Change of bases being done with orthogonal projection derived from defining scalar product of two functions via finite integral of product of two functions: f dot g = integral f(x)g(x)dx over some interval.
This is common misconception. The truth is that in HE every plaintext can be encrypted to (exponentially iirc) many different ciphertexts. During encryption one of those is chosen randomly. This makes dictionary attacks practically impossible.
Edit: HE scheme (lwe) works on individual bits. Meaning there are only two plaintexts (0,1). Each has exponentially many ciphertexts, only one chosen at random. They also share ciphertext space, meaning each ciphertext could be either encrypted zero or one.
Maybe I'm missing something, but surely a dictionary based attack will work because you have to be able to know that your key has already been submitted by another user. That's the point of the application.
1) Initial report is filed.
2) Second report is filed by a user who only knows the attackers details.
3) Match is found
Therefore you can just keep iterating through names till you get a match.
Another way of saying it is that the application won't work if a second user can't tell that the first user has entered an attackers name.
The vulnerability is in the application specification, not HE.
"Elo suggested scaling ratings so that a difference of 200 rating points in chess would mean that the stronger player has an expected score (which basically is an expected average score) of approximately 0.75, and the USCF initially aimed for an average club player to have a rating of 1500."
I guess that means that Magnus has expected score of roughly 0.25^((3585 - 2864)/200) = 0.00675 against Stockfish 15, which is basically 1 in 200 games?
That is not quite the right calculation. To see this, try plugging 0.75 into the same formula to get Stockfish's expected score. The result is about 0.3545. If this were the correct formula, then the two expected scores should sum to 1, but in this case we only get 0.3612.
Instead, you should convert the 0.25 to "odds" form. 0.25 is 1:3 odds, represented by the number 1/3. (1/3)^((3585 - 2864)/200) is about 0.01905 (still in odds form). To convert this back to an expected score you would take 0.01905 / (1 + 0.01905) = 0.0187. So Magnus Carlsen's expected score is 0.0187.
Applying the same method to Stockfish, we have 3:1 odds, which is represented by the number 3. 3^((3585 - 2864)/200) is about 52.48. Converting back to expected score we get 52.48 / (1 + 52.48) = 0.9813. So Stockfish's expected score is 0.9813.
Our sanity check is to add 0.0187 + 0.9813. The result is 1.0, as it should be.
ELO is dependent upon the pool of players that one plays in. Engine ELO's have no relationship to human ratings because no humans play computers under normal conditions. For an example of this phenomena taken to extremes, Claude Bloodgood [1] was a strong amateur who ended up as officially rated as one of the top players in the world (and #2 in the US) simply because he was only playing against a pool of other prison inmates who were in turn playing only against each other. So all his rating reflected was his relative strength in the prison pool.
Computers are definitely much stronger than humans, but not 3600 better. Magnus would certainly be able to eek out plenty of draws, if not only because white can create "simplified" (as a euphemism for dead) positions in just about any variation if he really wants. And Magnus regularly plays these sort of positions literally at the level of supercomputers.
I'd also add that much of the dominance of computers is not based just on raw ability alone, but more psychological issues. Humans can become tilted, intimidated, frustrated, tired, and so on. One of the last major human vs computer events was Kramnik vs Fritz. Kramnik, in a relatively simple position, ended up blundering mate in 1 with plenty of time on his clock. It's unlikely he would have ever made the same mistake against a human. It's just very difficult to get in the same mindset when playing against a human as when playing against a computer. Chess, in spite of being a game of complete information, is still extremely influenced by psychology.
Winning chance is basically 0.0% against current strong engines for a human. There are possible draw lines but if engine is configured correctly draw chance is also 0.0%
As some other commenter calculated Stockfish wins in about ~>98% of the cases. The other ~<2% of the cases aren't Magnus winning, rather them drawing. I think no GM is known to beat a modern competitive AI in chess, however there are known/recorded instances of draws.
Machine and human ELO ratings are decorrelated. No games are played between a full strength engine and a human anymore. Even Magnus can’t win against a top engine. He might draw if he is lucky.
There is seemingly random interconnectedness in math, meaning that governments probably can't just throw money at some problem and force themselves much deeper than academia. For example you can hire 100 number theorists and ask of them to solve factorization (stupid example, i know), but it just might happen that the key insight to solving it comes from some random dude working in some seemingly disconnected problem in combinatorial algebra or something.
This is an inappropriate analogy. Chess at the highest levels is not drawish because the game is inherently drawish but because the meta of it is.
The easiest way to illustrate this is with openings, though it applies throughout the game in different ways. Against e4 the Najdorf defense was once the opening of champions, being a major part of the repertoire of players like Kasparov and Fischer. In modern times it's an increasingly rare guest at the highest level. It's not because it's considered unsound or even slightly dubious - it's a rock solid opening that gives black real winning chances. But the problem is it also gives him real losing chances. It's complex, difficult to play, and if you get outprepared by your opponent you may lose without him even having to make a single move himself, which is really one of the worst feelings in the world.
So instead the meta has largely shifted to openings that are more about minimizing risk where black, more or less, aims for a draw - and usually gets it. Changing the risk:reward ratios in a sufficiently extreme way is most certainly capable of changing the meta.
The current trend has to been to increase the value of any win, with things like 3 points for a win and only 1 point for a draw. In my opinion increasing the points based on color is an interesting idea, but it's unlikely to be seriously considered because of the fact that it would disrupt the "symmetry" in chess both the sort of abstract fashion, and also in the specific - such as requiring tournament pairing systems based on score to be completely reworked.
