rob and I've got Tom Magnum Thank you Minister, good afternoon My first question is about emergency cards for unpaid carers in Wales Viewers tell us that while these inform others in an emergency they don't critically help get urgent priority supplies in a lockdown Ceredigion Council has today announced the CARERS ID card that addresses this issue You've spoken supportively about unpaid carers in previous briefings so do you agree that your Welsh councils should have announced a similar measure much earlier and why? Well I'm not aware of the exact scheme you've referred to in Ceredigion I'd be very interested in learning more about it and of course not just in working with Ceredigion Council we're looking at other authorities I've looked at other ways to try to support unpaid carers so I'd be very interested in learning more about the data of the scheme and after we finish Tom I'll go and speak to my officials to understand what Ceredigion have announced because actually myself and the Deputy Minister for Health and Social Services have a meeting later this week with all of the cabinet members for social care across Wales across all 22 authorities so it's one of our regular points to understand the pressures they're dealing with locally and how we can try and share good practice to see what works because in a small country like Wales we should be able to see to try and what works to see if it really does work and then to transfer it across the country so it's going to be useful and helpful example Tom Thank you We'll certainly pass on the details after this briefing but it's not just about ID cards there seems to be a lack of understanding of grassroots one viewer in south Wales echoes the experience of others she tells me that under a screen saying if you're a carer and make yourself known we can help she was told by the senior triads nurse in A&E you won't get seen any quicker because you're a carer alone I can assure you with that so how will you address this issue of grassroots services apparently having little or no respect for unpaid carers despite carer digging and council and your government's best efforts I think Tom we need to understand the particular circumstance of each service and what it's trying to do in a healthcare setting if you need to prioritise people it should be on the basis of their clinical need but it may well be that someone who isn't a carer has a much greater clinical need and I think if we try to intervene and interrupt that we'll end up causing a much wider series of injustices but in the broader way in which our services support carers that is what we're actively trying to do so the practical example you've given her about caradigion we're looking to support unpaid carers in a different way as well and I hope the deputy minister will be able to make announcements about that in the coming few weeks on active part of learning and as I said before we are tremendously grateful for all those people who are unpaid carers the work they've done and the extra responsibility they've had for the first six months of this pandemic and frankly the reality that over the course of the autumn and winter we're probably going to ask more of our unpaid carers people like you and me I guess Tom who look after people in our family and friends circle and do it because of how we feel about them but it's how we support those people to do that and the real benefit that brings not just to the carer but to the loved one or the friend that they're actually caring for as well thanks Tom we've now finally got Dan Barnes from the South Wales Argus very much going back to local lockdown can I ask whether the local lockdowns in Blaenau Gwent and to Vine are going to be reviewed and if so what's the time frame for that we review all of the local restrictions after the first two weeks and have a regular weekly review on the positive side it's possible that we're starting to see a levelling off in the growth numbers we saw quite high growth you'll be aware and Blaenau Gwent in particular but I've got some more optimism about the figures we've published over the weekend that we're seeing a levelling off in growth still growth but if we've managed to slightly flatten the curve that's good news and it shows the restrictions are working and they're working because people are changing the way that behave and thinking again about the number of contacts and how they see people and again trying to restrict that indoor content which I know comes at a cost but actually in terms of avoiding a much greater cost of real harm being done in our communities it's the right thing to do so again I'm very grateful to people on Blaenau Gwent, to Vine and across Wales we'll review those measures after two weeks and we'll then have a regular weekly review if the measures do continue very much that kind of leads into my second question just on a wider level across Gwent do you think that all the areas in local lockdown are on the right track and what would be a message to the people living there? Well actually we're seeing in Gwent some positive moves in all four of the counties that currently have measures they're still above the line where we'd want to take action to ease restrictions but it really is to thank people again who have followed and supported the restrictions that we put in place to avoid further harm from coronavirus your efforts and the way you're choosing to live your lives are making a difference we're asking for some more from people to keep on following the restrictions because that is the path that would then allow us to ease some of those restrictions for the future so we'll continue to review this with leaders of local authorities with Public Health Wales and with the Anarang Bevan University Health Board we'll look for a way where we can safely ease those restrictions to give people more freedoms to live their life and to have human contact but it all comes with the challenge about then understanding the risks that we have and as I said doing the small things wearing a face covering where we need to making sure that we wash our hands regularly and making sure that we avoid contact with other people where we don't need it in particular that point about social distancing and in this I'm tremendously grateful to all that are done by our front line health and social care professionals but equally as I say very grateful to the public who are following the rules thank you very much Dan that concludes today's press briefing I look forward to seeing you all in the future as we're answering questions again


