Sorry for rehashing, I'm trying to process this. Confirmed: 1. Russia is a major American adversary. 2. Russia wanted Trump to win. 3. Russia actively worked in various ways to support Trump's campaign. 4. Trump's campaign found out in the middle of the campaign that Russia was actively working to support them. 5. Trump's campaign continued to accept and even publicly request assistance from Russia. 6. Trump denied that Russia was interfering or helping, after his campaign met with Russian agents hoping to receive dirt on Clinton and after they were passing voter data to them. 7. After Trump wins the election, he meets repeatedly with Putin alone, with nobody else allowed to be present. And he publicly sides with Putin over his intelligence agencies. How can we possibly let this happen? How can people like EVD and Ram think this is in any way acceptable? You are Putin's best defense. I understand bias, I understand being defensive because you've put yourself out there for so long now. I don't understand why people like that can't just be quiet and stop thinking this is all a conspiracy by the left when it's so crystal clear there's a major problem here. Won't even go into all the other crap happening (the constant lying, resignations by generals and top cabinet members, cronyism, campaign finance felonies, capitulation (DPRK), obstruction...), just looking at the behavior with Russia... how?! And although I agree it will be monumentally difficult to impeach with what we have right now, it shouldn't be. And if it doesn't happen, what precedent does that set? That as long as you cheat well enough to get elected, you can basically do whatever you want as long as you have Fox and friends behind you? I feel like it must happen, and if it doesn't, that's truly how the country is lost. It won't be long before every vulture in the world is knocking on our doors knowing they can do whatever they like. Seriously, this is all happening as blatantly as possible and we still act like there's a question about what's happening. If Trump was a competent person I highly doubt we would have even found out. It's troubling. 20 years ago half the country was willing to impeach someone for lying about cheating. If what's happening today isn't enough, what would be?
This is why obstruction is a big deal. Basically, lots of smoke, but not quite enough to prosecute. If there was no obstruction (i.e. if Stone, Manafort, even Cohen would tell the full truth) then there very well could be provable crimes. As it is, the Trump campaign wanted the same thing the Russians did and it's only likely that they somehow helped each other or at least didn't try to stop it - as evidenced by numerous links between Russians and Trump's campaign, and their admitted willingness to receive hacked material. If Trump had to go under oath, or if he hadn't dangled pardons, or if he didn't have half the Congress actively working to kill the investigation... who knows.
why do you guys debate him? He doesn't listen and if he does, nothing registers. Yabo is the only one who can get even remotely close to making a point with those 2. Edit, ok I see why, it's really hard not to. Ram, my mom exonerated me, so obviously that means everything is alright, right?? ok, now where's the block button?
P. 288: So... now we have it from Sanders herself. We are not to believe the White House. But the press is the problem right?
Every time you're told, you seem to forget... Just for fun, brushing aside Papadopolous, here's a hint: Edit: Page 57: "Within approximately five hours of Trump's statement [on July 27, 2016 that Russia find Clinton's emails], GRU officers targeted for the first time Clinton's personal office."
I'll double down on how ignorant I am of corporate taxes, however I do know that a company that is labeled as paying $0 taxes is nearly always contributing a lot of revenue through dozens of different avenues. Whether it's property taxes, its employees income taxes, sales tax, etc... they certainly are contributing. Many of the tax laws that can result in a big business not paying anything in federal income make sense individually, it's just when you have this tweet that says hey look xyz company is big and didn't pay taxes, it looks bad. But dig into why they had a 0 liability this year, and all the pieces make sense. E.g. it's a good thing to incentivize R&D; it seems reasonable to allow some years of losses and to apply that loss against the taxes you owe in the future (within a reasonable time frame). These things can help businesses make it, which can help areas the businesses are in and people the businesses employ. I do think they corporations should pay a little more in taxes - but I think the shocking 'xyz paid 0$ in taxes' tweets are just like what Trump and Fox do when they take something out of context, even when there's some truth, and then use it to attack or manipulate. Edit: As far as employees paying taxes @bfwissel. I do struggle with that a little. But in the end, if a company is paying really well, and that cuts down on their profit, which in turn cuts down on the taxes they owe, it's just transferring it to the employees and to their tax burden. So, the way a company compensates their employees does matter and factor into the equation for how much benefit the company is for an area/country. I think a lot of people would prefer a company pay employees more and have the employees have the tax burden and the benefit of making more money.
When I was younger I used to see the 0$ in taxes paid arguments and get upset at how unfair the system is. Now I see it and get upset that whoever is telling me that is obviously preying on my ignorance of the tax system and leaving out relevant context.
Seems like they know what they're doing. They're not terrified of her if they give a previously little known rep this much attention. I would assume they think they can use her to solidify their base and divide the left.
That's an opinion article written by a hyper partisan man who for years has left out crucial pieces of exculpatory evidence or key context to frame the left in the worst possible way. He shouldn't have a platform to continue to spout his nonsense, but he does. Regardless, we shouldn't give him the time of day. If he's the only one talking about it, you can nearly be sure it's missing context or is simply not true. Years of crying wolf should have consequences.
I'm a little confused by this - is there a citation to where you're getting this information? And what do you mean they're hoping they're amenable to having those votes bought? Are you insinuating this is how it currently works and they want more people to partake? If so, would be interested in seeing the source of this info.
So the Special Counsel's report got filtered by a political appointee. One that happens to be hand picked by the subject of the special counsel's investigation. Totally makes sense, very believable. I think we should all just move on and trust that this political appointee is telling us all the truth. And republicans should give a standing ovation to Trump because obviously is wasn't his and his colleagues actions that triggered the investigation - it was just crazy democrats. And Trump conducted himself like a true gentleman this entire time, despite the constant despicable attacks by those saying lets just let the investigation occur without interference and lets see the results unfiltered. Basically sums up the last few days right?


Nigerian Musician Lamboginny Proposes To Girlfriend.
p style="box-sizing:border-box; margin:0px0px2.6rem; padding:0px; color:rgb(0,0,0); font-family:raleway,sans-serif; font-size:16px; font-style:normal; font-variant-ligatures:normal; font-variant-caps:normal; font-weight:300; letter-spacing:normal; orphans:2text-indent:0px; text-transform:none; white-space:normal; widows:2; word-spacing:0px; -webkit-text-stroke-width:0px; background-color:rgb(255,255,255); text-decoration-style:initial; text-decoration-color:initial; text-align:left;">Nigerian musician Lamboginny proposed to his girlfriend Taccara Rae and he revealed she said "yes" on his instagram page.
Lamboginny a former prison officer turned musician has done several records with the biggest music names in Nigeria, his latest single being Gbaladun, a song which features Olamide. His girlfriend who is is a blogger, actress, and model featured in his video for his"Give Me Love"single.


text:
Improving health and lives: the learning disabilities public health observatory
Eric Emerson, Gyles Glover, Sue Turner, Rob Greig, Chris Hatton, Susie Baines, Alison Copeland, Felicity Evison, Hazel Roberts, Janet Robertson, Victoria Welch
Centre For Excellence For Children'S Care And Protection
The purpose of this paper is to describe the first 15 months of operation of an innovative specialist national public health observatory for intellectual disability.
The paper provides a narrative account of aims and achievements of the service.
Findings – In the first 15 months of operation the observatory has: made available to those involved in commissioning health and social care services, a wealth of information on the health needs of people with intellectual disabilities; identified specific improvements that could viably be made to increase the quality of future information; and begun working with local agencies to support them in making the best use of the available information. People with intellectual disabilities experience significant health inequalities. This paper describes an innovative approach to helping local agencies make the best use of available information in order to commission services that may reduce these inequalities.
Advances in Mental Health and Intellectual Disabilities
Dive into the research topics of 'Improving health and lives: the learning disabilities public health observatory'. Together they form a unique fingerprint.
Learning Disabilities Medicine & Life Sciences 100%
Disabled Persons Medicine & Life Sciences 60%
Public Health Medicine & Life Sciences 57%
Social Work Medicine & Life Sciences 30%
Delivery of Health Care Medicine & Life Sciences 16%
Emerson, E., Glover, G., Turner, S., Greig, R., Hatton, C., Baines, S., Copeland, A., Evison, F., Roberts, H., Robertson, J., & Welch, V. (2012). Improving health and lives: the learning disabilities public health observatory. Advances in Mental Health and Intellectual Disabilities, 6(1), 26 - 32. https://doi.org/10.1108/20441281211198835
Emerson, Eric ; Glover, Gyles ; Turner, Sue et al. / Improving health and lives : the learning disabilities public health observatory. In: Advances in Mental Health and Intellectual Disabilities. 2012 ; Vol. 6, No. 1. pp. 26 - 32.
@article{e2601d658a4f46a08f97c3b89d17383e,
title = "Improving health and lives: the learning disabilities public health observatory",
abstract = "The purpose of this paper is to describe the first 15 months of operation of an innovative specialist national public health observatory for intellectual disability. The paper provides a narrative account of aims and achievements of the service. Findings – In the first 15 months of operation the observatory has: made available to those involved in commissioning health and social care services, a wealth of information on the health needs of people with intellectual disabilities; identified specific improvements that could viably be made to increase the quality of future information; and begun working with local agencies to support them in making the best use of the available information. People with intellectual disabilities experience significant health inequalities. This paper describes an innovative approach to helping local agencies make the best use of available information in order to commission services that may reduce these inequalities.",
keywords = "learning disabilities , public health, National Health Service , personal health ",
author = "Eric Emerson and Gyles Glover and Sue Turner and Rob Greig and Chris Hatton and Susie Baines and Alison Copeland and Felicity Evison and Hazel Roberts and Janet Robertson and Victoria Welch",
journal = "Advances in Mental Health and Intellectual Disabilities",
publisher = "Emerald Publishing Limited",
Emerson, E, Glover, G, Turner, S, Greig, R, Hatton, C, Baines, S, Copeland, A, Evison, F, Roberts, H, Robertson, J & Welch, V 2012, 'Improving health and lives: the learning disabilities public health observatory', Advances in Mental Health and Intellectual Disabilities, vol. 6, no. 1, pp. 26 - 32. https://doi.org/10.1108/20441281211198835
Improving health and lives : the learning disabilities public health observatory. / Emerson, Eric; Glover, Gyles; Turner, Sue et al.
In: Advances in Mental Health and Intellectual Disabilities, Vol. 6, No. 1, 2012, p. 26 - 32.
T1 - Improving health and lives
T2 - the learning disabilities public health observatory
AU - Emerson, Eric
AU - Glover, Gyles
AU - Turner, Sue
AU - Greig, Rob
AU - Hatton, Chris
AU - Baines, Susie
AU - Copeland, Alison
AU - Evison, Felicity
AU - Roberts, Hazel
AU - Robertson, Janet
AU - Welch, Victoria
N2 - The purpose of this paper is to describe the first 15 months of operation of an innovative specialist national public health observatory for intellectual disability. The paper provides a narrative account of aims and achievements of the service. Findings – In the first 15 months of operation the observatory has: made available to those involved in commissioning health and social care services, a wealth of information on the health needs of people with intellectual disabilities; identified specific improvements that could viably be made to increase the quality of future information; and begun working with local agencies to support them in making the best use of the available information. People with intellectual disabilities experience significant health inequalities. This paper describes an innovative approach to helping local agencies make the best use of available information in order to commission services that may reduce these inequalities.
AB - The purpose of this paper is to describe the first 15 months of operation of an innovative specialist national public health observatory for intellectual disability. The paper provides a narrative account of aims and achievements of the service. Findings – In the first 15 months of operation the observatory has: made available to those involved in commissioning health and social care services, a wealth of information on the health needs of people with intellectual disabilities; identified specific improvements that could viably be made to increase the quality of future information; and begun working with local agencies to support them in making the best use of the available information. People with intellectual disabilities experience significant health inequalities. This paper describes an innovative approach to helping local agencies make the best use of available information in order to commission services that may reduce these inequalities.
KW - learning disabilities
KW - public health
KW - National Health Service
KW - personal health
JO - Advances in Mental Health and Intellectual Disabilities
JF - Advances in Mental Health and Intellectual Disabilities
Emerson E, Glover G, Turner S, Greig R, Hatton C, Baines S et al. Improving health and lives: the learning disabilities public health observatory. Advances in Mental Health and Intellectual Disabilities. 2012;6(1):26 - 32. doi: 10.1108/20441281211198835

