It is now time for oral questions, and I recognize the leader of the Majesty's Royal Opportunity. Thanks so much, Speaker. Speaker, my first question this morning is to the Premier. You know, throughout the spring, the Premier insisted that he was sparing no expense in the fight against COVID-19. Yet, Saturday's Toronto Star reports that people running Ontario's labs and testing were pleading with the Ford government to bump up the investment throughout the spring and summer. They were pleading with the government to expand the testing capacity, only to be told that the government just didn't want to spend the money. In the midst of an unprecedented health crisis, why has the Premier been trying to cut corners and save a buck? Deputy Premier and Minister of Health. Well, I would say quite the opposite to the leader of the official opposition. We have been strengthening our lab capacity since the beginning. Just as a matter of history, I remind the member that when we started out, we were able to do 5,000 tests through Public Health Ontario. Since then, we've expanded that capacity to be able to do over 40,000 tests per day in Ontario. And we have a connected lab network that includes Public Health Ontario, hospitals, universities, community labs, and we're still expending more. In fact, we have just indicated that as part of our Public Health plan of keeping Ontarians safe, we're spending over a billion dollars to increase our testing, our lab capacity and our contact tracing. That is something that we're going to continue to do because we know more testing needs to be done to respond to this curve and be able to deal with the second wave of COVID. A supplementary question. Well, Speaker, nobody in Ontario that has been dealing with the long lineups and the consistently changing statements from the Premier believes that the government was ready. Labs had been informed that the government was aiming to conduct 100,000 tests by October. Now that deadline has been quietly moved back to January. Canada's largest city is scaling back on contact tracing. We're in an ever-worsening second wave now and in chaos because the Premier refused to spend the money that was necessary to protect the people of Ontario. Will we ever see a real plan and the investment necessary to back it up? Minister of Health. I would say through you, Speaker, to the leader of the official opposition, that is absolutely not the case. We have been increasing our testing capacity since March and we're going to continue to increase it. We did not ever indicate that we would be doing 100,000 tests by the end of October. What we did indicate was that we were going to be increasing from 40,000 to 50,000. We're well on our way. We routinely do more than 40,000 tests per day in Ontario. We are changing our capacity with respect to testing because we listened to what you were saying all last week, indicating that people are waiting outside in long lineups. It is getting colder. We responded quickly. So now we're changing to appointment-based testing so that people will have a better idea when they need to go. They don't have to stand outside in lineups for long periods of time and they will be screened before they come in to make sure that they are eligible to receive a test. So we are responding to changing conditions, to changing weather. Thank you. Thank you. And the final supplementary. Speaker, for the last week or so, the Premier has demeaned and derided doctors, hospitals, and medical experts who have spoken on his failure to ask. And now this Minister of Health just threw the labs under the bus. Only this Premier will call an ER doctor an armchair quarterback in the middle of a pandemic. These frontline health experts are telling us that it is, and I quote, 100% true, that the Premier's attempt to save a buck back in June, the way this government loves to do, it's all about saving, saving, saving. But that decision... Stop the clock. Government's side will come to order. Start the clock. Please place your question. 100% true that the Premier's attempt to save a buck back in June has left us scrambling today. That's what the experts are saying. So at what point will this government, will this Premier admit that his government's decision to focus on cost cutting instead of fighting COVID-19 has led to exploding numbers and us scrambling to try to react? Minister of Health to reply. Thank you, Speaker. In fact, let's speak actual facts. In fact, this government has spent an incredible amount of money into the issues that we absolutely need to deal with in order to deal with COVID-19. First of all, we've spent over $935 million to our hospital sector this year, a 5.5% increase which is greater than any hospital funding in a decade. Secondly, we have allowed for $458 million to go into home and community care so that people who can be cared for at home who don't need to be in hospitals can be cared for where they want to be. We've also put in $283 million to support dealing with the backlogs of surgeries and procedures that we've needed to deal with and a billion dollars to test, trace and isolate cases. Moreover, we've done all of that, expended those sums of money with the incredible assistance of our public health experts the best in the world. We have a team of people that we've spoken to. We held over 45 sessions with over 300 health experts in bringing forward our plan, keeping Ontarians safe. Thank you. The next question, the leader of the opposition. Well, it seems interesting that the minister says one thing and the financial accountability says the other about how much this government was prepared to invest over the summer to get us ready for a second wave. This weekend, Speaker, my questions to the Premier. This weekend, Speaker, Toronto Public Health actually had to stop its contact tracing. The biggest city in our country had to stop its contact tracing, saying that they can no longer notify close contacts of people who have tested positive with COVID-19 because they simply don't have the resources to keep up otherwise. Toronto's medical officer of health has been pleading, pleading with the Ford government for action that will help her do her job. Where is the action? Why hasn't it come? Minister of Health? Well, in actual fact, there has been significant action taken, which is why we released our plan, which is why with the pillars that we have in it, we want to reduce the number of cases that we're seeing, which is why we've also targeted Peel, Toronto and Ottawa with special measures to make sure that you don't have more than 100 people in restaurants. I believe Toronto has reduced that to 75. No more than six people at a table. These are places where we're seeing the outbreaks and we are responding to the remarks and concerns that have been expressed by Dr. de Villa and the breakouts in these areas. We have also allotted more than 200 contact tracers specifically to the City of Toronto to allow them to catch up and to be able to carry on the contact management. We are very aware of their concerns and we're responding to the concerns that have been expressed by both Mayor Tory as well as Dr. de Villa. The supplementary question. Well, Speaker, really what the Minister is saying is once again they're scrambling to catch up instead of getting out ahead of what's happening with COVID-19. People are desperate for clear direction and leadership from this government. And the Premier has failed to deliver, Speaker, even simple questions about what we should be doing in terms of celebrating Thanksgiving with our loved ones didn't even get a clear answer. The Medical Officer of Health in Ottawa describes the situation as a crisis. In Niagara they're scrambling to try to redeploy public health staff. The Medical Officer in Toronto, of course, we know is pleading with the province for some help. Will the Premier admit that his plan to save money isn't working for the second wave? It didn't work for the first wave, which is why we had over 1,800 seniors die in long-term care from COVID. So will he just admit that his saving money isn't the right thing to do and start listening to public health and medical experts who are pleading for the government's help? Minister of Health. Thank you. What we have prepared for the second wave, we've been preparing for it throughout this summer months with the consultations that we've held with public health experts and experts in other areas of health. That's why we came forward with our plan Keeping Ontarians Safe, which is a comprehensive $2.8 billion program. We are spending the money and we are spending the money in the right areas. We're going to maintain the public health measures that we need to maintain. We're going to bring forward the most comprehensive flu campaign in Ontario's history. We're going to make sure that we can identify and manage outbreaks in COVID-19. We're going to keep up with the backlogs of surgeries and procedures that were postponed in the first place. We're going to recruit and deal with health and human resources to make sure that we have the right people in place to manage COVID-19. And we are ready for surges to be able to increase hospital capacity and community capacity in areas where we do see the outbreaks. And I'll supplement you. Well, Speaker, consultations are useless if you don't take the advice of the experts. They're useless. Here's what families actually see. A second wave of COVID-19 is spreading in our schools, in our long-term care homes, across our communities. People on the front lines of our labs, in hospitals and in public health say that they've been ignored for months and they are pleading for action. And the Premier is attacking experts and insisting that his plan is working where everybody sees that it is not. The government needs to do better, Speaker. When will the Premier finally admit that the government failed to make the investments needed over the summer and start listening to frontline health care experts pleading with the government for some help? Thank you. In fact, we have been making the investments. I believe I answered in your previous question to indicate that we're spending $2.8 billion. That's a lot of money to increase our resources to make sure that we can test, trace and isolate new cases of COVID-19. We are continuing to do that. Our plan is working compared to most other countries and most other jurisdictions. We are doing extremely well per cases per 100,000. This is an outbreak that's not just happening in Ontario. This is happening throughout the world. Opposition, come forward. We have been prepared since the early summer and we are moving forward making those investments in our testing capacity, in our lab capacity, in increasing the number of contact managers from 2,750 by 1,000 to 3,750. Response? We are making the necessary improvements in our home and community care to make sure we have more money to respond to more people that are going to be home and to expand our hospital capacity when we need to. Thank you. Thank you. Good morning, Mr. Speaker. My question is to the Premier. There were outbreaks in 114 long-term care homes before the government stopped the practice of personal support workers working in multiple homes. We knew then, as we know now, that the risk of transmission increases exponentially when workers move between multiple groups of people. That's why it was so disheartening, Mr. Speaker, to read in the Toronto Star that classes in up to six schools in the Toronto Catholic District School Board were exposed to COVID-19 through a staff member who is still required to rotate through different classes and different schools. Speaker, why is the government putting students and staff at risk ignoring the lessons of the first wave of COVID-19? Thank you, Speaker, and member opposite for the question.  