The Slutsky Identity What is the Slutsky Identity? Up till now, we have Studied that the consumer demand changes in response to three facts Number one its own price So what will be the effect if the price of that good of Which we are going to study if it's going to change What will be the effect of the other goods on The quantity demanded what will be the effect of the income change on the commodity demand when we have to study the effect of the change either of price of x either of price of y or Either the income we have to study only that factor keeping all other factors constant in the position of Thus change in the quantity demanded we can come up with the various solutions So under that various solutions we have studied up till now that the change in the quantity demanded it can be on one path the Direct demand curve and we can solve this directly through the direct demand curve and at the same time this Optimal decision we can tackle this through the indirect utility fund so the total change in the quantity demanded of Change in x is the change due to change in price holding all other factors Constant then we can say that this change is equal to this when there is only Change that we have Included is the change in price when this change in the price It is decomposed or broken into two separate components. We have up till now Achieved that one part will be the substitution effect and the other part that will be the income effect when we add up the income and the Substitution effect we come up with the total effect mean the total change in the quantity Demanded due to change in the price So when we have to answer this type of the question We know that the signs of the income effect and the signs of the substitution effect they matter a lot and The substitution effect it will always be negative means It will be opposite to the change in the price if price will increase Substitution effect will be negative if price will decrease Substitution effect will be positive But the income effect it can be positive if we are dealing with the normal good But the income effect it can be negative if we are dealing with the inferior good. So the total effect in the form of the Normal good it can be positive means both Substitution and the income effect they reinforce each other But in the form of Inferior they both can offset each other So the slutski identity for a normal good if we come up We can say this is a method or the situation when we Calculate the change in the commodity demanded to the price change and Including the substitution and the income effects simultaneously So when we see that if there will be the decrease in the price and We are dealing with the normal good. We see that the price it decreases and Response to this demand it increases and This is the substitution effect and the real income because it Increases due to change in price and due to increase in real income now the demand will increase It means this is the income effect and this to reinforce the total effect will be here and when There will be the increase in the price. It will again will be the substitution effect and Income effect and the total reinforced effect will decrease in the commodity Demanded and here it was increased in the commodity Demanded now when we deal with the commodity the inferior There is only one change for the inferior when price it decreases the consumer Not only Responds in a manner that it decrease its demand. That is a very exceptional case so in this this Substitution effect this and now real income due to the Decrease in the price there will be some other form of the income left with the consumer to spend So due to that really income Here it increases so we are having both Substitution effect and we are having both Income effect, but one form there will be that Substitution effect and the income effect that they will be into the opposite direction to the each other so decrease in the price here It will not have the negative effect of the substitution rather Decrease in price will have the decrease in the demand mean Substitution effect will be positive. So when this will be the shape Here again, not only their sign rather the magnitude of change It will affect because if income effect will be more Then the substitution effect then it will have Total effect means change as compared to their sign and if the magnitude of the income effect will be less Then the sign of substitution effect will prevail and the result and total effect will be different And when we are dealing with the given good then again, we see that now Due to the change in the price decline demand has decreased and Due to the change in the price increase demand has increased so Substitution and the income effect they both move into the opposite direction but Particularly in this case We see that the income effect if it is quite large enough It can offset or it can outweigh the effect of the Substitution effect and the total effect it can be positive. Thus this necessities or it tells that a given good Whenever we are going to study for this it is sure that it should be inferior good But for an inferior good, it is not necessary that it will be a given good so for a given good the Necessary point that it should be an inferior means that it should have our reflection Related to the price response and at the same time we can say that It should not have any type of the substitute or the near substitute Available now if we explain this in the form of our diagram We can see that mostly the directions of the chain They are similar but with certain exceptions that it retains to the level of the magnitude If the diagram on this left-side panel we see that the consumer is having this original budget line and With this original budget line now we see That if price of commodity x1 it decreases So when it decreases we can have income divided by a price of x it means now Due to the smaller level of denominator we will have more of the x1 so Consumer it shifts from this budget line to this budget line and When it moves to this budget line it has different slope when equal to this budget line slope we draw a parallel Hypothetical budget line on the previous indifference curve and on the same indifference curve Having the Hypothetical or the expected budget line of the slope now the consumer shift from this point a To this point B in the first stage That is the pivot movement that will be reflected through substitution effect But after this substitution effect now the consumer Has shifted to this point B in a manner That now consumer has shown not only increase in the income rather Consumer has shown a total decrease in the backward form here and Likewise, we see that The similar nature of the response is shown in this panel where it was the original budget line and Consumer shift to this budget line due to decline in the price of x1 and With the hypothetical budget line consumer shifted from a to B and from a to B Shift was due to substitution effect and from B to C the shift was due to Income effect, but we see that in this panel from B to C there is Similar nature of the change But the magnitude of this change is smaller in the form of only inferior goods But in the form of the given case this magnitude is larger as Compared to the inferior it means for the given goods It is necessary that the given it should be inferior also But for the inferior commodity it is not compulsory for the commodity to be given