redpajama_set_name: RedPajamaCommonCrawl


text:
Team 7315 - Giga Knights (2022)
Deerfield Windsor School
From: Albany, Georgia, USA
District: Peachtree
On The Web: http://www.deerfieldwindsor.com/
2023: CHARGED UP℠ pr...
2022 Sponsors: Deerfield Windsor School
Qualification 49 20 (41%) 28 (57%) 1 (2%)
Playoff 14 9 (64%) 5 (36%) 0
Total 63 29 (46%) 33 (52%) 1 (2%)
GACOL
GAALB
GACMP
CMPTX
District Ranking
District Championship Winner
FIRST Championship Qualifying Award Peachtree District State Championship
PCH District Columbus Event
March 17 to March 19, 2022
This event counted towards Team 7315's District Ranking Points for 2022. A summary of district rankings points earned specifically at this event is shown here.
Qualification 3 Fri 3/18 - 11:29 AM 4188 7315 4701 3091 6905 8100 50 0
Qualification 6 Fri 3/18 - 11:58 AM 5632 3091 4701 7315 3635 6905 19 23
Qualification 9 Fri 3/18 - 12:28 PM 1648 4701 8865 3344 8849 7315 28 0
Qualification 13 Fri 3/18 - 1:01 PM 3635 5293 1261 2415 5828 7315 23 6
Qualification 19 Fri 3/18 - 2:34 PM 8849 6910 5828 3635 8815 7315 0 38
Qualification 22 Fri 3/18 - 3:04 PM 6925 6919 7315 4701 3635 8100 23 40
Qualification 54 Sat 3/19 - 9:50 AM 4509 1648 2415 8865 7315 3344 20 11
Qualification 58 Sat 3/19 - 10:23 AM 2415 1261 1414 7315 4509 4112 53 15
View Playoffs Event Page
Record: 0 Wins and 2 Losses
Alliance Selection: Round 2 of Alliance 6
Quarterfinal 4 4701 3635 8865 6910 8100 7315 39 16
Quarterfinal 8 4701 3635 8865 6910 8100 7315 36 6
PCH District Albany Event
Rank: 8 of 26
Qualification 5 Fri 4/1 - 11:41 AM 1002 3344 3635 4701 8815 7315 50 37
Qualification 8 Fri 4/1 - 12:04 PM 6471 7315 6829 7104 5632 6925 43 26
Qualification 12 Fri 4/1 - 12:52 PM 7470 7315 1002 1795 7538 3329 32 24
Qualification 41 Sat 4/2 - 9:37 AM 7104 3329 6919 7315 3091 5219 66 5
Alliance Selection: Captain of Alliance 5
Quarterfinal Tiebreaker 2 3329 7470 8815 5632 7315 5828 47 56
Semifinal 1 6829 1414 1002 5632 7315 5828 40 45
Semifinal Tiebreaker 1 6829 1414 1002 5632 7315 5828 69 46
Peachtree District State Championship
April 7 to April 9, 2022
18 3 90 0 111
Qualification 2 Fri 4/8 - 9:27 AM 7451 6705 1746 4701 832 7315 54 41
Qualification 16 Fri 4/8 - 11:19 AM 1771 1261 8083 4188 7315 6919 115 63
Qualification 38 Fri 4/8 - 3:09 PM 7315 1414 8080 6340 8866 2974 51 101
Qualification 45 Fri 4/8 - 4:04 PM 8736 6340 1414 7451 8849 7315 126 22
Event Champions!
Quarterfinal 1 Sat 4/9 - 12:34 PM 2974 1771 7315 6341 832 5632 124 73
Quarterfinal 5 2974 1771 7315 6341 832 5632 116 60
Semifinal 1 2974 1771 7315 6829 1746 1648 124 88
Final 1 2974 1771 7315 4188 1261 8080 120 88
FIRST Championship - FIRST Robotics Competition - Hopper Division
Qualification 12 Thu 4/21 - 10:00 AM 3478 148 11 7315 5454 624 123 123
Qualification 20 Thu 4/21 - 10:55 AM 3794 1706 3322 4213 8745 7315 137 64
Qualification 29 Thu 4/21 - 12:04 PM 7848 8701 2832 2202 7315 5806 77 57
Qualification 39 Thu 4/21 - 2:37 PM 2614 7315 8575 2202 2052 6135 71 121
Qualification 55 Thu 4/21 - 4:35 PM 302 1678 7315 1156 8575 488 139 68
Qualification 67 Fri 4/22 - 8:36 AM 7315 4926 1807 2502 8159 177 83 94
Qualification 78 Fri 4/22 - 9:55 AM 2443 3538 2905 7315 2883 8037 84 64
Qualification 88 Fri 4/22 - 11:05 AM 2052 180 7315 5505 7021 1111 110 91
Qualification 112 Fri 4/22 - 3:02 PM 2905 7315 4522 6420 597 8822 52 94
Qualification 121 Fri 4/22 - 4:08 PM 5817 4063 829 1574 1625 7315 99 91
FIRST Championship - FIRST Robotics Competition
No Matches Yet This team is not yet listed in a Qualification or Playoff schedule at this event.
District teams are ranked against all teams in their district and the final rankings, once all district events have been played, are used to determine eligibility for District Championships and/or the FIRST Championship. More Info...
Total Points: 167
Rank: 15 of 101 teams in the Peachtree District (Full Rankings )
Qualified for District Championship: Yes (DistrictPoints)
Qualified for FIRST Championship: Yes (DCMPEventWinner)
Team 7315 had the following district points activities in the 2022 season:
Event/Activity
PCH District Columbus Event 12 6 0 0 -- 18
PCH District Albany Event 16 12 10 0 -- 38
Peachtree District State Championship 18 3 90 0 -- 111
If Team 7315 is tied with any other Peachtree teams in total District Ranking Points, the tie will be broken using the following criteria:
Level 1 Total Playoff Round Performance Points 100
Level 2 Max Playoff Round Performance Points 90
Level 3 Total Alliance Selection Points 21
Level 4 Max Alliance Selection Points 12
Level 5 Total Qualification Points 46
Level 6 Highest Match Score 126
Level 7 Second Highest Match Score 124
Level 8 Third Highest Match Score 124
Level 9 Random Sort ---


The Moscow Region Government sees great potential in implementing projects in the area of goat milk production in the region, said Andrei Razin, Minister of Agriculture and Food of the Moscow Region, in the Government House of the Moscow Region at the V International Agricultural Dairy Forum.
"According to our estimates, at least plus 100 thousand heads of goats is definitely the line that the market will swallow and will not even notice. Thus, the livestock of goats in the Moscow region can grow at least twice," said Andrei Razin.
According to the minister, goat milk is hypoallergenic, a number of breeds of these animals give milk, which differs little in taste and smell from cow milk.
"Goat milk production projects are also beneficial because goats require less space and feed than cattle," said Andrei Razin.
Earlier, Andrei Razin reported that at the beginning of October 2018, the goat livestock in the Moscow region amounted to 65.2 thousand heads, of which over 34.5 thousand heads (52.5%) are in private subsidiary farms.


La prochaine séance du Séminaire Epistémè aura lieu à la Maison de la Recherchede la Sorbonne Nouvelle, 4 rue des Irlandais, Paris 5e, le lundi 8 avril à 17h00. Il prendra cette fois la forme d'une conférence-concert.
La séance sera suivie d'un pot.
While the relation between dress and gender, disguise and identity-building, and the importance of the numerous sumptuary laws in the shaping of social identity has been largely explored, much less attention has been devoted to the relation between dress and the ecological environments for which dress was devised. Whether worn by the poor, the middling sort, or the nobility, clothes need to be looked at not only in the relation to broad social, cultural, and material contexts, but also in relation to climactic or geographic environments. Because clothes protect the human body and serve as an interface between the body and the environment, dress can be considered as the most immediate locus for the establishment of any sort of ecology, in its etymological sense of a "discourse" or "science" of the oikos, that is of the home, or human habitat. From their production down to the way they are worn, clothes interweave natural materials and artifice, the human body and the social body, the weather conditions and the culture in which they are born and those to which they adapt. They come to materialize and epitomize identity in its various inclinations and inflections. Conversely, they participate in shaping the environments or the landscapes for whose diversity they stand metonymically.
In staging climes through costumes, Shakespeare and his contemporaries invite us to decentre our perspective by, first, looking beyond clothes as an object and clothes as a means of fashioning personal identity and personae (or impersonations) on stage, toward clothes as a privileged space of "eco-logy," and second, by adopting an anthropological gaze on early modern English dress and culture. It is through a confrontation with foreign dress, that is with the materials of difference, that English identity can be better gauged. Ultimately, this conference aims at exploring how dramatic text and textile enrich each other in the early modern period, and how dress and costume are essential in England's attempt to define its own cultural identity within a new global space inclusive of many different climes reflected on stage.
We are seeking proposals that inquire into the complex ecology, economy and anthropology of dress, drawing notably on the material history of concrete elements such as pigments, dies, and raw materials (sometimes imported from distant regions and climes) used to make clothing and costumes. We also invite papers with more literary approaches that look at the ways in which dress on stage becomes a means to negotiate the self or same in relation to the other or embodies contemporary understandings of climes and the environment. Proposals may focus on a specific costume or a specific dramatic corpus by Shakespeare or one of his contemporaries. Comparative approaches, drawing on European and global materials and practices, are also encouraged.
The conference will include outreach activities, such as workshops and round-tables open to the general public. We welcome proposals in English from established scholars, doctoral students, curators and other professionals working on or with early modern dress and more contemporary costumes representing that period.
Vendredi 16 novembre 15h30-17h30, Réunion de rentrée, Institut du Monde Anglophone, salle 15. 17h30-19h30, Séminaire Epistémè / PEMS. Avec Anne-Valérie Dulac (Sorbonne Université) et Céline Cachaud (Diplômée de l'Ecole du Louvre et de l'EPHE) : 'Nicholas Hilliard en 2019. Autour du numéro 35 d'Etudes Epistémè (printemps 2019)'. Institut du Monde Anglophone, salle 16.
11 juin 2018, 14h-18h, Institut du Monde Anglophone, salle 16.
Les deux ateliers pratiques organisés par PEARL le 11 juin prochain seront centrés sur la matérialité de l'épistolaire et des correspondances de la période moderne (XVIe-XVIIIe). Etudiants et enseignants-chercheurs seront initiés à la paléographie de la période et aux différentes techniques pratiquées pour garantir à la fois la confidentialité et l'authenticité des lettres.
Jana Dambrogio is Thomas F. Peterson (1957) Conservator at MIT Libraries. She has held positions at the National Archives, United Nations, and Vatican Secret Archives, is a recipient of a Booth Family Rome Prize Fellowship in historic preservation and conservation, and was recently elected a member of the Grolier Club. She coined the term letterlocking in 2009 to describe the systems of deliberate folds, slits, locks, and seals that build security, privacy, and authentication enhancements into letters. Dambrogio's specialization is developing tools and treatment techniques to conserve material culture and the secrets they contain.
Daniel Starza Smith is a lecturer in early modern English Literature at King's College London. He is author of John Donne and the Conway Papers (OUP, 2014), and co-editor of Manuscript Miscellanies in Early Modern England (Ashgate, 2014). He most recently published on a newly discovered John Donne manuscript at Westminster Abbey (https://doi.org/10.1093/res/hgx135).
Dambrogio and Smith are co-founders of the Unlocking History research group, co-editors of the Dictionary of Letterlocking and letterlocking.org, and both work on the international project Signed, Sealed, & Undelivered (http://brienne.org/).
Workshop participants will unlock models of various historical locked letters, then learn to make their own. We will discuss these letters in terms of their security and aesthetic features, and consider them alongside images of real archival originals by historic figures such as John Donne and Elizabeth I.
Travelling players scoured Europe in the late sixteenth and early seventeenth centuries. Accounts of the visits of the English actors show that their performances "attracted great crowds" of young and old, men and women, city fathers and "educated professionals." Henslowe, entrepreneur and purveyor of costumes and stage properties to the London theatre, apparently numbered the continental troupes among his clients. The traveling players employed not only English actors, but foreign comedians as well, and thus fostered a theatre that was multi-lingual and what we today term transnational. Texts too were mobile. Long before "global Shakespeare," before, in fact, the First Folio saw print, booksellers were peddling their intellectual property in Shakespeare internationally. Early advertisements of the First Folio offered for sale to a European market at the Frankfort Book Fair, evidence of the visits of English players to the Continent, and the presence of Shakespeare in various libraries suggest that the continental presence of Shakespeare and early modern English drama has been under-estimated and undervalued.
Karen Newman is Owen Walker '33 Professor of Humanities and Professor of Comparative Literature and English at Brown University. She has written widely on early modern English and continental letters and culture and on Shakespeare and Renaissance drama. Books include Fashioning Femininity and English Renaissance Drama; Fetal Positions: Individualism, Science, Visuality; Cultural Capitals: Early Modern London and Paris and Essaying Shakespeare. Recent collections include Early Modern Cultures of Translation, co-edited with Jane Tylus, and This Distracted Globe: Worldmaking in Early Modern Literature, edited with Jonathan Goldberg and Marcie Frank. She is currently working on early modern translation and on the reception of Shakespeare in Europe.