For specialized teachers like French, like cited in the members in the article noted by the member or music teachers, we've asked them to undergo strict screening before they enter schools. The Chief Medical Officer of Health, who's improved our plan and endorsed our plan, is constantly reviewing the protocols to make sure that they are adhering to the guidelines we are open to any adaptations he may make to further improve them. But that's why, Speaker, you will note in the summer, we asked when it comes to these specialized teachers, our union partners, to work with us when it comes to managing prep time, to bundle it, to start at the beginning of the day, the end of the day, to minimize these impacts on the grant. This could have been avoided, Speaker, but I do believe there is a way forward working with our unions. This is a proof positive that these issues are being asked to work with us to do just that. Supplementary question. Thank you, Mr. Speaker. Well, here we go again. Blame the boards. Blame the teachers. Blame the unions. Refuse to take any responsibility out of the room. Speaker, you do not need to be an epidemiologist to know that moving staff from class to class and school to school is going to increase risk. There are six schools impacted, Speaker, and those families can't even get a test today because testing centers are closed. It is not just itinerant teachers I want to point out, Mr. Speaker. Education workers of all kinds, EAs, custodians are still being asked to work in multiple settings and it is on this government and their failure that they are still doing that. The government has ignored repeated warnings. Speaker, will the Premier direct his minister to act now to limit the risk to students and staff and bring forward the urgently needed funding to keep these workers and students safe? Minister of Education. Mr. Speaker, the government has provided a $1.3 billion investment to school boards. The issue before a speaker has absolutely nothing to do with funding that has access to educators and the member knows that we have a limit of French teachers in the province of Ontario, so when the Premier said he is open to ideas from the opposition, I look forward to that member bringing forth ideas how we are going to create thousands of French teachers just overnight as the member opposite pretends that that could happen. What we know Speaker, what we know Speaker is that itinerant teachers have been asked to enhance screening. We have asked school boards to limit the contact, the indirect direct contact for students to 100. School boards have very clear protocols on how to administer and we will work with our school boards as well as the public health agencies to make sure we limit the spread in our schools. We are seeing incredible I want to thank our school boards, our public health educators for doing everything they can to reduce the risk within our schools. The next question, the member from Mississauga streets. Good morning Speaker, my question is to the Deputy Premier. This past week we received a real wake-up call. We saw from the modelling update that if we don't act now to halt these trends, we could see a thousand new cases a day by mid-October. We could have 200 to 300 people a day arriving in our hospital ICUs. In fact, Speaker, there were 732 new cases just last Friday. We're in a second wave of COVID-19. Throughout this pandemic we have made some tough but necessary decisions always based on the best medical advice and scientific evidence available. Speaker, can the Deputy Premier please share with the legislature about the new public health measures that our government in consultation with the chief medical officer and the command table announced to stop this virus from spreading any further. Thank you. The Deputy Premier. Thank you Speaker and thank you to the member from Mississauga streets for the question for your leadership. Effective this past Saturday we're making the masking policies already in place in many communities across the entire province. That means we're bringing in additional masks when shopping, when taking public transit and at work if you can't keep two meters between you and your colleagues. As much as possible we also need to limit close contact with anyone outside our households. We also have to extend the pause on the reopening of any other businesses or facilities for another 28 days. And we are bringing in additional targeted measures in Ottawa, Peel and Toronto. The number of thousands of night clubs in Ottawa and Peel must limit their capacity to 100 customers or less. Toronto Public Health has already decided to limit capacity to 75 with no more than 6 people per table and each customer's contact information must be collected for contact tracing. A supplementary question. Thank you Speaker and thank you to the Deputy Premier. I also put my supplementary question to the Deputy Premier. I know that these strong measures are necessary and builds on additional new requirements for gyms, fitness centres, banquet halls and event spaces. Deputy Premier, I know that Ontario has been a major leader when it comes to testing. We have over 4 million individuals tested. We are a leader in Canada and North America. But with the weather getting colder this will lead to more of my constituents returning indoors and the community spread and a surge in testing. Additionally, we can't continue to have people waiting in the cold and outdoors to get tested. Deputy Premier, can you please share with my constituents about the new enhanced process our government has implemented on testing to help streamline the process and provide better service and certainty to everyone in the province. Thank you. And the Deputy Premier to reply. Yesterday, assessment centres will no longer be accepting walk-ins as well as assessment centres are now moving to an appointment-based testing only system. The vast majority of the testing at assessment centres in Ontario is already offering testing by appointments and they will continue to operate as usual while we make this important transition. Up to 80 pharmacy locations in Ontario will also continue to provide access by appointment. With more pharmacies to offer tests in the coming weeks while they transition, the assessment centres will also take the opportunity to conduct a deep cleaning. This will also give labs a critical opportunity to clear any testing backlogs. Colleagues, these changes are absolutely necessary. We're working with our partners in hospitals and assessment centres to make sure this transition to as possible. Thank you, Speaker. Thank you. The next question, the Member for Nicolette. Thank you very much, Mr. Speaker. My question is to the Premier. Thousands of Ontarians are anxiously waiting to get their COVID test result because we have this huge backlog, the result of government in action. The public testing centres have shut down their walk-in services. We have a number of Ontarians to book on website Evan Bright, often days in advance to get an appointment. Ottawa Public Health calls it a crisis, but today the Globe and Mail reports that if people have the means to pay hundreds of dollars, they can jump that queue. The Premier calls it healthcare a free market and is letting private companies from Ontario help in charge hundreds of dollars for a COVID test. Why is the Premier letting private corporation profit from COVID testing? Thank you, Speaker. I would like to assure the member opposite and the people of Ontario that this is not happening. This is not allowed. We are not allowing companies to receive private swabs that we are using in our public health system. We are not allowing private companies to use to conduct tests at hundreds and hundreds of dollars. This is not happening. Our tests are public tests available to people without charge. That is what our public health system is modeled on. That's the way we are conducting tests. And while companies can buy their own swabs elsewhere in Ontario, we are not allowing private companies to use to conduct tests. Supplementary question. Thank you, Speaker. One of the company offering these for-profit tests is switch health. The Premier knows them well. He allowed their private mobile tests on farms earlier this year. Now, because he calls healthcare a free market, healthcare is a free market for all the people around the world. We are not allowing private companies to use public health. We are not allowing public health to use the medical system. We are not allowing private companies to use the medical system. We are not allowing private companies to use system has what it needs to test Ontarian in a timely manner. Mr. Health. Thank you, Speaker. Well, our government is not doing what the member has suggested. In fact, Ontario Health has been directed not to provide swabs to any providers known to charge patients and we're currently determining further action to ensure that this activity is discontinued. This is something that private employers can do if they wish, as I indicated earlier. If they wish they can conduct testing for their employees free of charge, but they need to buy their own swabs and they need to have their own equipment. What we are focusing on is testing in the public sphere. You're right, that's what our public system is based on and we are following that. Anyone who comes to any of our testing at assessment centres or pharmacies does not have to pay for a test. They will be given a test. That is our responsibility and our duty to the people of Ontario and that is what we are doing. No private testing is being used using Ontario swabs that belong to the public system. Thank you, Speaker. My question is for the Minister of Health. The messaging on testing has been all over the place. The public was told that anybody who needed a test could get a test. But testing capacity was not there to meet demand, even though we knew demand would go up when schools and businesses reopened. So I understand why the government has reacted to move to appointment-based testing to deal with the backlog that they created. But I'm worried, Speaker, that the appointment-only testing will make it even less accessible for vulnerable people. There are many people, for instance, people who are homeless or who may have disabilities who may not be able to access the technology to set up an appointment to get a test. So Speaker, can the Minister tell us what has been done to ensure that vulnerable Ontarians have access to testing? Minister of Health. Thank you. Well, I thank the Member very much for the question. This is a very important point because we want to make sure that anyone in Ontario who needs a test will get a test. And the move to the appointments is, in fact, a response to long lineups, changes in weather. In fact, all of the Northern Assessment Centers are already doing appointment-based testing. That hasn't shown to be a problem. However, we do recognize there may be some people who might be in that situation who might need to have home and community care come and visit their homes to be able to do their testing for them or primary care through the physician's office. There are other ways that we can do this testing without necessarily having people being tested in the assessment centers themselves. We have allowed for that, and that is happening as we speak. The supplementary question. Thank you, Minister, for the answer. And I hope the government communicates that clearly with the people of Ontario. If we have learned anything about this pandemic, it is that speed is of the essence. The federal government has entered into conditional contracts for purchasing millions of rapid testing to deal with these long lineups, even though some of these companies haven't been approved by Health Canada. The federal government has done this, Speaker, because they know we have to increase capacity for rapid testing. There's a company in my writing that is waiting on imminent approval for rapid testing, and they have said that if they can have a conditional contract, they can roll that out quickly. So, Speaker, knowing that time is of the essence to deal with the testing backlog, will the Minister commit to entering into conditional contracts with companies who are on the brink of receiving Health Canada approval for rapid testing so there is no excuse for further testing delays? And the Minister of Health. Thank you. Yes, we are looking at the new point of care testing that's coming forward. We've had a brief conversation about the company that's in your writing. We're always looking to follow new innovations because, as the Premier has indicated, that is a game changer. If we can do point of care testing and get results in 10 or 15 minutes, it's a big difference from what we're dealing with now, where we try to receive test results within 24 hours, but it doesn't always happen for a variety of circumstances. So we are moving forward. We are in contact with Health Canada and with the Federal Ministry of the Government of Health, Department of Health. They are obtaining the orders, and then we will receive the orders from them. So that is happening, and we are looking for other creative and innovative solutions, but we are working directly with the federal government on this and with Health Canada, hoping to be encouraging them to have these tests approved as quickly as possible, while making the necessary testing requirements and making sure they do their due diligence. Thank you. Next question, the Member for Willowdale. Thank you very much, Mr. Speaker. This morning, the Liberals, at the direction of their new leader, Stephen Del Duca, brought forward an unprecedented private members motion outlining their confidence in this government. The motion, sponsored by the Member for Scarborough-Gilwood, expresses her party's desire to see the government continue the great work it's done for a full four-year term. Later today, Speaker, there will be an opportunity for all members to vote on this motion and record their support for this government. Speaker, would the government House Leader please update this house on the great work this government has done and will continue to do for the people of this awesome province? Government House Leader. Thank you very much, Mr. Speaker. I appreciate the tough question from the Member for Willowdale. Today was a very historic day for many reasons, Mr. Speaker. Not only is today the first day that the opposition was provided two additional questions during question period to hold government accountable. It was also the addition of another private members bill or motion, Mr. Speaker. But as historic as that is, it pales in comparison to the fact that today, for the first time in the least in my 35 years of being involved in politics, an opposition party has tabled a motion of confidence in the government, Mr. Speaker. And as I said in my speech this morning, as I said in my speech this morning, I thank the Honourable Member. I know how difficult it must have been to sit around the table with her new leader and to put the very first motion that they've had to debate in months, Mr. Speaker, and to make it a motion of confidence in the government. And I urge all members of this legislature to do the right thing and vote with us and vote with the Liberals. Show your support for this government and vote in favour of that confidence motion. Thank you. It's a supplementary question. Thank you very much, Speaker. And I agree with the member from Scarborough-Gildwood that this government has done a great job since coming into office in 2018. We returned fiscal sanity and accountability to Queen's Park and started tackling the Liberals out of control $15 billion deficit. We made Ontario open for business, invested $26 billion to build better subway transit in Toronto, reduced small business taxes, increased housing supply to make the dream of homeownership a reality for families and young people, invested in education, healthcare, mental health and supports for the skilled trades, and created over 300,000 jobs before the pandemic hit. Speaker, though I am still a little surprised at this unprecedented motion of confidence brought forward by the Liberals today, usually opposition parties bring forward motions that challenge or criticize the government. Speaker, I think the government House Leader mentioned that in his 35 years in politics, he's never seen anything like this. Can you expand on that? Can you tell the House if you've ever seen a motion of this kind of an opposition party? I want to thank the Honourable Member for that question. As I said, not only my 35 years have I never seen an opposition bring forward a motion of support and confidence in the government, I've actually asked the legislative library to find out if in the history of parliamentary democracy, if an opposition party has ever gone out of its way to use the time that it has been given to hold government accountable to show support in the government. As I said in my remarks this morning, that truly shows the Ontario spirit, Mr. Speaker. When the Liberal Party can come forward with a motion of support in the government and confidence and ask for the support of all members of this legislature for that, Mr. Speaker, it speaks volumes, not only of the job that we are doing as a government and that this Premier has done, Mr. Speaker, but all legislatures, all legislators in this place. So I thank the Honourable Member for making history. I thank the Liberal Party for the support that they've had, and I encourage all members, vote in favour of this confidence motion, join with us in the Liberals and make sure that this government... Start the clock. The next question, the member for Muskegawak, James Bay. Good morning, Mr. Speaker. My question is for the Premier. While the Minister of Education is refusing to do all in this power to protect our students, we have a lack of staff in the north. The school board is asking other people to step in and work in schools. Last Thursday, we heard that immersion programs might close because there are no French teachers in the province. Despite the fact that the boards are asking for changes in months, Mr. Speaker, does the Premier really think that we can face this pandemic with very weak measures like online portals? Thank you, Speaker. The province, Ontario and governments across the country have been facing a shortage of French language educators. But, Speaker, I'm very proud of just days ago last week, the Minister of Francophone Affairs, the Parliamentary Assistant Minister of Education, and I announced a plan to reestablish the French language working group, which is bringing together boards, unions, leaders in education to confront this issue, head on and address it meaningfully with action after 15 years of inaction by the former government that got us to this place where we do not have a sufficient supply of educators. The issue is not financing, Speaker. It's literal access to qualified educators in our province. We are working with the French government in trying to attract skilled educators from France and the Francophonie across the world. We're working with Laurentian University, working with this specific program to foster more graduates from the French education. We're doing everything we can, Speaker, knowing that this issue is acute, recognizing, Speaker, you can't just create them overnight. We're going to keep working very hard on this, Speaker. Mr. Speaker, this government clearly doesn't see the problem. We cannot solve this lack with employment boards. It's a long-term lack. In 2015, the Liberals were just ignoring this problem by reducing the funds for education, including French training in Ontario. And this government also does not understand that teachers are giving up on this employment because of the conditions, the employment conditions. So this government should stop neglecting the teachers. He has to listen to them and take aggressive measures. Are you going to take clear measures adapted to Franco-Ontarian needs? Yes or no? Thank you, Speaker. We do agree with the member opposite. For the former government, there was absolute inaction, which has really resulted in this shortage at the security of that problem. But, Speaker, that is why just days ago to the member opposite, we literally just relaunched the working group on French second language to ensure we work with all partners, with federations, with school boards, and with faculties of education to encourage more individuals to pursue French language education and to teach within our schools. We relaunch it because we recognize the problem, and we're working very hard aggressively with a limited time timeline to get results for your community, for all communities across this province. Next question, the member for Scarborough-Gildewood. Start the clock. Once again, the member for Scarborough-Gildewood. Thank you, Speaker. My question is for the Premier. Speaker, the first wave of the pandemic was devastating on our small businesses and local economies. The second wave shows every sign of being worse than the first, and our testing and contact tracing infrastructure is unprepared and overwhelmed. Yet, the response from the Premier has been slow and reactionary. Many small businesses fear that they will not survive a second wave lockdown. Just this week I was contacted by Michael Wood, a small business owner in Ottawa, whose message is clear. Small businesses need support. They need their government to act. They need real rent relief and more. They're worried about closing that will devastate the livelihoods of their families. Speaker, through you to the Premier, will the Premier step up to provide real relief to small businesses or will he wait for them to collapse before he acts? The parliamentary system to the Minister of Finance. Thank you very much, Mr. Speaker, and it certainly has been a tough time for our small business community out there, Mr. Speaker, and that's why this government responded from the very beginning of the pandemic with real relief for individuals and businesses. $3.7 billion announced in March increased to $11 billion in those direct supports in August, Mr. Speaker. $241 million was put towards direct rent relief and it's been almost 591,000 employees who've been helped by this program, Mr. Speaker. That's almost 60,000 commercial tenants throughout this province. But businesses are attacking in a number of ways, not just on paying their rent but their hydro bills, and that's why this government has provided $175 million in relief there, Speaker. $355 million in employer health tax cuts. WSIB premium reductions, Mr. Speaker. This is a tough situation for small businesses and that's why we're there to support our hardworking small business