text:
\section{Introduction}

\input{intro}

\subsection{Related work}\label{related-work}

The full-information model (also known as the \emph{perfect information model}) was introduced by Ben-Or and Linial \cite{BL85}
to study the problem of collective coin-flipping when no secret communication is possible between honest players.

\paragraph{In the static setting.}
Protocols for collective coin-flipping in the presence of static corruptions have been constructed in a series of works
that variously focus on improving the fault-tolerance, round complexity, and/or bias of the output bit.
Feige \cite{Fei99} gave a protocol that is $(\delta^{1.65}/2)$-secure\footnote{A
coin-flipping protocol is $\varepsilon$-secure against~$t$ static corruptions if for any static adversary that corrupts up to~$t$ parties,
the probability that the protocol outputs~$0$ is at least $\varepsilon$.}
in the presence of $t=(1+\delta)\cdot n/2$ static corruptions for any constant $0<\delta<1$.
Russell, Saks, and Zuckerman \cite{RSZ02} then showed that any protocol that is secure in the presence of linearly many corruptions must
either have at least $(1/2-o(1))\cdot\log^*(n)$ rounds, or communicate many bits per round.

Interestingly, nearly all proposed \emph{multi-round} protocols for collective coin-flipping
first run a \emph{leader election} protocol in which one of the $n$ players is selected as a ``leader'', who then outputs a bit that is taken as the protocol outcome.
We remark that this approach is inherently unsuitable for adaptive adversaries, which can always corrupt the leader after he is elected,
and thereby surely control the protocol outcome.

\paragraph{In the adaptive setting.}
The study of coin-flipping protocols has been predominantly in the static setting.
The problem of adaptively secure coin-flipping was introduced by Ben-Or and Linial \cite{BL85}
and further examined by Lichtenstein, Linial, and Saks \cite{LLS89} as described in the previous section.
In addition, Dodis \cite{Dod00} proved that through ``black-box'' reductions from \emph{non-adaptive} coin-flipping, it is not possible tolerate
significantly more corruptions than the majority protocol. The definition of ``black-box'' used in \cite{Dod00}
is rather restricted: it only considers sequential composition of non-adaptive coin-flipping protocols, followed by a (non-interactive)
function computation on the coin-flips thus obtained.

\paragraph{In the pairwise-channels setting.}
An adversarial model bearing some resemblance to our strong adaptive adversary model was introduced and analyzed by Hirt and Zikas \cite{HZ10}
in the \emph{pairwise communication channels} model, rather than the full-information model. In their model, the adversary can corrupt a party $P$ based on
some of the messages that $P$ sends within a round, then the adversary controls the rest of $P$'s messages in that round (and for future rounds).
Unlike in our strong adaptive model, the adversary of \cite{HZ10} cannot ``see inside all players' heads''
and overwrite arbitrary honest messages based on their content before they are sent.

Interestingly, a separation has been shown between standard adaptive adversaries and the stronger adversaries of Hirt and Zikas:
\cite{HZ10} shows that broadcast is impossible to achieve for $t>n/2$ corruptions in their stronger adversarial model,
whereas Garay et al. \cite{GKKZ11} showed that broadcast is achievable for any $t<n$ corruptions in the standard adaptive adversarial model.

\paragraph{In the computational setting.}
The problem of generating a shared random bit has also been studied in the setting where players are computationally bounded,
and in different communication network models. Blum \cite{Blu81} introduced the coin-flipping problem in the two-player computational setting; and
Goldreich, Micali, and Wigderson \cite{GMW87} subsequently showed that it is possible to efficiently generate a shared bit with negligible bias, in the presence of static adversaries.

Another line of work shows that the existence of any coin-flipping protocol for computationally bounded players which achieves a sufficiently small bias
implies the existence of one-way functions. The latest result in this line of work, due to Berman, Haitner, and Tentes \cite{BHT14}, proves
that if there exists a two-player coin-flipping protocol that achieves any constant bias, then one-way functions exist.

\section{Preliminaries}

We consider coin-flipping protocols in the \emph{full-information model} (also known as the \emph{perfect information model}),
where $n$ computationally unbounded players communicate via a single broadcast channel.
The network is synchronized between rounds, but is asynchronized within each round (that is, there is no guarantee on message ordering within a round,
and an adversary can see the messages of all honest players in a round before deciding his own messages).

In this work, we focus on \emph{one-round} protocols, and we consider protocols that terminate (and produce an output) with probability 1.
In particular, we focus on \emph{coin-flipping} protocols, which are defined as follows.

\begin{definition}[Coin-flipping protocol]\label{def:coinFlipping}
A \emph{coin-flipping protocol} $\Pi=\{\Pi_n\}_{n\in\mathbb{N}}$ is a family of protocols
where each $\Pi_n$ is a $n$-player protocol which outputs a bit in $\{0,1\}$.
\end{definition}

\paragraph{Notation.}
We write $\overset{s}{\approx}$ for statistical indistinguishability of distributions.
We denote by $\Pr^\Pi(b)$ the probability that an honest execution of $\Pi$ will lead to the outcome $b\in\{0,1\}$.
We denote by $\Pr^{\Pi,\mathcal{A}}(b)$ the probability that an execution of $\Pi$ in the presence of an
adversary $\mathcal{A}$ will lead to the outcome $b\in\{0,1\}$.
The probability is over the random coins of the honest players and the adversary.

For one-round protocols, we write $\Pi_n(r_1,\dots,r_n)$
to denote the outcome of the protocol $\Pi_n$ when each player $i$ sends message $r_i$.
(The vector $(r_1,\dots,r_n)$ is a protocol \emph{transcript}.)

\subsection{Properties of protocols}

\begin{definition}[Symmetric protocol]
A protocol $\Pi$ is \emph{symmetric} if the outcome of a protocol execution is the same
no matter how the messages within each round are permuted. In particular, a one-round protocol $\Pi$ is symmetric
if for all $n\in\mathbb{N}$ and any permutation $\pi\in[n]\rightarrow[n]$,
$$\Pi_n(r_{1},\dots,r_{n})=\Pi_n(r_{\pi(1)},\dots,r_{\pi(n)}).$$
\end{definition}

We remark, for completeness, that in the multi-round case,
the outcome of a symmetric protocol should be unchanged even if different permutations are applied in different rounds.

\begin{definition}[Single-bit/multi-bit protocol]
A protocol is \emph{single-bit} if each player sends at most one bit over the course of the protocol execution.
Similarly, a protocol is \emph{$m$-bit} if each player sends at most $m$ bits over the course of the protocol execution.
More generally, a protocol which is not single-bit is called \emph{multi-bit}.
\end{definition}

\begin{definition}[Public-coin protocol]
A protocol is \emph{public-coin} if each honest player broadcasts all of the randomness he generates
(i.e. his ``local coin-flips''), and does not send any other messages.
\end{definition}

\subsection{Adversarial models in the literature}\label{sec:advModels}

The type of adversary that has been by far the most extensively studied in the coin-flipping literature
is the static adversary, which chooses a subset of players to corrupt \emph{before} the protocol execution begins,
and controls the behavior of the corrupt players arbitrarily throughout the protocol execution.

A stronger type of adversary is the \emph{adaptive} adversary, which may choose players to corrupt at any point during protocol execution,
and controls the behavior of the corrupt players arbitrarily from the moment of corruption until protocol termination.

\iffalse
\begin{definition}[Static rushing adversary]
The adversary chooses all the players that he wishes to corrupt, before the protocol starts.
Then, within each round, the adversary sees the messages of all honest players before
deciding the messages of the corrupt players.
\end{definition}
\fi

\begin{definition}[Adaptive adversary]
Within each round, the adversary chooses players one-by-one to send their messages;
and he can perform corruptions at any point during this process.
\end{definition}

\subsection{Security of coin-flipping protocols}

The security of a coin-flipping protocol is usually measured by the extent to which an adversary can, by corrupting a subset of parties,
bias the protocol outcome towards his desired bit.

\begin{definition}[$\varepsilon$-security]
A coin-flipping protocol $\Pi$ is \emph{$\varepsilon$-secure} against $t=t(n)$ adaptive (or static or strong adaptive)
corruptions if for all $n\in\mathbb{N}$, it holds that
for any adaptive (resp. static or strong adaptive) adversary $\mathcal{A}$ that corrupts at most $t=t(n)$ players,
$$\min\left({\Pr}^{\Pi_n,\mathcal{A}}(0),{\Pr}^{\Pi_n,\mathcal{A}}(1)\right)\geq\varepsilon.$$
\end{definition}

We remark that this definition of $\varepsilon$-security is sometimes referred to as \emph{$\varepsilon$-control} or \emph{$\varepsilon$-resilience} in other works.
We next define a \emph{secure} protocol to be one
with ``minimal'' security properties (that is, one where the adversary does not almost always get the outcome he wants).

\begin{definition}[Security]
A coin-flipping protocol is \emph{secure} against $t=t(n)$ corruptions
if it is $\varepsilon$-secure against $t$ corruptions for some constant $0<\varepsilon<1$.
\end{definition}

In this work, we investigate the maximum proportion of adaptive corruptions that can be tolerated by \emph{any} secure protocol.