community, Mr. Speaker. Message is clear again, small businesses out there. We have your back. The supplementary question. Speaker, Mr. Wood presented at the Standing Committee on Finance and Economic Affairs on August 17th. He needs action from this government now. This government is asleep on the issue of support for small businesses. The COVID pandemic is making the cost of doing business in Ontario unaffordable for small businesses. Insurance companies are denying coverage or hiking rates to unreasonable and unaffordable levels in industries where the risk of transmitting COVID-19 is elevated or uncertain. I asked the Minister of Finance about this issue on July the 8th. He said he would look into it. This issue affects small businesses including restaurants, some nonprofits, in fact even film and television production is seeing escalating costs that are quickly becoming unaffordable. Ontario's economy has been hard hit. Many small businesses are at risk of closing with no credible release package in sight from this government. Winter is coming, Mr. Speaker, and we need action now. Will the Premier step in to address the issues of insurability caused by COVID-19 or will he wait until these businesses close? The Associate Minister of Small Business and Red Tape Production. Thank you very much, Mr. Speaker. I want to thank the member opposite for the question. I actually had an opportunity to host a round table with the individual referenced by the member opposite, Michael Wood, who's been a great advocate for small businesses across the provinces. It was with consultation like individuals, we've held over 90 round tables. It was due to consultations with business owners like Michael Wood that this government worked together with our federal partners to put together the commercial emergency rent relief program. It is through these conversations that this government was able to put forward programs like electricity rate relief to the tune of $175 million. It is through the conversations with these small business owners that this government was able to put forward tackling the barrier's website that looked at temporary red changes that could help support small businesses like allowing restaurants to deliver alcohol with takeout food. This government will continue to support small businesses, whether it was before the pandemic, during the pandemic, or during this very difficult time. Thank you, Speaker. My question to the parliamentary assistant to the Minister of Labor, Training and Skills Development. Speaker, the people in my riding of Durham have been hit hard by COVID-19 like so many across the province of Ontario. While it's encouraging to see more people returning to work as we safely reopen the economy, some people in my community do not have jobs to return to. Recently, the Premier and Minister of Labor Training and Skills Development announced that our government's helping people retrain and gain the skills they need to work in the skilled trades. Speaker, would the government please tell this House how this investment will help increase skilled trades jobs in our community? Parliamentary Assistant to the Minister of Labor Training and Skills Development. Thank you so much, Speaker, and thank you to the member for Durham for the question. Last month, the Premier and Minister announced 37 million to support over 80 projects across Ontario. This funding will help more than 15,000 people to train for new jobs in high demand fields. In Durham Region, this includes investing in construction, automotive, aerospace, and transportation jobs. These jobs not only help provide for people in their families, they are the foundation for long lasting, meaningful careers that will be indeed for years to come. For example, we're investing in a project that will bring together industry partners with the most talented students to help them find hands on work placements that will teach them the skills they need to succeed. We've got a plan to get people back to work, and I look forward to speaking to the supplementary. Thank you, Speaker, and thank you to the Parliamentary Assistant for that answer. This is great work that's being done to train and upskill people so that they have the talent they need to connect the jobs of today. This investment is giving Ontario Tech University Durham College and Loyalist College the tools they need to deliver work integrated learning like placements and apprenticeships and we know how important those are to young people today. Speaker, as we know the pandemic has impacted different people in different ways. Particularly, it's created an additional barrier for those who are already unemployed or underemployed. Can the government please also share with this how these investments are supporting these groups that need help. Thank you to the member for the follow-up question. Speaker, I'm happy to share that we are enriching school training programs to better align with the needs of employers and to help get workers back into the workforce. Take for example our driving success commercial driver training program with the Ontario Truck Training Academy. This program provides future drivers with the training opportunities and experiences they need to thrive in the current and future workplace. It has been specifically designed to enhance the student's knowledge, increase job retention and improve communication between training organizations and candidates and employers. Speaker, we're on a mission to get people into good jobs to support them, their families and their communities. We will not stop until this important work is done. Thanks, Speaker. The next question, the member for University Rose Hill. Thank you, Speaker. My question is to the Premier. It's been five months. The Premier promised to build an iron ring around long-term care homes. This weekend, Vermont Square, a long-term care home in my riding reported that 26 residents and 15 staff members have tested positive for COVID-19. That's 45 people. We have known for months that our province was in for a second wave set to begin in the fall. Now here it is and families are reliving the nightmare. Seniors at Vermont Square want to know why is the Premier failing to protect them from the second wave? The minister of long-term care. Thank you, Speaker. And thank you to the member opposite for that question. Looking at the outbreaks in our long-term care homes and some of the homes are particularly affected. A small number. There are only 12 homes out of 626 homes that currently have resident cases. Vermont being one of them. We look at the measures being taken in conjunction and coordination with the hospitals, with the IPAC teams, the infection prevention and control teams, working with the medical officers of health, adding more and more layers of protection. And this is something that's been ongoing since January looking at ways to protect our long-term care homes. And the reality is we have an invisible intruder that is in the communities and you've heard Ottawa, Toronto, Peele these areas have higher numbers of community cases. And so we are adding more measures in addition to what we've done already to support our homes and we will continue to do that with great vigilance and care. Thank you. The supplementary question. Thank you, Speaker. Back to the Premier. The Ford government was urged to prepare long-term care homes for COVID-19 as early as January. Chronic understaffing, overcrowding and funding cuts have made long-term care homes particularly vulnerable. Instead of acting this Premier has chosen to do the minimum. And when the first way of hit we know long-term care homes were devastated. And we are now well into the second wave and it's looking like it will be worse than the first. There are now 43 outbreaks in long-term care homes. 43. What, why is the government failing to properly prepare long-term care homes for a second wave? Mr. Long-term Care. Thank you, Speaker. And thank you for the question. I completely disagree with your premise of the question. In fact, we have been preparing all along working as a coordinated effort with the Ministry of Health as the lead for the command table. Public Health Ontario, Ontario Health our medical officers of health and the outbreaks again I will I will clarify an outbreak in a long-term care home includes a staff member that might be isolating at home when there is no case in the long-term care home. And as I said a small number about 12 homes right now have any resident cases that's out of 626 homes. 98% of our long-term care homes have no resident cases. And the dollars that we put behind our plans also matter $243 million for surge capacity Sponsors? Staffing infection control screening Over half a billion dollars that we've just announced in addition to hundreds of millions of dollars that have gone to this sector we will continue to work with our sector to Thank you. Thank you. The next question the member for Ottawa Van Niel. Thank you Mr. Speaker my question Thank you Mr. Secretary of Education. The return to school has been the source of much anticipation and anxiety. Unfortunately the fears and concerns of Ontario and surrounding the back to school plan are becoming facts. As we see an increasing number of persons in schools testing positive we are learning with this May that the province is decreasing testing instead of expanding it. Our education workers are now working under impossible stress. Families were waiting for hours to have their children tested to then wait days to get results because of the backlogs. Dr. Vera Etches of the Ottawa Public Health at our health system is in crisis and on the edge of collapse. Communities families and students are looking up at the government for support. What is the minister's plan to tackle the wave of COVID-19 spreading through our education system? Minister of Education. Well thank you Speaker and thank you to the member opposite for the question I should start Speaker with a recognition today as World Teacher Day and I think all members will join me in thanking our frontline educators who every day make a difference in our province and inspire the next generation to pursue higher learning and to be people of good character and we're very grateful for their leadership every day particularly now. With respect to Ottawa region I was pleased to assemble a call with Dr. Etches with this board chairs directors and the public health nurse leadership within Ottawa with a single mission to understand how the protocols are working. What I can tell you is in Ottawa region is clear with higher rates of community transmissions quitting some impact in our schools but our educators our administrators and public health are working very quickly decisively to get into those schools contact trace isolate when required test these children the minister of health has unveiled a robust plan a national leading plan to increase capacity so that all parents and all kids get the test when they need it. Elementary question Again to the minister of education school boards are asked to perform miracles without adequate resources Education workers and especially teachers are stretched too much between teaching in class teaching online teaching to additional students because of other teachers withdrawing for health reasons although while looking after the well being of the students they care for the increasing number of person in schools being exposed to the virus requires testing capacity to have an efficient contact tracing if we want to avoid school closures the education workers are the front line workers of our education system the economic recovery cannot happen without the recovery of our education system now is not the time to limit testing now is not the time to try and save money the health of our communities has no price will the minister ensure that education workers have access to priority testing and access to timely results Minister of Education Thank you very much Mr. Speaker we are fully committed to all citizens of this province having access to having access to testing within their community the minister of health has unveiled a variety of steps of increasing capacity as well as the assessment centers pharmacies more points of contact I mean this is a reflection speaker that we all are doing our very best and our school boards on the ground are being provided with significant funds literally over a billion dollars of investment to help ensure every layer of protection is in place and speaker when I reflect on how Ontario is doing in this federation when you look at what other opposition members or governments are saying across the country the opposition in in Quebec for example the Liberal Party a member Gregory Kelly said Ontario is taking back to school safely seriously Premier Ford quote is miles ahead of us in protecting kids and teachers and supports have money's been invested for distance learning school ventilation and PPE response Quebec must do more to reassure parents and we respect Quebec's role in our country but we will acknowledge speaker we lead this nation investment in a very serious comprehensive protocol we will do whatever it takes to keep kids safe The next question the member for Waterloo Thank you very much my question to the Premier speaker this past Friday spoke with Ben Grazi he and his family own and operate the doctors house event venue in Kleinberg in the riding of King Vaughan they have been in the event venue business for over 35 years they employ 130 people and they host up to 300 events a year due to the economic effects of the pandemic businesses like the doctors house are hanging on by a thread they've written to this government asking for help and they haven't heard anything from anyone on that side of the house they have done their part to prevent the spread of COVID-19 they closed their doors and they are paying the highest price they've asked for direct direct government support without it their businesses has no future and Secra ends today speaker will the government work with Mr. Grazi and the other event venues to support them financially through this difficult time the Minister of Heritage Sport, Tourism and Culture Industries thank you very much I appreciate the member opposite bringing this important issue to the floor of the legislature through the ministry we continue to engage with our stakeholders last week alone we had over 700 stakeholders including those in the event business talk to the ministry about some of the challenges they're facing as a result of COVID-19 I had the opportunity earlier in the summer to visit Kleinberg with the Minister of Education and the local member for that area and I can also tell you that on Thursday evening I had a significantly long conversation with the federal minister of responsible for economic development and all of my federal provincial territorial colleagues on the requirement for us to have a national approach to supporting banquet event spaces such as the one that you're talking about I'd be more than happy to take your direct concerns today with my staff and meet with the individual later on today or later on this week but we recognize these have been challenging times and we want to make sure that right now we're taking their concerns to the federal government and we'll continue to do that response and the supplementary question just to be clear Mr. Speaker these businesses are asking for provincial support that's what they need Speaker Mr. Grassy formed a group with his colleagues collectively they represent over 100 event venues and banquet halls and employ 5,000 people after drastically reduced 2020 season most of the 1100 event venues across the province won't be able to survive even two more months without government support these venues wrote to the government explaining this problem and asking for support in the form of bridge financing rent and mortgage relief and real tax relief and they have heard nothing Speaker Ontario's banquet halls and event spaces employ approximately 50,000 people in this province at the very least the government should have directly responded to the concerns the government can't just give up on these businesses it should not be allowed what is the government's plan to support event venues and banquet halls in the province of Ontario they deserve our support we should show up for them Minister of Heritage this is the second question in a week where the NDP asked the question I responded and then they forgot to pivot speaker the reality is this ministry this government continues to work with event and banquet halls we have continued to set up multiple sectoral tables including within my ministry 14 ministerial advisory committees alone focused on how we can best pivot to support these sectors as I said my ministry had a a large gathering by a telephone town hall last week with over 700 of our stakeholders we continued to meet over the weekend with our theme park operators last week with our professional sports and we continue to meet with those like Mr. Grassie and I'm happy to take his concerns and meet with him later today and set up a table as well for him if that's what he thinks with his stakeholders but the reality is the member opposite doesn't seem to understand that there's no point in Ontario duplicating something that's already going to be coming from the federal government and I had those conversations just last Thursday I just wish for once they would live for the next question the member for Ottawa sir very much to speak or my questions for the deputy premier so on Friday Ottawa's chief medical officer of health fair etch has said our health care system is in crisis our hospitals are at 100 percent we're on the edge that same day there was a testing backlog of 92,000 tests that meant people were waiting but what's worse is you might have to get retested because your test went bad and then we find Toronto Public Health is shutting down contact tracing because they can't keep up for a month now we've been unable to keep up demand experts have been telling the government they've got two million kids going back to school you've got to get to 100,000 tests so my question is two questions through you why aren't we there and when are we going to get there and testing in Ontario the minister of health well with respect to the issue of testing we have been increasing tests we started at 5,000 at the beginning of the pandemic we're typically doing around 40,000 tests per day in Ontario right now which has been noted by the premier outperforms all the other provinces and territories in Canada combined compared to Ontario we've already tested over four million Ontarians and we're increasing that testing capacity we're getting up to 50,000 that should happen within the next few weeks and we're going to increase from there because we know with the second wave testing tracing and isolating cases is key but along with the testing strategy of course you have to have the lab capacity to be able to do the tests testing in a timely basis we're increasing the lab capacity as well we're reaching out to other universities that have testing facilities we're working on establishing a network which we did not have when we started with us so we are increasing both of them and we will be able to continue to increase our volumes as time goes on supplementary question well speaker British Columbia did more tests than they've ever done of 11,000 which is roughly equal to the share of the population and you know what their backlog was zero zero so maybe the previous should make that comparison because that's what we're talking about here so we're still in a situation where we have huge demand and huge test backlogs and on the weekend in the Toronto Star experts are saying to get to 100,000 tests it's going to take us to three to four months but they just announced the money last week so we have the time the advice and the money in Ontario to do what needed to be done and it didn't happen the Premier took his eye off the ball and as a result we're not going to be where we need to be right now so my question back to the previous when are you going to get to 100,000 tests here in Ontario to make sure that we can meet the demand that's necessary to manage this pandemic Minister of Health thank you we have been meeting our targets every step along the way remember we started at 5,000 tests per day we're now at 40,000 tests we will get to 50,000 tests within the next week or so we are going to increase from there to get up to 78,000 80,000 and then on from that but we are meeting those targets and we've been working on this since the pandemic began we brought out our plan keeping Ontario in safe which is dealing with testing, tracing, isolating but also managing all the volumes of tests and procedures and surgeries that were postponed during the first wave we're also getting ready for flu season we're going to have the largest flu immunization campaign in Ontario's history so we are moving forward on all fronts we are meeting our targets we are doing everything that we can to keep Ontarians safe the next question the Member for Toronto sent thank you Speaker my question is for the Premier families in my riding are frustrated with this government's failed online learning plan elementary students in my riding have been waiting weeks to start school one of my constituents Helen was so excited to start grade seven that she woke up at 5 30 in the morning on the first day of school she logged into her virtual classroom only to be disappointed that she didn't actually have a teacher yet it wasn't until in fact last Friday 11 days after she had been scheduled to start school that she finally met her teacher and her classmates 11 days speaker Helen's mother Sandy is frustrated at this government's botched back to school plan which left school board scrambling to hire teachers for virtual classes it meant that her daughter spent days isolated and disappointed unsure of when she would actually get to start school this year what does this Premier have to say to parents like Sandy who are fed up with this government's failed online learning plan that left students in my riding without a teacher for weeks Thank you Thank you well thank you speaker when I'd say to Sandy if it was up to her MPP you wouldn't even have the choice of online learning in this province of Ontario so the irony is not lost on me the irony is not lost on progressive conservatives who stood alone in this house who stood alone in this house and now the member thinks that that individual collected lecture this government when we've said a nationally leading standard 75% of online learning must be done live synchronous learning in Quebec for example from grades one to three it is a matter of I believe four to six hours a week we do that in a day we have educated by mandate and compulsory every educator must undergo professional development when it comes to online learning we've ensured virtual learning environments in every school high school and internet and internet in every high school we're taking this seriously speaker we're creating a save the art platform we're doing it for the first time in this province and we're proud of the work that we're doing in this province that concludes our question period for this morning we now have a deferred vote