\section{Our results}

\subsection{Strongly adaptive adversaries}

In this work, we propose a new, stronger adversarial model than those that have been studied thus far (see section \ref{sec:advModels}),
in which the adversary can see all honest players' messages within any given round,
and \emph{subsequently} decide which players to corrupt. That is, he can see all the messages that the honest players ``would have sent'' in a round,
and then selectively intercept and alter these messages.

\begin{definition}[Strong adaptive adversary]
Within each round, the adversary sees all the messages that honest players would have sent,
then gets to choose which (if any) of those messages to corrupt (i.e. replace with messages of his choice).
\end{definition}

This notion is an essential tool underlying the proof techniques in our work.
Moreover, we believe that the notion of \emph{strong adaptive security} gives rise to a natural and interesting new adversarial model
in which to study multi-party protocols, which is of independent interest beyond the scope of this work.

\subsection{Corruption tolerance in secure coin-flipping protocols}

Our main contributions consist of the following three results.
These can be viewed as partial progress towards proving the 30-year-old conjecture of \cite{BL85}.

\begin{theorem}\label{thm:main}
Any one-round coin-flipping protocol $\Pi$ can be secure against at most $t=\widetilde{O}(\sqrt n)$ strong adaptive corruptions.
\end{theorem}

\begin{theorem}\label{thm:minmax}
For any symmetric one-round coin-flipping protocol $\Pi$ secure against $t=t(n)$ adaptive corruptions,
there is a symmetric one-round coin-flipping protocol $\Pi'$ secure against $\Omega(t)$ strong adaptive corruptions.
\end{theorem}

\begin{corollary}
Any symmetric one-round coin-flipping protocol $\Pi$ can be secure against at most $t=\widetilde{O}(\sqrt n)$ adaptive corruptions.
\end{corollary}
\iffalse
\begin{proof}
Suppose, for contradiction, that there exists a symmetric one-round coin-flipping protocol $\Pi$ which is secure against more than $\widetilde{O}(\sqrt n)$
adaptive corruptions. Then, by Theorem \ref{thm:minmax}, there exists a one-round coin-flipping protocol $\Pi'$ which is secure
against more than $\widetilde{O}(\sqrt n)$ \emph{strong} adaptive corruptions. This contradicts Theorem \ref{thm:main}.
\end{proof}
\fi

In the next sections, we proceed to give detailed proofs of the theorems.

\subsection{Proof of Theorem \ref{thm:main}}

We begin by recalling the result of Lichtenstein et al. \cite{LLS89} which proves that the maximum number of adaptive corruptions for any secure \emph{single-bit}
coin-flipping protocol is $O(\sqrt n)$. Note that the \emph{majority protocol} is the one-round protocol in which each player broadcasts a random bit,
and the majority of broadcasted bits is taken to be the protocol outcome.

\begin{theorem}[\cite{LLS89}]\label{thm:lls}
Any coin-flipping protocol in which each player broadcasts at most one bit can be secure against at most $t=O(\sqrt n)$ corruptions.
Moreover, the majority protocol achieves this bound.
\end{theorem}

Next, we establish some definitions and supporting lemmas.

\newcommand{{\sf dist}}{{\sf dist}}
\begin{definition}[Distance between message-vectors]
For vectors $\vec{r},\vec{r'}\in{\cal M}^n$, let ${\sf dist}(\vec{r},\vec{r'})$ be equal to the number of coordinates $i\in[n]$
for which $r_i\neq r'_i$.
\end{definition}

\newcommand{{\sf Robust}}{{\sf Robust}}
\begin{definition}[Robust sets]
Let $\Pi$ be a one-round coin-flipping protocol in which each player sends a message from a message space ${\cal M}$.
For any $n\in\mathbb{N}$ and $b\in\{0,1\}$, define the set ${\sf Robust}^{\Pi_n}(b,t)$ as follows:
$${\sf Robust}^{\Pi_n}(b,t)=\left\{\vec{r}\in{\cal M}^n ~:~ \forall\vec{r'}\in{\cal M}^n \mbox{ s.t. } {\sf dist}(\vec{r},\vec{r'})\leq t, ~\Pi_n(\vec{r})=\Pi_n(\vec{r'})=b\right\}.$$
\end{definition}

\begin{lemma}\label{lem:robustSets}
Let $\Pi$ be a one-round coin-flipping protocol in which each player sends a random message from a message space ${\cal M}$.
$\Pi$ is secure against $t=t(n)$ strong adaptive corruptions if and only if
there exists a constant $0<\varepsilon<1$ such that for all $n\in\mathbb{N}$ and each $b\in\{0,1\}$,
$$\Pr_{\vec{r}\leftarrow{\cal M}}\left[\vec{r}\in{\sf Robust}^{\Pi_n}(b,t)\right]\geq\varepsilon.$$
\end{lemma}
\begin{proof}
{\sc (``if'')}
Suppose that there exists a constant $0<\varepsilon<1$ such that for all $n\in\mathbb{N}$ and all $b\in\{0,1\}$, it holds that
\begin{align}\label{eqn:probGoodConstant}
\Pr_{\vec{r}\leftarrow{\cal M}^n}\left[\vec{r}\in{\sf Robust}^{\Pi_n}(b,t)\right]\geq\varepsilon.
\end{align}

Let $\mathcal{A}$ be any strong adaptive adversary making up to $t$ corruptions.
For $n$-vector of (honest) messages $\vec{r}\in{\cal M}^n$, let $\mathcal{A}(\vec{r})\in{\cal M}^n$ denote the
corresponding corrupted message-vector, where up to $t$ of the messages have been modified by $\mathcal{A}$.
By the definition of the set ${\sf Robust}^{\Pi_n}(b,t)$,
it holds that
\begin{align}\label{eqn:advLoses}
\Pr_{\vec{r}\leftarrow{\cal M}^n}\left[\Pi_n(\mathcal{A}(\vec{r}))=b ~|~ \vec{r}\in{\sf Robust}^{\Pi_n}(b,t)\right]=1.
\end{align}

Combining equations \eqref{eqn:probGoodConstant} and \eqref{eqn:advLoses}, it follows that for each outcome $b\in\{0,1\}$,
$$
\Pr_{\vec{r}\leftarrow{\cal M}^n}\left[\Pi_n(\mathcal{A}\left(\vec{r})\right)=b\right]\geq\varepsilon.
$$

We have shown that for each $b\in\{0,1\}$, $\Pr^{\Pi,\mathcal{A}}(b)\geq\varepsilon$, as required.

{\sc (``only if'')}
Suppose, on the other hand, that there is no constant $0<\varepsilon<1$ such that for all $b\in\{0,1\}$, it holds that
$\Pr_{\vec{r}\leftarrow{\cal M}^n}\left[\vec{r}\in{\sf Robust}^{\Pi_n}(b,t)\right]=\varepsilon$.
That is, there exists some $\varepsilon'=o(1)$ such that for some $b\in\{0,1\}$ and infinitely many values of $n\in\mathbb{N}$, it holds that
\begin{align}\label{eqn:fewGoods}
\Pr_{\vec{r}\leftarrow{\cal M}^n}\left[\vec{r}\in{\sf Robust}^{\Pi_n}(b,t)\right]\leq \varepsilon'.
\end{align}
Without loss of generality, let $b=0$ be the bit for which equation \eqref{eqn:fewGoods} holds.
By the definition of ${\sf Robust}^{\Pi_n}(b,t)$, it holds that for any $\vec{r}\notin{\sf Robust}^{\Pi_n}(b,t)$,
there exists a vector $\vec{r}_{bad}\in{\cal M}^n$ such that ${\sf dist}(\vec{r},\vec{r}_{bad})\leq t$ and $\Pi_n(\vec{r})\neq\Pi_n(\vec{r}_{bad})$.
In other words, if the honest players' messages $\vec{r}$ do not fall in ${\sf Robust}^{\Pi_n}(0,t)$, then
it is possible for a strong adaptive adversary $\mathcal{A}$ to \emph{force} the outcome to be 1, by doing as follows:
$$\mathcal{A}(\vec{r}) = \begin{cases}
\vec{r} & \mbox{ if } \Pi_n(\vec{r})=1 \\
\vec{r}_{bad} & \mbox{ if } \Pi_n(\vec{r})=0
\end{cases}$$
Note that since ${\sf dist}(\vec{r},\vec{r}_{bad})\leq t$, it is always possible for the adversary to change from $\vec{r}$ to $\vec{r}_{bad}$
using $t$ or fewer corruptions. Moreover, if $\Pi_n(\vec{r})=0$, then it must be that $\Pi_n(\vec{r}_{bad})=1$, by construction of $\vec{r}_{bad}$.
Hence,
\begin{align}\label{eqn:advWins}
\Pr_{\vec{r}\leftarrow{\cal M}^n}\left[\Pi_n(\mathcal{A}(\vec{r}))=1 ~|~ \vec{r}\notin{\sf Robust}^{\Pi_n}(0,t)\right]=1.
\end{align}
Combining equations \eqref{eqn:fewGoods} and \eqref{eqn:advWins} (for $b=0$), we obtain:
$$
\Pr_{\vec{r}\leftarrow{\cal M}^n}\left[\Pi_n(\mathcal{A}(\vec{r}))=1\right] =
\Pr_{\vec{r}\leftarrow{\cal M}^n}\left[\vec{r}\notin{\sf Robust}^{\Pi_n}(0,t)\right] \geq 1-\varepsilon'.
$$
Hence, $\Pr^{\Pi,\mathcal{A}}(1)\geq 1-\varepsilon'$, and so $\Pr^{\Pi,\mathcal{A}}(0)\leq\varepsilon'=o(1)$.
Therefore, $\Pi$ is not secure against $t$ strong adaptive corruptions.
The lemma follows.
\end{proof}

Since players are computationally unbounded and we consider one-round protocols,
we may without loss of generality consider public-coin protocols\footnote{This is without loss of generality:
each player can simply send his random coin tosses, and security holds since we are in the full-information model.}:
for any one-round protocol $\Pi$ in the full-information model,
there is a protocol $\Pi'$ with an identical output distribution (in the presence of any adversary),
in which honest players send random messages in $\{0,1\}^k$ for some $k=\mathsf{poly}(n)$.

The following lemma serves as a stepping-stone to our final theorem.
\iffalse
It shows that any coin-flipping protocol secure against $t=t(n)$ strong adaptive corruptions
can be converted to a protocol in which players' messages are of polylogarithmic length, and which tolerates the same number $t$ of strong adaptive corruptions.
We remark that the communication-reduction technique of Lemma \ref{lem:manyBitsMatrix} is by no means specific to coin-flipping protocols,
and we consider the lemma to be of some independent interest towards characterizing how much communication is needed in strongly adaptively secure protocols in general.
\fi

\iffalse
Before the lemma, we recall the statement of the Chernoff bound.

\begin{theorem}[Chernoff bound]
Let $X_1,\dots,X_n$ be independent random variables taking values in $\{0,1\}$, which all have the same expectation $\mu=\mathop{{}\mathbb{E}}[X_i]$. Then, for every $0<\varepsilon<1$,
$$\Pr\left[\left|\frac{1}{n}\cdot\sum_{i\in[n]} X_i-\mu\right|\geq\varepsilon\right]\leq 2e^{-2n\varepsilon^2}.$$
\end{theorem}
\fi

\begin{lemma}\label{lem:manyBitsMatrix}
For any one-round multi-bit coin-flipping protocol $\Pi$ secure against $t=t(n)$ strong adaptive corruptions, and any constant $\delta>0$,
there is a one-round $\ell$-bit coin-flipping protocol $\Pi'$ that is secure against $t$ strong adaptive corruptions,
where $\ell=O(\log^{1+\delta}(n))$.
\end{lemma}
\begin{proof}
Without loss of generality, we consider only public-coin protocols,
and assume that each player sends a message of the same length (say, $k=k(n)$ bits).
Let $\delta>0$ be any constant, let $\ell=O(\log^{1+\delta}(n))$, and let $\ell'=2^{\ell}$.