All right, so hello everyone, my name is Sam Kim, and I'll be presenting key homomorphic pseudo-random functions from LWB with a small modulus. So the topic of this talk is a pseudo-random function. So let's begin by discussing what these are. So we're all probably familiar with this concept already, a pseudo-random function or a PRF is an efficiently computable deterministic function that has some associated key space, an input space and an output space. So we say that a pseudo-random function is secure if no efficient adversary can distinguish oracle access to the PRF for a randomly chosen key to an oracle access to a completely randomly generated function. So pseudo-random functions are very useful objects in cryptography, they are of course very fundamental objects in symmetric cryptography and can be used for symmetric encryption or max. So even beyond symmetric crypto, they are often incorporated into many cryptographic constructions and protocols. So when we need to generate random bits, for instance, PRFs are very useful. When we want to de-randomize a randomized algorithm, PRFs are also very useful in these settings. And so due to these useful properties, pseudo-random functions are widely studied in the literature today. So in terms of constructions, block ciphers such as AES are generally modeled as PRFs for analysis. We can also construct provably secure PRFs from specific algebraic assumptions. So using the classical result of Goldwright, Goldwasser and McCauley, we know how to construct PRFs from any one way functions. We also know how to construct PRFs from the DDH assumption by the result of Naur and Reingold from 95. And we also know how to construct PRFs from the learning with errors assumptions or LWE from the result of energy pattern frozen from 2012. So in this talk, we will focus on a lot of space PRFs. So in the world of lattices, there are two main families of PRF constructions. So first there are constructions that follow the GGM construction with a lattice-based one way function. So one way functions based on hard problems like SIS or LWE. So these constructions are very simple and quite elegant, which is a big plus. At the same time though, these constructions are not the most property efficient and they also require evaluation circuits that are inherently sequential. This means that even if we had designated hardware devices for evaluating these PRFs, hardware acceleration will be somewhat limited, which is not ideal for objects like PRFs that are very fundamental. The other family of PRF constructions from lattices are direct constructions from LWE. So these PRFs generally have better concrete efficiency compared to PRFs that follow generically from one way functions and they also have shallow evaluation circuits. And even more, they have many useful algebraic structures like key homomorphism, which has a number of nice applications. So since one of the results in this work is about key homomorphic PRFs, let me define them in a little bit more detail. So the concept of key homomorphic PRFs was introduced by Naur Atel from 1999 and was formalized by Monet Atel in 2013. So we say that a PRF is key homomorphic if it satisfies the following properties. So first we require that the key space and the output space form groups. So the key space and the output space don't necessarily have to be the same group, but let me just denote the group operations with additions for this talk. So then we require that for any two keys, K1 and K2, if we evaluate the PRF on the sum of these keys, then this is equal to the sum of the PRF evaluation on each of these keys on the same input. So key homomorphic PRFs have many useful applications. So they give rise to update of encryption, proxy encryption, distribution of PRFs, and many others. So from LWEE, we can directly construct pseudo-random functions that are efficient, have shallow evaluation circuits, and have useful algebraic structures like key homomorphism, which is very nice. However, all the existing constructions of PRFs in this family have one downside to them. So they all rely on LWEE, where the size of the modulus must be super-pollinally big with respect to the security parameter. To explain the issue with the modulus, let me recall the LWEE assumption. So the LWEE assumption, a problem is parameterized by three parameters, the dimension N, the modulus Q, and a noise distribution that's usually denoted by a chi. So in the LWEE problem, our goal is to distinguish the following two distributions. So in the first distribution, we're given a set of uniformly sampled vectors, A1 through AM in ZQ to the N, and it's noisy inner product with a uniformly sampled LWEE secret vector S. In the second distribution, we're provided with a set of uniformly sampled vectors, A1 through AM, and just uniformly random elements in ZQ. So the LWEE assumption states that these two distributions are computationally indistinguishable. So generally, for most applications, including PRFs that rely on the LWEE assumption, it is always better to set the modulus Q to be as small as possible. So we always want Q to be small. So the first reason for this is related to efficiency. So generally, applications that rely on LWEE operate under the same ring, ZQ. So as Q gets bigger, the keys in these applications also become bigger, and so the operations also become slower. And the second reason is related to the quality of the LWEE assumption. So generally, the hardness of the LWEE assumption is related to what is called the noise to modulus ratio. The greater the size of the modulus is compared to the size of the errors, the easier the LWEE problem becomes. And so we want to set Q to be as small as possible to base our security on the hardest variant of the LWEE assumption. So all non-GGM-based allotus PRFs rely on LWEE with Q that must scale super-polynomially in the security parameter. So a natural open problem in this area has been whether we can construct an LWEE-based PRF that relies on a polynomial-sized modulus Q. And this problem has actually been a big open problem in this area for quite a while, ever since the first direct LWEE-based PRF instruction of Banerjee-Pike-Grid and Rosem from 2012. Given the fact that we know how to construct seemingly more powerful primitives like polymorphic encryption or latch-based encryption on LWEE with only a polynomial-sized modulus Q, the state of affairs has been even more puzzling that we cannot construct PRFs using a polynomial-sized modulus Q. So let me summarize the main results of this work. So we provide two new PRF constructions that relies on a polynomial-sized modulus Q. So in the first construction, we focus on building a low-depth PRF. So we construct a new PRF that can be evaluated by circuits in NC2 and works over a polynomial-sized modulus Q. So in our second PRF construction, we focus on key homomorphism. So we build a PRF that has high depth evaluation circuits but is key homomorphic and works over a polynomial-sized Q. So both of our constructions can be extended to the ring setting as well for better efficiency. So the security of these constructions can be based on the ring LWEE as opposed to LWEE. And I think the most interesting part of this work is actually not on the final result of our paper but actually the technique that we use. So both of our PRF constructions use a technique called chaining, which I will talk about later on in the talk. I think chaining can have more applications even beyond the context of PRFs. So in the remainder of the talk, let me give an overview of our PRF constructions and explain what this chaining technique is. But before I get started, let me briefly comment on the limitation of our first PRF construction, the low depth PRF that we construct. So in order to prove that our construction is secure, we provide a reduction from LWEE. So any adversary that can break our PRF can be translated into an attacker that breaks the LWEE assumption. However, the success probability of our reduction will be quite low. So it will be barely noticeable when the model's Q is set to be a polynomial in the security parameter. So more precisely, what we get is basically a trade-off between the loss in our security reduction and the size of the model's Q. So the bigger the Q is, the smaller the reduction loss becomes. So asymptotically speaking, the parameter settings that we achieve for our first PRF construction is not much better than any GGM-based LWEE PRF. But when we consider the concrete deficiency of our PRF, it will be much better than a GGM-based PRF. So I will refer to our paper for the full discussion on this issue with reduction loss. And let me move on to describing an overview. Let me move on to giving an overview of our PRF construction. All right, the main intermediate abstraction that we're going to use to construct our PRF is a special family of pseudo-random generators called pseudo-random synthesizers, which is a concept that was introduced by Nauri Reingold from 95. So a pseudo-random synthesizer that we will denote by S in this talk is a deterministic two-to-one function that takes in a pair of elements from some domain D and outputs a single element in the same domain. So one can think of this as a pressing function that compresses two elements into just a single element. So we say that a pseudo-random synthesizer is secure if for any polynomial m, if we sample two m-uniform and random elements, a1 through am and b1 through bm and apply the synthesizer to each pair of ai's and bi's, then the result is computationally indistinguishable from m-squared independent uniformly random elements in the same domain. So a pseudo-random synthesizer is basically a PRG that allows us to take two m elements in the domain as a seed and expand it into m-squared elements in the domain. All right, so Nauri Reingold showed that if you have a secure pseudo-random synthesizer, then we can construct a low-dive PRF as follows. So suppose that we want to construct an n-bit PRF where n is a part of two integer. Then the PRF-t is going to consist of two n-random elements from the domain of the synthesizer. And so let's denote these elements as a1-0 all the way through a-n-0 and then a-1-1 all the way through a-n-1, okay? Then we're going to define the PRF evaluation on an input x using a tree structure as follows. So each leaf of the tree is going to be associated with a pair of elements in our key that are associated with each bit of our PRF input. So the first leaf position is going to be associated with a-1-0 and a-1-1. And the second leaf position will be associated with a-2-0 and a-2-1 and so on, okay? And then to evaluate the PRF on a specific input x, we will take either a-1-0 or a-1-1 depending on the first bit of our input. And we will take either a-2-0 or a-2-1 depending on our second bit of our input and so on. And then we will apply the synthesizer to each pair of these elements one by one, okay? So at each level of the tree, we're basically compressing our elements into a half and eventually we will land on a single element in the domain which is going to be our final PRF output. We can argue about the security of this PRF recursively. So it can be rarely checked that at level one of the tree, each synthesizer is basically a