For an $\ell'\times n$ matrix of messages $M\in(\{0,1\}^k)^{\ell'\times n}$, we define the protocol $\Pi^M$ as follows:
each player $P_i$ broadcasts a random integer $a_i\leftarrow[\ell']$, and the protocol outcome is defined by
$$\Pi^M_n(a_1,\dots,a_n) = \Pi_n(M_{(a_1,1)},\dots,M_{(a_n,n)}),$$
where $M_{(i,j)}$ denotes the message at the $i^{th}$ row and $j^{th}$ column of the matrix $M$.
For notational convenience, define $\vec{M}(a_1,\dots,a_n)=(M_{(a_1,1)},\dots,M_{(a_n,n)})$.
Notice that by construction of the protocol $\Pi^M$, it holds that for any message-vector $\vec{a}\in[\ell']^n$,
\begin{align}\label{eqn:robustSetsSame}
\vec{M}(\vec{a})\in{\sf Robust}^{\Pi_n}(b,t) ~\Longrightarrow~ \vec{a}\in{\sf Robust}^{\Pi^M_n}(b,t).
\end{align}

Suppose each entry of the matrix $M$ is a uniformly random message in $\{0,1\}^k$.
Note that the length of each player's message in $\Pi^M$ is $\log(\ell')=\ell$.
We want to show that $\Pi^M$ is a secure coin-flipping protocol against $t$ strong adaptive corruptions, for some $M$.
By Lemma~\ref{lem:robustSets}, it is sufficient to show that there exists $M\in(\{0,1\}^k)^{\ell'\times n}$ such that for all $b\in\{0,1\}$,
\begin{align}\label{eqn:notReallyRandom1}
\Pr_{\vec{a}\leftarrow[\ell']^n}\left[\vec{a}\in{\sf Robust}^{\Pi^M_n}(b,t)\right]\geq\varepsilon,
\end{align}
where $0<\varepsilon<1$ is constant.
Using implication \eqref{eqn:robustSetsSame}, it actually suffices to prove:
\begin{align}\label{eqn:notReallyRandom2}
\exists M\in(\{0,1\}^k)^{\ell'\times n} \mbox{ s.t. } \forall b\in\{0,1\},~~
\Pr_{\vec{a}\leftarrow[\ell']^n}\left[\vec{M}(\vec{a})\in{\sf Robust}^{\Pi_n}(b,t)\right]\geq\varepsilon.
\end{align}

Suppose the matrix $M$ is chosen uniformly at random. Let $\vec{a}_1,\dots\vec{a}_n$ be sampled independently and uniformly from $[\ell']^n$.
Since, the number of matrix rows $\ell'=2^{O(\log^{1+\delta}(n))}$ is super-polynomial,
it is overwhelmingly likely that $\vec{a}_1,\dots\vec{a}_n$ will be composed of distinct elements in $[\ell']$.
That is, to be precise,
$$\Pr_{\vec{a}_1,\dots,\vec{a}_n}\left[\forall (i,j)\neq(i',j')\in[n]\times[n], ~ (\vec{a}_i)_j \neq (\vec{a}_{i'})_{j'}\right]\geq 1-\mathsf{negl}(n).$$
If $\vec{a}_1,\dots,\vec{a}_n$ are indeed composed of distinct elements, the message-vectors
$\vec{M}(\vec{a}_1),\dots,\vec{M}(\vec{a}_n)$ are independent random elements in $(\{0,1\}^k)^n$.
Thus,
\begin{align}\label{eqn:statIndist}
(\vec{M}(\vec{a}_1),\dots,\vec{M}(\vec{a}_n))\overset{s}{\approx}(\vec{r}_1,\dots,\vec{r}_n),
\end{align}
when $M$ is a random matrix in $(\{0,1\}^k)^{\ell'\times n}$,
the (short) message-vectors $\vec{a}_1,\dots,\vec{a}_n$ are random in $[\ell']^n$,
and the (long) message-vectors $\vec{r}_1,\dots,\vec{r}_n$ are random in $(\{0,1\}^k)^n$.

Since $\Pi$ is a secure coin-flipping protocol, there is a constant $0<\varepsilon'<1$ such that for all $n\in\mathbb{N}$ and $b\in\{0,1\}$ and $i\in[n]$,
$$\Pr_{\vec{r}_i}\left[\vec{r}_i\in{\sf Robust}^{\Pi_n}(b,t)\right]\geq\varepsilon'.$$
The rest of the proof follows from a series of Chernoff bounds.

For $i\in[n]$ and $b\in\{0,1\}$, let $Z_{i,b}$ be an indicator variable for the event that $\vec{r}_i\in{\sf Robust}^{\Pi_n}(b,t)$.
Since the $\vec{r}_i$ are independent, we apply a Chernoff bound to obtain the following (for all $b\in\{0,1\}$):
\begin{align}\label{eqn:chernoff}
\Pr_{\vec{r}_1,\dots,\vec{r}_n}\left[\frac{1}{n}\cdot\sum_{i\in[n]}Z_{i,b}<\varepsilon'- \varepsilon''\right]\leq \mathsf{negl}(n),
\end{align}
for any constant $0<\varepsilon''<\varepsilon'$.

Let $Y_{i,b}$ be an indicator variable for the event that $\vec{M}(\vec{a}_i)\in{\sf Robust}^{\Pi_n}(b,t)$.
It follows from \eqref{eqn:statIndist} and \eqref{eqn:chernoff} that with overwhelming probability over the choice of the random matrix $M$,
it holds for all $b\in\{0,1\}$ that
\begin{align}\label{eqn:chernoffSet}
\Pr_{\vec{a}_1,\dots,\vec{a}_n}\left[\frac{1}{n}\cdot\sum_{i\in[n]}Y_{i,b}<\varepsilon'- \varepsilon''\right]\leq \mathsf{negl}(n).
\end{align}

For $b\in\{0,1\}$, let $\alpha_b$ denote the probability $\Pr_{\vec{a}_i}\left[\vec{M}(\vec{a}_i)\in{\sf Robust}^{\Pi_n}(b,t)\right]$.
Note that for any given $b\in\{0,1\}$ the variables $Y_{i,b}$ are independently and identically distributed, each taking value 1 with probability $\alpha_b$
and value 0 with probability $1-\alpha_b$.
By a Chernoff bound, for any constant $0<\varepsilon'''<1$, it holds that (with overwhelming probability over the choice of $M$):
\begin{align}\label{eqn:realChernoff}
\Pr_{\vec{a}_1,\dots,\vec{a}_n}\left[\left|\frac{1}{n}\cdot\sum_{i\in[n]}Y_{i,b}-\alpha_b\right|\geq\varepsilon'''\right]\leq\mathsf{negl}(n).
\end{align}
From \eqref{eqn:chernoffSet} and \eqref{eqn:realChernoff}, it follows that with overwhelming probability over the random choice of $M$,
for all $b\in\{0,1\}$ and any constant $0<\varepsilon''<1$ and $0<\varepsilon'''<1$,
$$
\Pr_{\vec{a}_1,\dots,\vec{a}_n}\left[\alpha_b<\varepsilon'-\varepsilon''-\varepsilon'''\right]\leq\mathsf{negl}(n).
$$
By taking $\varepsilon''+\varepsilon'''\leq\varepsilon'/2$, we have that with overwhelming probability over $M$, it holds that $\alpha_b<\varepsilon'/2$ for all $b\in\{0,1\}$.
Finally, the $\alpha_b$ correspond exactly to the probability expression in \eqref{eqn:notReallyRandom2}, so
we have shown statement \eqref{eqn:notReallyRandom2} as required.
\end{proof}

Having reduced the length of players' messages to $\mathsf{polylog}(n)$ in Lemma \ref{lem:manyBitsMatrix},
we now prove the following lemma which reduces the required communication even further, so that each player sends only one bit.
This comes at the cost of a polylogarithmic factor reduction in the number of corruptions.

Before the lemma, we recall the statement of the Chernoff bound.

\begin{theorem}[Chernoff bound]
Let $X_1,\dots,X_n$ be independent random variables taking values in $\{0,1\}$, which all have the same expectation $\mu=\mathop{{}\mathbb{E}}[X_i]$. Then, for every $0<\varepsilon<1$,
$$\Pr\left[\left|\frac{1}{n}\cdot\sum_{i\in[n]} X_i-\mu\right|\geq\varepsilon\right]\leq 2e^{-2n\varepsilon^2}.$$
\end{theorem}

\begin{lemma}\label{lem:multiToOneBit_strong}
For any one-round $\ell$-bit coin-flipping protocol $\Pi$ secure against $t=t(n)$ strong adaptive corruptions,
there is a one-round single-bit coin-flipping protocol $\Pi'$ that is secure against $t/\ell$ strong adaptive corruptions.
\end{lemma}
\begin{proof}
Let $\Pi$ be any one-round $\ell$-bit coin-flipping protocol secure against $t=t(n)$ strong adaptive corruptions.
We define our new single-bit protocol\footnote{We remark that the protocol $\Pi'$ that we construct does not strictly adhere to
Definition~\ref{def:coinFlipping}, because $\Pi'=\{\Pi_n\}_{n\in\ell\cdot\mathbb{N}}$ does not define an $n$-player protocol for every $n\in\mathbb{N}$.
We consider this to be a very minor technical detail that we bury for clarity of exposition.
} $\Pi'$ as follows, for each $n\in\mathbb{N}$:
\begin{gather*}
\Pi'_{n\cdot\ell}(r_1,\dots,r_{n\cdot\ell})= \\
\Pi_n\left((r_1||\dots||r_\ell), (r_{\ell+1}||\dots||r_{2\ell}), \dots, (r_{(n-1)\cdot\ell+1}||\dots||r_{n\cdot\ell})\right),
\end{gather*}
where the messages $r_i\in\{0,1\}$ are bits and $||$ denotes concatenation.
Informally speaking, there are $n$ groups of $\ell$ players in the single-bit protocol $\Pi'_{n\cdot\ell}$, each of which
``corresponds to'' a single player in the protocol $\Pi_n$.

We show that $\Pi'$ is secure against $t/\ell$ corruptions.
Let $G_i$ denote the $i^{th}$ group of $\ell$ players: to be precise, $G_i=\{i\cdot\ell+1,\dots,(i+1)\cdot\ell\}$.
If all of the players in the set $G_i$ are honest, then the $i^{th}$ ``combined message'' $(r_{i\cdot\ell+1}||\dots||r_{(i+1)\cdot\ell})$
is distributed identically to an honest message of the $i^{th}$ player in the protocol $\Pi_n$.
By the construction of the protocol $\Pi'$, it follows that for any $b\in\{0,1\}$ and $n\in\mathbb{N}$,
\begin{align}\label{eqn:robustSingleBit}
\Pr_{\vec{r}\leftarrow\{0,1\}^{n\cdot\ell}}\left[\vec{r}\in{\sf Robust}^{\Pi'_{n\cdot\ell}}(b,t(n))\right]\geq
\Pr_{\vec{r'}\leftarrow(\{0,1\}^\ell)^n}\left[\vec{r'}\in{\sf Robust}^{\Pi_n}(b,t(n))\right].
\end{align}
By Lemma \ref{lem:robustSets}, since $\Pi$ is secure against $t$ strong adaptive corruptions,
there is a constant $0<\varepsilon<1$ such that for all $b\in\{0,1\}$ and $n\in\mathbb{N}$, the right-hand side of inequality \eqref{eqn:robustSingleBit}
is at least $\varepsilon$. Hence we obtain
$$
\Pr_{\vec{r}\leftarrow\{0,1\}^{n\cdot\ell}}\left[\vec{r}\in{\sf Robust}^{\Pi'_{n\cdot\ell}}(b,t(n))\right]\geq\varepsilon.
$$
It follows (by applying Lemma \ref{lem:robustSets} again) that $\Pi'$ is secure against $t/\ell$ strong adaptive corruptions.
\end{proof}