secure two-bit PRF simply by the definition of a secure synthesizer. So in this example, the left synthesizer is a secure two-bit PRF and the right synthesizer is a secure two-bit PRF and this means that if you apply the synthesizer on the combined outputs of these two synthesizers, then the result is indistinguishable from an output of a secure four-bit PRF. So we can continue this argument to show that the entire construction is a secure PRF. So in order to construct a synthesizer from the other assumption, we must figure out a way of generating errors in some deterministic fashion. So the work of Banerjee Pack and Rosen from 2012 showed a way to do this in an elegant way. So instead of adding noise to the product of vectors or matrices, A times S, we will instead map them into some small subsets or buckets in ZQ. So here's a picture example of Z mod of 24. And so if the inner product AS lands in this blue subspace then we're going to assign the value zero up to the inner product. If it lands in this green subspace then we're going to assign the value one to the inner product and so on. If it lands in the red, then we're gonna assign the value two. So this way we're rounding a value in Z mod 24 to an assigned partition in the space that represented by the subgroup of Z mod three in this example, okay? So in algebraic form we denote this operation by the rounding location. So we're basically mapping elements in ZQ to elements in Z mod P. So intuitively when we add some noise to the inner product of AS then we're basically masking the lower bits of the result with this noise. So basically what rounding is doing here is it's basically just completely removing the lower bits from the result. So intuitively this rounding and this adding nodes has basically the same effect of completely masking the lower bits from an adversary. So given this intuition Banerjee et al introduced a new computational assumption called the learning with rounding assumption which is parametrized by N and Q as in LWD but instead of the noise distribution it has an associated modulus P. So the LWR assumption states that for uniformly random public matrices A1 through AM and secret matrix S the rounding of the matrix products AIs times S so here we're applying the rounding operation component wise. Then this distribution is computational and distinguishable from the rounding of uniformly random matrices. For using the LWR assumption then it is now straightforward to construct a natural synthesizer that takes in two matrices A and S, multiplies these two matrices and then round the result down. So it is straightforward to show that the synthesizer is indeed secure from the LWR assumption. And here the synthesizer only consists of matrix multiplication and rounding operation. And so the synthesizer can be computed by a low depth circuit. There is a psych-technicality here with the domain of the synthesizer. So the synthesizer takes in a square matrices from ZQ and maps them into square matrices in ZP. So the domain of the synthesizer and the range of the synthesizer is actually different but this is really a minor technicality that can be fixed in a number of ways and I will just refer to the paper for more details on how we can actually do that. All right, so we now know that number one a secure random synthesizer gives us a low depth up here at construction. And number two, the LWR assumption gives us a secure pseudo random synthesizer. So the remaining question is whether the LWR assumption can be shown to hold assuming that the LWR assumption holds. Banerjee et al showed that this is indeed the case via a hybrid argument. So we want to show that given uniformly random matrices, AIs and the rounded matrix products of AIs and S, the resultant distribution is computationally indistinguishable from a bunch of uniformly random matrices. So as an intermediate hybrid distribution, we're going to introduce a distribution that provides the product of AIs with S but with both noise and rounding applied to it. So it provides AI times S plus some noise and then round the result. If the rounding modulus Q is sufficiently greater than the rounding modulus P, then with overwhelming probability, we can show that the added noise goes away with the rounding. The intuition is that if you round away enough the load of bits from the output, then some small noise in the load of bits do not really impact the final outcome. So these two distributions are statistically indistinguishable. So once we have this statistical indistinguishability, we can show that the hybrid distribution is computationally indistinguishable from a uniform distribution. This follows just before we leave from the LWR assumption. And this is how we basically show that the LWR assumption holds as long as the LWR assumption holds. However, the apparent limitation of this reduction is the requirement that the modulus Q must be much greater than the rounding modulus P. So roughly speaking, we can show that for uniformly random matrices, A and S, the probability that the rounding of A times S is equal to A times S plus some noise E is roughly P over Q multiplied by the size of the noise E. So this means that even if we set the noise and the value of P to be very, very small, a Q has to be super polynomial greater than these values for the Z quality to hold with overwhelming probability. So this is where our new idea of chaining comes in. As a thought experiment, let's modify the LW distribution in some funny way. So this distribution is going to be defined with respect to some positive integer, tau, which we will call the chaining parameter. We're also going to make use of an explicit error sampler or a Gaussian sampler that takes in a uniform C at R and deterministically produces a noise matrix E. Then we're going to define an oracle that we will refer to as the hybrid oracle as follows. So the hybrid oracle will take in a uniformly random a square matrix A and a set of uniformly random square matrices S1 through S tau. Then it will produce the following distribution. So first, it will sample a noise matrix E1 from the noise distribution and then compute the matrix A times S1 plus E1 as in an LW sample. Then it will round the result down and feed it into the error sampler to generate the noise for the next iteration of the chain, E2. Next, it will move on to compute the next iteration of the chain. So it will compute A times S2 plus E2 round the result down and use the result as a C to generate the error E3 for the next iteration of the chain. So it'll basically continue on this procedure for tau iterations. And in the tau iteration, it will just return the C of the result. So what we're basically doing is chaining multiple LW distributions with multiple keys together and deriving a new distribution from it. So it is actually not too hard to show that the output distribution of this oracle is computational indistinguishable from Uniform. So by the LW assumption, we can first show that the first iteration of AS1 plus E1 is indistinguishable from Uniform by the LW assumption. So this implies that R1 is uniformly distributed, meaning that E2 is also properly distributed on noise matrix. So this in turn implies that the second iteration of AS2 plus E2 is computationally indistinguishable from Uniform by LW E. And you can basically repeat this process for the entire chain to show that the final output is uniformly random. Okay, so this is all nice and fine, but why are we even defining this oracle in this fashion? This oracle is still a randomized oracle that most samples the noise matrix E1 in a randomized way. So it doesn't really appear to help in building a deterministic synthesizer. But the key observation that we make is that even if this initial noise matrix E1 is said to be zero, the pseudo-randomness of this oracle distribution is still preserved. So to see this, let's define another oracle that we'll refer to as the LWE or LWRE oracle. So this LWRE oracle works in exactly the same way as the hybrid oracle that we defined previously. The only difference is going to be that this first noise matrix E1, we're just going to set this to be a zero matrix. So in the first chain, the oracle computes A times S1 plus zero, so plus zero. And derive a seed for E2 by rounding this result down. That's it, so everything else stays the same. So the first chain here, the rounding of A times S1 is basically an LWRE sample. So we're essentially chaining an LWRE sample with a bunch of LWRE samples. And so this is why we refer to this oracle as the LWRE oracle, which stands for learning with rounding and errors oracle. So we can actually show that the distribution produced by the hybrid oracle is actually statistically indistinguishable from the LWRE oracle, even when the model is Q is said to be small. So recall that the probability that the rounding of the exact product of two uniform random matrices, A and S, is equal to the rounding of their noisy product, AS plus E, is roughly proportional to the size of the noise times P over Q. So let's say that we set the modulus Q to be just big enough so that this product is roughly one half. So then we can actually make the following observation. At each iteration of the chain in the hybrid and LWRE distributions, the probability that the Cs that we use to derive the noise matrices, so these RIs or oracle is going to be one over two. So no matter what noise matrix that we get from the previous iteration, the probability that A times S i plus noise will coincide in the two distributions is basically one over two. By the way, we set the modulus Q. Next, we note that it's at least one of the result of the chain in the two distributions produced the same seed RIs. Then for all the sequence iterations of the chain, the Cs will coincide in the two distributions. And this is just because the Gaussian sampler, the error sampler is deterministic. So there is basically one over two chance that the seed will be different in any iteration of the chain. But if there is one chain for which the seed is the same, then all the sequence iteration will produce exactly the same result. And this means that the probability that the output of these two oracles will differ on any given input will be at most one over two to the tau. So for tau that is super logarithmic, this probability is basically negligible. All right, so this LWRE oracle induces a natural computational problem that we call the learning with rounding and errors problem. So it basically adds an adversary to distinguish the outputs of the LWRE oracle with those of a uniform sampler. So using the same argument that we discussed in the previous slide, we can show that the LWRE problem is hard assuming that the LWRE assumption holds. So for this, we can consider the intermediate distribution that provides the output of the randomized hybrid oracle from the previous slide. So we showed that as long as the parameter tau is big enough compared to the modulus q, the outputs of the LWRE oracle and those of the hybrid oracle are statistically indistinguishable. Then via a standard hybrid argument, we can use the LWRE. We can use LWRE to switch each output of hybrid oracle to uniformly random values, to uniformly random outputs. So then using the LWRE assumption, we can naturally construct a synthesizer in a straightforward way. So the synthesizer takes in a matrix A and a set of matrices, S1 through S tau and return the output of the LWRE oracle on these inputs. So the security of the synthesizer follows very straightforwardly