Finally, we bring together Lemmas \ref{lem:manyBitsMatrix} and \ref{lem:multiToOneBit_strong} to prove the theorem.

\begin{customthm}{\ref{thm:main}}
Any one-round coin-flipping protocol $\Pi$ can be secure against at most $t=\widetilde{O}(\sqrt n)$ strong adaptive corruptions.
\end{customthm}
\begin{proof}
Suppose, for contradiction, that there exists a one-round coin-flipping protocol $\Pi$ which is secure against $t$ corruptions,
where $t=\omega(\sqrt{n}\cdot\mathsf{polylog}(n))$.
Then, by Lemma \ref{lem:manyBitsMatrix}, there is an $\ell$-bit one-round coin-flipping protocol $\Pi'$
that is secure against $t$ strong adaptive corruptions, where $\ell=\mathsf{polylog}(n)$.
By applying Lemma \ref{lem:multiToOneBit_strong} to the protocol $\Pi'$,
we deduce that there is a single-bit one-round coin-flipping protocol $\Pi''$ which is secure against $t/\ell=\widetilde{\Omega}(t)$ strong adaptive corruptions.
Since a \emph{strongly adaptive} adversary can perfectly simulate
any strategy of an \emph{adaptive} adversary, it follows that $\Pi''$ is secure against $\widetilde{\Omega}(t)$ adaptive corruptions.
Since $\Pi''$ is single-bit, this contradicts Theorem \ref{thm:lls}.
\end{proof}

\subsection{Proof of Theorem \ref{thm:minmax}}

In this section, we show that for any symmetric one-round coin-flipping protocol secure against $t$ \emph{adaptive} corruptions,
there is a one-round coin-flipping protocol secure against $\Omega(t)$ corruptions by \emph{strong adaptive} adversaries.
That is, one-round strong adaptively secure protocols are a more general class than one-round symmetric, adaptively secure protocols.

\begin{remark}
In fact, Theorem \ref{thm:minmax} holds even if the protocol $\Pi$ is just statically secure: the proof does not make use of the fact
that $\Pi$ is adaptively, rather than statically, secure.
Our theorem statement refers to $\Pi$ as an adaptively secure protocol because this is exactly what we need in order to obtain our final result
that any one-round symmetric coin-flipping protocol can be secure against at most $O(\sqrt{n})$ corruptions.
\end{remark}

The Minimax Theorem -- a classic tool in game theory -- will be an important tool in our proof.
The statement of the Minimax Theorem and supporting game-theoretic definitions are given below.

\begin{definition}[Two-player strategic game]
A \emph{two-player finite strategic game} $\Gamma=\langle (A_1,A_2),(u_1,u_2)\rangle$ is defined by:
for each player $i\in\{1,2\}$,
a non-empty set of possible \emph{actions} $A_i$ and
a \emph{utility function} $u_i:A_1\times A_2\rightarrow\mathbb{R}$.
\end{definition}

\begin{definition}[Zero-sum game]
A two-player finite strategic game $\Gamma=\langle (A_1,A_2),(u_1,u_2)\rangle$ is \emph{zero-sum} if
for any pair of actions $a_1\in A_1$ and $a_2\in A_2$, it holds that
$u_1(a_1,a_2)+u_2(a_1,a_2)=0$.
\end{definition}

\begin{theorem}[Minimax \cite{vN44,Nash}]\label{thm:originalMinMax}
Let $\Gamma=\langle (A_1,A_2),(u_1,u_2)\rangle$ be a zero-sum two-player finite strategic game.
Then
$$\max_{a_2\in \Delta(A_2)} \min_{a_1\in \Delta(A_1)} u_2(a_1,a_2) = \min_{a_1\in \Delta(A_1)} \max_{a_2\in \Delta(A_2)} u_1(a_1,a_2),$$
where $\Delta(A_i)$ denotes the set of distributions over $A_i$ (in game-theoretic terminology, this corresponds to the set of ``mixed strategies'' for player $i$.)
\end{theorem}

\iffalse
\begin{theorem}[Minimax \cite{vN44}]
Let $\Gamma=\langle (A_1,A_2),(u_1,u_2)\rangle$ be a zero-sum two-player finite strategic game.
Then
$$\max_{a_2\in A_2} \min_{a_1\in A_1} u_2(a_1,a_2) = - \min_{a_2\in A_2} \max_{a_1\in A_1} u_1(a_1,a_2).$$
Moreover, $a_2\in A_2$ solves the problem $\max_{a_2\in A_2} \min_{a_1\in A_1} u_2(a_1,a_2)$
if and only if it solves the problem $\min_{a_2\in A_2} \max_{a_1\in A_1} u_1(a_1,a_2)$.
\end{theorem}
\fi

\begin{customthm}{\ref{thm:minmax}}
For any symmetric one-round coin-flipping protocol $\Pi$ secure against $t=t(n)$ adaptive corruptions,
there is a symmetric one-round coin-flipping protocol $\Pi'$ secure against $s=t/2$ strong adaptive corruptions.
\end{customthm}
\begin{proof}
Let $\Pi$ be a symmetric one-round coin-flipping protocol secure against $t=t(n)$ adaptive corruptions,
and define $s(n)=t(n)/2$.
We define a new protocol $\Pi'=\{\Pi'_n\}_{n\in\mathbb{N}}$ as follows:
$$\Pi'_n(r_1,\dots,r_n)=\min_{\textcolor{NavyBlue}{r'_1,\dots,r'_s}}\max_{\textcolor{WildStrawberry}{r''_1,\dots,r''_s}}\Pi_{n+2s}\left(\rs{1}{n},\textcolor{NavyBlue}{r'_1,\dots,r'_s},\textcolor{WildStrawberry}{r''_1,\dots,r''_s}\right),$$
where $s=s(n)$ and honest players in $\Pi'_n$ must send messages according to the same distributions as in $\Pi_{n+2s}$.

Observe that $\Pi_{n+2s}$ is secure against $t(n+2s(n))>t(n)$ corruptions.
We show that $\Pi'_n$ is secure against $s(n)=t(n)/2$ strong adaptive corruptions.

{\sc Case 1.} Suppose that the adversary aims to bias the outcome towards $0$.
By the security of $\Pi_{n+2s}$, there is a constant $0<\varepsilon<1$ such that
$\Pr^{\Pi_{n+2s},\mathcal{A}}(1)\geq\varepsilon$ for any adaptive adversary $\mathcal{A}$ that corrupts up to $t=2s$ players.
Without loss of generality (since the protocol is symmetric), suppose that the adversary corrupts the last $2s$ players in $\Pi_{n+2s}$.

We say that the honest players' messages $\rs{1}{n}$ ``fix'' the outcome of
$\Pi_{n+2s}$ to be $1$ if for any possibly malicious messages $\rhats{1}{2s}$,
it holds that $\Pi_{n+2s}(\rs{1}{n},\rhats{1}{2s})=1$.
Then, with probability at least $\varepsilon$, the honest players' messages $\rs{1}{n}$ ``fix'' the outcome of $\Pi_{n+2s}$ to be $1$.
(To see this: suppose not. Then there would exist an adversary which could set the corrupt messages $\rhats{1}{2s}$
so that the protocol outcome is $0$ with probability $1-\varepsilon$.
But this cannot be, since we already established that $\Pr^{\Pi_{n+2s},\mathcal{A}}(1)\geq\varepsilon$.)

Define the set
$R_1\overset{{\rm def}}{=}\left\{(\rs{1}{n}):\forall\rhats{1}{2s},~~\Pi_{n+2s}(\rs{1}{n},\rhats{1}{2s})=1\right\}$
to consist of those honest message-vectors that fix the output of $\Pi_{n+2s}$ to be $1$.

Take any $(\rs{1}{n})\in R_1$.
We now show that the outcome of $\Pi'_n$ when the honest players send messages $\rs{1}{n}$ is equal to $1$,
even in the presence of a strong adaptive adversary $\mathcal{A}'$ that corrupts up to $s$ players and aims to bias the outcome towards $0$.
Without loss of generality, suppose that $\mathcal{A}'$ corrupts the first $s$ players in $\Pi'_n$, and replaces their honest messages $\rs{1}{s}$
with some maliciously chosen messages $\rhats{1}{s}$.
In this case, the outcome of $\Pi'_n$ is
\begin{align*}
\Pi'_n&(\hat{r}_1,\dots,\hat{r}_s,r_{s+1},\dots,r_n) \\
& =\min_{\textcolor{NavyBlue}{r'_1,\dots,r'_s}}\max_{\textcolor{WildStrawberry}{r''_1,\dots,r''_s}}\Pi_{n+2s}\left(\rhats{1}{s},\rs{s+1}{n},\textcolor{NavyBlue}{r'_1,\dots,r'_s},\textcolor{WildStrawberry}{r''_1,\dots,r''_s}\right) \\
& \geq\min_{\textcolor{NavyBlue}{r'_1,\dots,r'_s}}\Pi_{n+2s}\left(\rhats{1}{s},\rs{s+1}{n},\textcolor{NavyBlue}{r'_1,\dots,r'_s},\rs{1}{s}\right) \\
& =\min_{\textcolor{NavyBlue}{r'_1,\dots,r'_s}}\Pi_{n+2s}\left(\rs{1}{n},\rhats{1}{s},\textcolor{NavyBlue}{r'_1,\dots,r'_s}\right)\tag{by symmetry} \\
& =1,
\end{align*}
where the last line follows from the definition of $R_1$, since we started with $(\rs{1}{n})\in R_1$.

We already established that the probability that the honest players' messages fall in $R_1$ is at least $\varepsilon$.
Thus we deduce that with probability at least $\varepsilon$, the outcome of the new protocol $\Pi'_n$ is equal to $1$,
even in the presence of a strong adaptive adversary corrupting $s$ players and aiming to bias towards $0$.

{\sc Case 2.} Suppose instead that the adversary $\mathcal{A}'$ aims to bias the outcome towards $1$.
We apply the Minimax Theorem to a zero-sum game where player 1 chooses the messages $\textcolor{NavyBlue}{r'_1,\dots,r'_s}$ and player 2 chooses the messages $\textcolor{WildStrawberry}{r''_1,\dots,r''_s}$,
and player 1 ``wins'' if the protocol outcome is 0, and player 2 wins otherwise.
By the Minimax Theorem,
$$\Pi'_n(r_1,\dots,r_n)=\max_{\textcolor{WildStrawberry}{r''_1,\dots,r''_s}}\min_{\textcolor{NavyBlue}{r'_1,\dots,r'_s}}\Pi_{n+2s}\left(\rs{1}{n},\textcolor{NavyBlue}{r'_1,\dots,r'_s},\textcolor{WildStrawberry}{r''_1,\dots,r''_s}\right).$$

Given this new and equivalent definition of $\Pi'_n$, we can apply exactly
the same argument structure as that given for Case 1 above, to deduce that
\begin{itemize}
\item There is a constant $0<\varepsilon'<1$ such that $\Pr^{\Pi_{n+2s},\mathcal{A}}(0)=1-\Pr^{\Pi_{n+2s},\mathcal{A}}(1)=\varepsilon'$ for any adaptive $\mathcal{A}$ performing up to $2s$ corruptions,
and hence there is a non-empty set $$R_0\overset{{\rm def}}{=}\left\{(\rs{1}{n}):\forall\rhats{1}{2s},~~\Pi_{n+2s}(\rs{1}{n},\rhats{1}{2s})=0\right\}\mbox{,~~and}$$
\item by the adaptive security of $\Pi_{n+2s}$, the messages of honest players will fall in $R_0$ with probability at least $\varepsilon'$, and
\item if the honest players' messages fall in $R_0$, then the outcome of $\Pi'_n$ is equal to $0$,
even in the presence of a strong adaptive adversary corrupting $s$ players and aiming to bias towards $1$.
\end{itemize}