from the hardness of the LWRE problem. And basically the PRF that is implied by the synthesizer is our final low depth PRF. So here there is actually a domain mismatch. So the synthesizer takes in a single square matrix A and tau matrices S1 through S tau, just to produce a single square matrix in ZQ. So this domain mismatch is a slight technicality that can be addressed pretty easily in multiple ways. And so I will just refer to the paper for the details on how we can do that. So one thing to note here is that the depth of the synthesizer is basically determined by the chaining parameter tau for the LWRE problem. So this is where our trade-off between the depth of the PRF and the size of the models comes in. So if tau is large, then we can set Q to be very small and still have a tight reduction. So if Q is a little bit larger, then we can set tau to be much smaller and get a very low depth PRF. All right, so I think my time is almost up. So let me just briefly comment on our second PRF construction. So we achieve our second PRF, which is key homomorphic, by applying the chaining technique on top of already existing key homomorphic PRFs. So chaining existing key homomorphic PRFs allows us to prove their security with much better primer settings. So in particular, if we change the key homomorphic PRF of Banerjee and Piper from 2014, then we can get a key homomorphic PRF whose security can be based entirely on LWRE with a polynomial size modulus of Q. So I will refer to the paper for the full details on how we can do this. All right, so let me wrap up with some open problems. So the first nice open problem is whether we can construct a low depth PRF without the undesirable loss in the security reduction. So can we construct a PRF that can be evaluated by a circuit of poly logarithmic depth and still have a tight security reduction? So I think this is a very nice open problem. Also, can we construct PRFs with additional algebraic properties using just LWRE with a polynomial size Q? So for instance, can we use a chaining to base the security of existing constraint PRFs on LWRE with polynomial amount of SQ? And finally, the method of chaining may have additional applications outside the world of PRFs. So it would be interesting to find other applications where chaining may be useful. Okay, and with that, let me conclude. Thank you very much.


Please welcome to the stage Founder and Chairman of Intentional Media, Bob Caruso.  I hope you had a great first day of SoCAP, did we? I love it. But we are really excited for SoCAP 18 and to start the next decade of our journey. We want to thank all of you, the six of you that are here for everything that you do for SoCAP. It's amazing. I just want to start by thanking all of you because none of this could happen without all of your help and inspiration and motivation. And I have to thank our amazing team at SoCAP and Intentional Media for all the amazing work that they do throughout the year. We host dozens and dozens of events all throughout the country, put out really inspiring media, and it kind of all culminates here at SoCAP. So thank you guys so much for all of your incredible work. I also have to thank all of our amazing partners and sponsors and community members for all of their enthusiasm and encouragement and incredible generous support for all of what we do. It's really amazing and all the wisdom that they provide is really incredible and what the work that they do is truly inspiring. I also want to give a special thanks to Lindsay Smalling. Many of you know Lindsay. Lindsay is truly the SoCAP guerrilla glue that holds everything together. Her incredible work and dedication not only to SoCAP but to this industry more broadly is really inspiring. And I couldn't be more proud to have Lindsay as my partner. We're also so excited and you can join me in congratulating her on becoming the CEO of SoCAP this year.  So can I get a show of hands? How many people are at SoCAP for the first time? Okay, that's awesome. And how many people have been to SoCAP five or more times? Awesome. Well, we really appreciate it. So before we dive into intentional media, I just want to show you this quick video. Well, so can we all just turn to our neighbors and tell them be intentional? I know it's 8.30. We could do a little bit better than that. Come on. Let's try it again. Be intentional. Awesome. Well, like the video says that intentional media, we aim to turn moments into movements and thoughts into actions. Intentional media is a family of brands that's working at the intersection of money and business and meaning to really build a more inclusive economy. We're charged to catalyze a transition of a new economy where we ensure that our social and our environmental and our economic systems all thrive together and for everyone. And we want to leverage the intimacy and the amazing relationship building that happens at physical convenings as well as the power of storytelling content and media to help connect and to educate and to inspire millions of people to act much more intentionally about how they invest and how they consume and how they run their business for greater good and with a greater sense of purpose and in a much more inspired way. So, you know, in starting some prep for this, I was reading the JIN roadmap report, which I don't know if any of you are as dorky as I am, but it's actually pretty good. And there's a few thoughts that I wanted to share. Throughout history, pioneering leaders have led ambitious movements. They've challenged the status quo. They've sought transformational change in the way the world operates. And many of those impact pioneers are in this room today or will be on stage this week. And they've led this amazing impact investing movement over the last decade. And while the industry has achieved so much incredible growth and has really achieved widespread interest, it's still only generating a fraction of the impact that we know is possible and that is really required to address all these global challenges that are affecting all of our communities and our planet. And the current pace of the industry and the trajectory that the industry is on while it's impressive, it's not going to be enough to solve these global challenges. So, on intentional media, we want to help all of you drive bold and inclusive and innovative action to usher in a new economy where capital and businesses align to address some of the most critical issues of our time with the goal of driving a much more inclusive economy. And we need a future, we need to envision a new future where impact and intentionality is embedded and embodied in every business decision and every investment decision and in every consumption decision that we make. We were so lucky in 2016 to have the opportunity to buy SoCAP and to re-envision what's possible with an inflow of capital and an inflow of ideas and some new people. And earlier this year, we launched intentional media. And intentional media, as you can see, is a family of five brands that each address a different market segment and demographic. And our family of brands includes SoCAP, SoCAP 365, Conscious Company Media, Good Capital Project, and What Will It Take? movements. And it's so important that each of our brands has its own distinct voice and engages with their audience in a very genuine and authentic way. We cannot have a one-size-fits-all message because we truly need to meet people where they are and where they are on their own impact journey. So I just want to show our amazing team, our executive team that we have in intentional media. Hopefully that comes up on the slides. There we go. You know, we are so extremely proud of our total lack of gender diversity. All of our brand's CEOs are women, which we're so excited about. Every brand is run by an amazing woman CEO, and 75% of our intentional family is women-led. So we are really, really proud of that. So with the launch of intentional media, we want to bring together this very fragmented and sub-scale impact economy and really tell all of these beautiful stories that exist out there today because so often the most influential people out there are the ones that are least heard and least known. You know, Darren Walker, the head of the Ford Foundation, recently said, all inspiring work and storytelling of such work is the key to promoting greater compassion and building and rebuilding bridges in our society between cities and rural counties, between the poor and the prosperous, and between the past and the present and the possible. Because when we do impactful work, we hold the mirror up to ourselves and our society. We not only experience our shared humanity, but we arrive at our shared obligation to humanity, which is our connection with an obligation to all people who suffer and struggle and seek to be heard. And that's really one of our main goals at Intentional. So in order to create this future that we all collectively and desperately want and we can ensure that all voices are heard, we need to engage and connect with people that are far beyond the boundaries that our current industry engages with. We need to create compelling emotional connections through media and powerful storytelling content to recapture the sense of awe and re-spark and rekindle this excitement that we had before that, honestly, we've lost a little and we can't come at this with a sense of despair or with a sense of defeat. And we need to encourage people to stop talking about what's normal and what's the status quo and what's accepted, and we need to start talking about what is possible. Anyone? All right. So in this future that we all want, it should be totally unacceptable to make any business or investment or consumption decisions without regard for impact on people and on the planet. And as we and the industry embark on our next decade, we need to grow our community. We need louder and stronger voices and we need to engage much more mainstream audiences to amplify all of the amazing work that's happening out there in the field so we can accelerate this positive change that our communities and our planet so desperately need. We also need to connect and grow our big tent. We need to aggressively seek out and partner with those who may not see themselves as part of our community. And as Lindsay so eloquently says, a big tent accelerates the market so much faster than a velvet rope and that convening valuable strangers creates unlikely allies. And I don't know about all of you, but I think in this country and in this world there's nothing we need more than unlikely allies. So intentional media is designed to amplify your stories and your great work and your impact and your research and your challenges and your insights. We need to continue to act by example. We need to live intentionally in everything that we do and we need to keep pushing and we can't agonize about the problems. We need to organize around the solutions because the progress that the world needs so desperately to end poverty, to preserve natural resources, to halt climate change, to create more gender and racial equity and to build a much more equitable and sustainable future deeply, deeply depends on it. So I want to close with this African proverb. If you want to go fast, go alone and if you want to go far, go together. And we want to go far and we want to go really, really far and we want to go far together with you. And it is time. And we are so excited and we know the next decade holds even more promise than the first. So thank you all so much.