We have established that both outcomes 0 and 1 occur with constant probability in $\Pi'_n$, even in the presence of
an arbitrary strong adaptive adversary corrupting up to $s$ players.
Therefore, $\Pi'_n$ is secure against $s=t/2$ corruptions.
\end{proof}

\section{Conclusion}

We have introduced a new adversarial model for multi-party protocols and an associated security notion, \emph{strong adaptive security}.
We have made use of a novel and widely applicable technique for reducing the amount of communication in a protocol,
to show that any one-round strongly adaptively secure coin-flipping protocol can tolerate at most $\widetilde{O}(\sqrt n)$ corruptions.
We believe that this work paves the way to a number of little-explored research directions.
We highlight some interesting questions for future work:

\begin{itemize}
\item To study the extent to which \emph{communication can be reduced in protocols in general}, and to extend our communication-reduction techniques to
the settings of multi-round protocols and/or adaptive security.
\item To apply the \emph{strong adaptive security notion} in the context of other types of protocols and settings,
and to design protocols secure in the presence of strong adaptive adversaries.
\item To consider whether adaptively secure \emph{asymmetric} coin-flipping protocols can be converted to adaptively secure \emph{symmetric} protocols, in general. This is not known even for the one-round case, and the question is moreover of interest since there are known one-round protocols which are not symmetric.
\item To extend this work to prove (or disprove) the long-open conjecture of Lichtenstein et al. \cite{LLS89} that \emph{any} adaptively secure coin-flipping protocol
can tolerate at most $O(\sqrt n)$ corruptions.
\end{itemize}

\printbibliography

\end{document}

\subsection{Our contribution}

We first define a new adversarial model of {\em strong adaptive} corruptions.
Informally, an adversary is strongly adaptive if he can corrupt players depending on the content of their messages. More precisely, in each round,
he can see all the messages that honest players ``would'' send, and then decide which of them to corrupt.
This is in contrast to a (traditionally defined) adaptive adversary who can, at any point in the protocol, corrupt any player who has not yet spoken based on the history of communication, but cannot alter the message of a player who has already spoken.
Thus, strong adaptive adversaries are more powerful than adaptive adversaries.

We believe that the notion of \emph{strong adaptive security} gives rise to a natural and interesting new adversarial model
in which to study multi-party protocols in general.
Indeed, it is a realistic concern in many settings that malicious parties may decide to stop or alter messages sent by honest players \emph{depending on message content},
and it is a shortcoming that existing adversarial models fail to take such behavior into account.

We consider our strong adaptive adversarial notion to be closely tied to the notion of a \emph{rushing} adversary in the setting of static corruptions.
A rushing static adversary can see the messages that the honest players
send in each round, before deciding the messages that the corrupted players will send in the same round.
The intuitive idea of a rushing adversary is that the adversary \emph{sees all possible information in each round, before making his move}.
We remark that a notion of ``rushing adaptive adversary'' has been previously proposed in the literature, but such an adversary is weaker
than our strong adaptive adversary\footnote{In particular, the ``rushing adaptive adversary'' from the literature can decide the order in which players
send messages in a round, and can decide to corrupt a player who has not yet sent a message within a round. However, unlike our strong adaptive adversary,
this adversary cannot decide to corrupt a player based on the content of the message which the player \emph{would send if uncorrupted}.}.
We argue that our strong adaptive adversary better captures the idea that the adversary \emph{sees all possibly relevant information in each round, before making his move}, since in the adaptive setting, the adversary's strategy must decide not only what messages to send, but also \emph{which players to corrupt}.

Our main result is that the conjecture of \cite{BL85} holds (up to polylogarithmic factors)
for any one-round coin-flipping protocol in the presence of \emph{strong adaptive} corruptions.

\begin{informalthm}
Any secure one-round coin-flipping protocol $\Pi$ can tolerate at most $t=\widetilde{O}(\sqrt n)$ strong adaptive corruptions.
\end{informalthm}

This is shown by a generic reduction of communication in the protocol: first, we prove that any strongly adaptively secure protocol $\Pi$ can be converted to one
where players send messages of no more than polylogarithmic length, while preserving the number of corruptions that can be tolerated.
Then, we show that any protocol with messages of polylogarithmic length can be converted to one where each player sends only a single bit,
at the cost of a polylogarithmic factor in the number of corruptions.
Finally, we reach the \emph{single-bit} setting in which the bound of Lichtenstein et al. \cite{LLS89} can be applied to obtain the theorem.
We believe that our technique of converting any protocol into one with short messages is of independent interest and will find other applications.

Furthermore, we prove that strongly adaptively secure protocols are a more general class of protocols than symmetric adaptively secure protocols.
A symmetric protocol $\Pi$ is a one that is oblivious to the order of its inputs: that is, where
for any permutation $\pi:[n]\rightarrow[n]$ of the players, it holds that the protocol outcome $\Pi(r_1,\ldots,r_n)=\Pi(r_{\pi(1)},\ldots,r_{\pi(n)})$ is the same.

\begin{informalthm}
For any symmetric one-round coin-flipping protocol $\Pi$ secure against $t=t(n)$ adaptive corruptions,
there is a symmetric one-round coin-flipping protocol $\Pi'$ secure against $\Omega(t)$ strong adaptive corruptions.
\end{informalthm}

Curiously, this proof makes a novel use of the Minimax Theorem \cite{vN44,Nash} from game theory, in order to take any symmetric, adaptively secure protocol and convert it to a new protocol which is strongly adaptively secure. This technique views the protocol as a zero-sum game between two players $\mathcal{A}_0$ and $\mathcal{A}_1$, where $\mathcal{A}_0$
wins if the protocol outcome is 0 and $\mathcal{A}_1$ wins if the outcome is 1.
We analyze the ``minimax strategy'' in which the players try to minimize their maximum loss, in order to deduce the strong adaptive security of the new protocol.
Whereas some prior works have made use of game theory in the analysis of (two-party) protocols,
this is the first use of these game-theoretic concepts in the \emph{construction} of distributed multiparty protocols.

Finally, using the above results as stepping stones, we return to the classical conjecture of \cite{BL85},  in the model of adaptive adversaries, and show that the conjecture holds (up to polylogarithmic factors)
for any \emph{symmetric} one-round protocol with arbitrarily long messages.

\begin{informalthm}
Any secure symmetric one-round coin-flipping protocol $\Pi$ can tolerate at most $t=\widetilde{O}(\sqrt n)$ adaptive corruptions.
\end{informalthm}

\subsection{Related work}\label{related-work}

The full-information model (also known as the \emph{perfect information model}) was introduced by Ben-Or and Linial \cite{BL85}
to study the problem of collective coin-flipping when no secret communication is possible between honest players.

\paragraph{In the static setting.}
Protocols for collective coin-flipping in the presence of static corruptions have been constructed in a series of works
that variously focus on improving the fault-tolerance, round complexity, and/or bias of the output bit.
Feige \cite{Fei99} gave a protocol that is $(\delta^{1.65}/2)$-secure\footnote{A
coin-flipping protocol is $\varepsilon$-secure against~$t$ static corruptions if for any static adversary that corrupts up to~$t$ parties,
the probability that the protocol outputs~$0$ is at least $\varepsilon$.}
in the presence of $t=(1+\delta)\cdot n/2$ static corruptions for any constant $0<\delta<1$.
Russell, Saks, and Zuckerman \cite{RSZ02} then showed that any protocol that is secure in the presence of linearly many corruptions must
either have at least $(1/2-o(1))\cdot\log^*(n)$ rounds, or communicate many bits per round.

Interestingly, nearly all proposed \emph{multi-round} protocols for collective coin-flipping
first run a \emph{leader election} protocol in which one of the $n$ players is selected as a ``leader'', who then outputs a bit that is taken as the protocol outcome.
We remark that this approach is inherently unsuitable for adaptive adversaries, which can always corrupt the leader after he is elected,
and thereby surely control the protocol outcome.

\paragraph{In the adaptive setting.}
The study of coin-flipping protocols has been predominantly in the static setting.
The problem of adaptively secure coin-flipping was introduced by Ben-Or and Linial \cite{BL85}
and further examined by Lichtenstein, Linial, and Saks \cite{LLS89} as described in the previous section.
In addition, Dodis \cite{Dod00} proved that through ``black-box'' reductions from \emph{non-adaptive} coin-flipping, it is not possible to tolerate
significantly more corruptions than the majority protocol. The definition of ``black-box'' used in \cite{Dod00}
is rather restricted: it only considers sequential composition of non-adaptive coin-flipping protocols, followed by a (non-interactive)
function computation on the coin-flips thus obtained.

\paragraph{In the pairwise-channels setting.}
An adversarial model bearing some resemblance to our strong adaptive adversary model was introduced and analyzed by Hirt and Zikas \cite{HZ10}
in the \emph{pairwise communication channels} model, rather than the full-information model. In their model, the adversary can corrupt a party $P$ based on
some of the messages that $P$ sends within a round, then the adversary controls the rest of $P$'s messages in that round (and for future rounds).
Unlike in our strong adaptive model, the adversary of \cite{HZ10} cannot ``see inside all players' heads''
and overwrite arbitrary honest messages based on their content before they are sent.

\medskip

A brief survey of the literature on coin-flipping in the setting of computationally bounded players
is given in Appendix \ref{appx:computationalSetting} due to space constraints.

redpajama_set_name: RedPajamaArXiv


text:
The Best MMORPG Of 2019 And Beyond
In Articles, Reviews, RPG Games
As the year comes to a close, we're highlighting the best MMORPG titles to look forward to next year. 2018 has been a great year for online gaming and MMORPG games in particular. A select few games in particular led the way with excellent content addition this year, while others that are still in development has teased great features for the future. Based on this, we're listing the best MMORPG of 2019 to keep an eye on. This features plenty of usual suspects based on their 2018 performance so far, including those that are set for release in 2019.
Best Triple A MMORPG 2019
Elder Scrolls Online (PC/PS4/XB)
The Elder Scrolls Online is having a fantastic year. Zenimax Online Studios have been keeping its 10 million players and counting happy this year. The game released a full Chapter (expansion) in Summerset in 2018, in addition to numerous DLC updates like Dragon Bones and Murkmire. Not only are they content rich – but also tweaks and adds plenty of quality of life improvements. For example, players are given log-in rewards, which let them earn DLC like Murkmire for free. Additionally, players on the ESO Plus subscription got more value for their money, as the monthly Crowns stipend increased and members are given additional perks like free items and exclusive discounts. Moreover, much more polish and other feature arrived this year too like the ability to finally make cosmetic outfit and save them in slots. PvP Battlegrounds became free for all, Morrowind got bundled into the base game purchase and much more. It's hard to argue that ESO is at the moment, on top of their game.
Elder Scrolls Online have led the way this year when it comes to releasing good content on top of great features for its players. Based on this, 2019 look set to be another great year for ESO if they can keep up the pace of putting out good content on time.
Final Fantasy XIV: Stormblood (PC/PS4)
Another MMORPG that can boast 10 million registered players in 2018 is Square Enix's Final Fantasy XIV. The Stormblood expansion released in 2017 is still going strong content-wise. This year, players have seen an incredibly cool crossover with Monster Hunter: World, in addition to the slew of updates that progresses the main story chain further. The game also received numerous group content updates which is crucial for an MMORPG to survive.
Final Fantasy XIV definitely saved the best for last. It was announced only two weeks ago that the game's third expansion, Shadowbringers, will arrive in summer 2019. The expansion gives players the opportunity to crossover from Warriors of Light to Warriors of Darkness, and brings more jobs and classes to the game. The cool thing about FFXIV is that with every new job, players don't need to roll alts as one character may pick up several jobs. The Blue Mage is the latest coming to the game. While FFXIV suffers from content drought and DDOS attacks from time to time, it's still a fantastic MMORPG and 2019 will be a big year for the game.
World of Warcraft (PC)
The world of Azeroth got bigger in 2018 as the war returned between the Alliance and the Horde. 2018 started with the winding down of 2016's Legion expansion, as players said goodbye to their Artifact Weapons. Everything went pear-shaped however, when Sylvannas attacks Darnassus due to the emergence of Azerite, the world's essence that's come to the surface thanks to that huge sword sticking out of the world's surface.
That led to the Battle for Azeroth expansion in August, which brought new maps and zones to quest in, including new dungeons and raids. Overall, BFA has been a great addition to the WoW universe, but not without it's fair share of issues. The Azerite system that replaced Artifact Weapons are far from perfect, but tweaks and improvements are coming and that will be the case leading into 2019. The expansion also apparently saw the most amount of WoW subscription numbers dropping, but people love the drama of a game losing players, so pay no attention to it if you're looking for a solid all rounded MMORPG to play in 2019. With fixes, tweaks, the continuation of the Horde vs Alliance story and more raids coming in 2019, the Battle for Azeroth expansion can only get better in time.
Guild Wars 2 (PC)
Rounding up the big AAA MMORPG titles to play in 2019 is ArenaNet's Guild Wars 2. It was a genre-changer when released in 2012, and six years later the game is still going strong on the heels of its 2017 Path of Fire expansion. Guild Wars 2 has become more accessible since making the base game free to play, and many gameplay systems have improved the game like Mounts. In 2018 the game saw numerous quality of life updates like changes to make underwater combat better, and making runes and sigil salvageable to save inventory space. For PvP players, Guild Wars 2 has added 2v2 arenas and and World vs World should get its major update in 2019 that removes the need for servers that limits playing with friends and guilds.
Additionally on the content front, 2018 saw a steady release of new maps and story content for the game's Living World Season 4 that continues the Path of Fire storyline. 2019 will definitely build on this, and we're expecting a big announcement later in 2019 regarding a new expansion or something else.
Best Upcoming MMORPG 2019
Anthem (PC/PS4/XB)
Of all the MMO titles currently in early access, only EA's Anthem has a set release date, and the game arrives for all platforms in February 2019. In Anthem, up to four players explore and battle in a dangerous, mysterious place populated by great characters and a unique BioWare story. Players will also create stories of their own while they play, so no two journeys are the same. Javelin exosuits make their pilots powerful heroes in the world, with awesome weapons and incredible special abilities that allow for combat and exploration in a hostile environment. Players will be able to customize and personalize their suits with unique paint jobs and gear so they'll have the right tools to confront almost any situation, and look good doing it.
So far, the multiplayer aspect excites more than going solo as we're all about online play. The game should give similar online titles with RPG elements like Destiny 2, Planetside 2, Defiance and more a run for its money. The downside? It's EA, and their reputation has worsened in 2018.
Ashes of Creation (PC)
Ashes of Creation is still the biggest hyped MMORPG yet to be released today. It garnered famed through its widely shared Kickstarter campaign that raised over $3million overnight, promising to make MMORPGs great again with complex systems, group play, player driven towns, economies and communities. Solo play will still be viable for those that prefer it, but the devs believe that MMOs are best when players come together. One of the first things we learn is that the base group size in Ashes is eight. It's a steady increase from other games, where four to six is usually the norm. 8-man groups allows the content to amplify party roles and take advantage of all class archetypes. Speaking of roles, Ashes of Creation will emphasize importance on the holy trinity. That is of course, the Tank, DPS and Healer. However, Support roles are as equally important via the secondary class system to make customizing player characters a key feature.
Topics: AAA mmorpg 2019 anthem Ashes of Creation best mmo 2019 best mmo games best mmorpg 2019 best upcoming mmorpg 2019 Elder Scrolls Online Final Fantasy XIV Guild wars 2 MMO mmo 2019 MMORPG triple a mmorpg 2019 world of warcraft
Adrian Tan
Web Editor of MMOsWorld. Serial video game critic and gaming YouTuber. I run on caffeine. I hit level cap two hours ago. I love going /afk.


The aviation industry in India has emerged as one of the fastest growing in the country during the last few years.
If all that wasn't good enough, a new study by consultancy firm KPMG now predicts that "India is all set to become the world's third largest aviation market by 2020 after the US and China".
Aviation in India has a long history spanning over 99 years. In 1912, during British rule, first Indian flight was inaugurated from Delhi to Karachi.
During that time, all flights were operated by Indian State Air Services in collaboration with Imperial Airways of United Kingdom. Then immediately after the independence, two main lines were established in India.
Air India and both were state owned entities.
Construction of new airports including refurbishment (or increased capacity) of existing airports in numerous Indian cities and associated infrastructure and facilities such as road networks, rail transportation and hotel accommodation.
Expansion of aviation services provided by the government owned airline carriers.
Modernization of airline services such as maintenance, repair and overhaul and safety at airports through latest instrumentation.
Rapid increase in business travel and global mobility. In essence, increase in demand for domestic and international travel owing to rapid economic growth.
Besides above, the recent statics indicate that there has been amazing increase in air traffic since year 2010.
In consideration of the above, it is certain that Indian aviation industry will certainly outperform other Asian countries in terms of air travel. Consequently, more and more international airlines will compete to get a slot in the Indian aviation sector. Thus shows that the Indian aviation industry is further poised to develop into next level depicting it as one of the rapidly growing aviation industry internationality and the in region.
It's well known that catalysing of the rapid growth of the industry was due to private investment friendly government policies that encouraged the international aviation investors to participate. As such, nearly 75% of the Indian aviation industry is dominated by the foreign and the domestic investors. Hence, in the near future India will become an attractive aviation market in the Asian region for foreign and local investors.
Developments of Indian tourist industry and overall economic progress have resulted in increasing inbound and outbound travel.
Revamp of existing aviation policies (by Indian government) to encourage local and international participation in the development of aviation industry. Also, such policies further encourage public-private partnerships to develop massive airport infrastructure development projects.
By looking at the above trends in the Indian Aviation Industry, one could see the tremendous potential it has in the coming years, in terms of rapid development and economic prosperity to the country.


When and why did the candy cane originate? How many different religions and cultures enjoy gingerbread and marzipan during the holidays? How do these traditions make us more the same than different? These and other tantalizing questions will be addressed below. Read on, my sweets.
Candy is always associated with special and happy occasions, and there are few times of the year more celebratory than Christmas. For people of all faiths and backgrounds, December marks the end of a year of evaluation, devaluation, thanks and no thanks for all that has preceded. It is the beginning of winter and festivals of lights and dazzling displays for Christians and Jews. Food is the one thing that unites all cultures and eating sweets is the pinnacle of that experience. Candy can be used as great Christmas gifts for your wife, warm wishes and for congratulations. Marzipan and gingerbread are just two of many "sweets" that are always associated with Christmas. They both have a long and interesting culinary history worth mentioning here.
Gingerbread dates back to eleventh century Europe. It was sometimes light, sometimes dark, once in a while a cookie and every now and then a bread. It was always, however, cut into the shapes of people and animals and colorfully decorated with a mold and dusted with white sugar. In Medieval England, gingerbread meant simply "preserved ginger," and was an adaptation of Gingebras, derived from the Latin name for the spice, Zingebar. It wasn't until the 15th century that the term came to be applied to a kind of cake made with treacle, an uncrystalized syrup drained from raw sugar during the refining process, and flavored with ginger.
Gingerbread, from its very inception, was always a fairground delicacy. Certain shapes were associated with different seasons: buttons and flowers were found at Easter fairs, and animals and birds were featured in autumn. At Christmas, gingerbread makes its most impressive appearance in the form of beautiful little houses. Germany, of all the other European countries, has the longest tradition of flat shaped gingerbread. Their practice of making lebkuchen houses never caught on in Britain as much as it did in North America. The settlers from Northern Europe brought it along with their customs and family recipes. By the nineteenth century, America had been baking gingerbread for decades.
The versatility of the frontier spirit rendered a practicality to American gingerbread. In New England, for example, maple syrup gingerbreads were made and in the South sorghum molasses was used. More and more regional variations developed as more and more settlers from different countries arrived. In Pennsylvania particularly, the German influence was great and many traditional gingerbreads re-appeared in this area, particularly around Christmas time. No where in the world is there a greater collection of gingerbread recipes than in America. With the vast choice of ingredients, baking aids and decorative items, the imaginative cook can create the most spectacular gingerbread houses and centerpieces ever.
Gingerbread houses can be found in the celebration of both Christian and Jewish festivals. Many bakeries make just as many Hanukkah gingerbread and candy houses as they do traditional Christian ones. It is a delicacy that transcends all cultures and traditions and well it should; for gingerbread is the king. The queen to share its sugary crown is marzipan, whose sweet story is the next to unfold.
Somewhere someone during these times created marzipan. People liked it so much that it flourished throughout Italy even after the famine passed. Travelers carried the recipe throughout various cities and eventually to other countries until it reached Lubeck and Hamburg, Germany. The marzipan made there is still the finest that can be found anywhere today. Look for "Made in Lubeck" or "Lubecker Marzipan" for the finest marzipan that money can buy.
The candy cane has its own unique and surprising history. (I always thought FW Woolworth created it as a tree decoration). Actually, its origins date back to more than 350 years, when candy-makers were making hard sugar-sticks. It was originally white in color and straight. The first historical reference to the familiar cane shape goes back to 1670, when the choirmaster at the Cologne cathedral in Germany, bent the sugar sticks into canes to represent a shepherd's staff. The all-white candy canes were dispensed to children to keep them quiet during long nativity services. The custom of the clergy handing out candy canes during Christmas services spread throughout Europe and later to America.
The first historical reference to the candy cane in America dates back to 1847, when a German immigrant, August Imgard, decorated the Christmas tree in his home in Wooster, Ohio with candy canes. No one knows who exactly invented the stripes, but it is known that Christmas cards prior to 1900 depicted only all-white candy canes. Around the same time, candy-makers added peppermint and wintergreen flavors to their candy canes and those flavors soon became traditional favorites. Gregory Keller, a Catholic priest, invented a machine to automate candy cane production in the 1950s.
There are other more religious legends surrounding the candy cane. Some say it is the secret symbol of Christianity used during times of more oppressive circumstances. The J shape is said to represent the name of Jesus and the red stripes Christ's blood and purity. The three red stripes symbolized the Holy Trinity and the peppermint flavor a reference hyssop; an herb referred to in the Old Testament. There is no historical evidence to support these claims and so like Tinkerbelle, they must live in the eyes of the believer.
Chocolates, almond cookies, macaroons of all kinds and sugarplums all dance around our high calorie heads. But they dance joyfully, for we all only live once and what would life be without chocolate? So indulge and enjoy. but remember this important holiday rule of the road. Your dentist is having a fine holiday too and your over-indulgence may well help him make his own ends meet in the coming year!
Happy Sweets And Holidays To All!
Egg Nog: Why Is It So?
Attention is lavished on Lindt chocolate making - starting with the selection of the finest raw materials and highest quality ingredients, and culminating in their final exquisite creations.
A beautiful book with easy recipes for things not seen very often, such as sugar plums. Contains some nice pictures. A must for the Christmas baker!
This Gingerbread book is the best I've seen. If you are a beginner you will have no problems succeeding with your first gingerbread houses. If you are a master, you will find that the detail and suggestions for expanding your craft are all in one book. It doesn't feel like Christmas until we've built our houses. You need a book like this which is long on ideas...have fun!