One tie break system, armageddon, does break the symmetry by giving black draw odds (he wins if the game is drawn) but less time in a blitz game. But it's very poorly regarded and generally only used as a this-game-MUST-have-a-decisive-result last resort.
There is a small chance that deep into the tree the first to play gets into zugzwang first, meaning position with no good moves. Is there any other scenario where playing black would give you theoretical edge?
So until proven otherwise it's still possible that its theoretical win for white, theoretical draw or theoretical win for black as i understand it.
Zugzwang is the only way for white to lose. Without Zugzwang and a winning strategy for black for chess with Zugzwang, white could skip the first move and now has the winning strategy.
Zugzwang means that you would be better off not moving any piece on your turn letting your opponent make two moves in a row. There are a number of endgames that depend on getting your opponent into a position where he has only one legal move and by making it you can then play the winning move. If your opponent could instead skip his turn you have no ability to win the game.
I know what zugzwang is. I was just asking what he meant by 'skipping turns' because skipping turns by simply refusing to make a move is illegal, so i was not sure what point was Gehinnn trying to make.
The question was if there is any other scenario where black could force a win other than by forcing a zugzwang.
I would say no by contradiction. Let's assume black could win without zugzwang.
Then white would win (and in particular not lose) by skipping the very first move and then playing blacks strategy (because now black has to make the first move and by white skipping the first move, the colors swapped).
If white would not skip the very first move and play an arbitrary move instead, white loses and black wins.
But this is the very definition of zugzwang! Thus, black can only win because of white's initial zugzwang, which contradicts our assumption.
I think I may share confusion with the other poster. I don't understand the following step:
> Then white would win (and in particular not lose) by skipping the very first move and then playing blacks strategy
I understand the other comment, that there do exist setups in which colors can be effectively switched by e.g. 1. e3 e5 2. e4, but that requires cooperation on black's part. How does white "skip" the first move? Thanks in advance.
Edit: it may be that the statement "without Zugzwang" implicitly (or perhaps by definition) means you are allowed to skip moves? If so, that clarifies my confusion.
Well, applying "zugzwang" means you only win because the opponent has to do a move and cannot skip their turn.
When black has a winning strategy, black already applies "zugzwang" for white's very first move: Black only wins because white has to make a move. If white could skip, black would not win.
> Edit: it may be that the statement "without Zugzwang" implicitly (or perhaps by definition) means you are allowed to skip moves?
Yes. It's not well defined, but I'd say a non-zugzwang win is a win (or rather a winning position) where you would also win when your opponent can skip their turn. A zugzwang win is a win that is not a non-zugzwang win.
Ahh okay I think I get it now, you are saying if chess is win for black, that immediately implies zugzwang for white from starting position.
In the same way, chess being a win for white implies zugzwang for black starting position.
So chess being a win for one side is equivalent to starting position being zugzwang for the other side.
It's obvious now, but so interesting to me, I never thought about it that way! Thanks for taking time for explaining yourself.
White could win without zugzwang!
Because white starts. After the first move, the position is no longer symmetric, so it doesn't help black to skip to switch sides.
Provably there's little that can be said about chess at large when it comes from opening to endgame. Heuristically speaking though matching moves against even a beginner is a very bad strategy for play, as a recent speedrun being done by GM Aman Hambleton on youtube has been showing.
Having that happen in one place deep in the tree wouldn't mean much: black would need to be able to get such a zugzwang (or other win) against anything white does.
"One might define simplicity as the length of the equation, say, and accuracy as how close the curve gets to each point in the data set, but those are just two definitions from a smorgasbord of options."
"..the algorithm evaluated candidate equations in terms of how well they could compress a data set. A random smattering of points, for example, can’t be compressed at all; you need to know the position of every dot. But if 1,000 dots fall along a straight line, they can be compressed into just two numbers (the line’s slope and height). The degree of compression, the couple found, gave a unique and unassailable way to compare candidate equations."
There is long history of science-fiction writers painting correct visions of, at the time seemingly impossible future. Sometimes You don't have to be an expert to notice the trends in certain domains. I'd go even further - it could be easier to notice the big picture without being overly bothered with nitty-gritty technical details - yes, to further the field they are necessary, but to comment the direction the field is heading in and about its implications for society they are not.
I don't necessarily agree with the author, I'm just making a general comment.
> "There is long history of science-fiction writers painting correct visions of, at the time seemingly impossible future"
That's a pretty weak argument. I love scifi, and I love for example Philip Dick's writing, yet I would not consider PKD's opinion on the future of AI/AGI particularly relevant.
James Cameron is not an authority on AI either.
If people said "Yudkowsky is a nice fanfiction author" it would be one thing. But he considers himself an actual AI researcher, and that's just not right. He is not qualified, and has no accomplishments in the area, other than writing fanfiction about it.
You keep hammering on with the same lazy slander. Yudkowsky was well known long before he wrote the popular Harry Potter fanfic, which incidentally is pedagogical / allegorical. Because his main roles are teacher and philosopher, and philosophers do that.
But you'll just say there's nothing of value there, and it's somehow figuratively "fan fiction", because he didn't go to college, and he doesn't work much on ML, which is clearly the end-all of AI.
"There is long history of science-fiction writers painting correct visions of, at the time seemingly impossible future."
Science fiction writers tell entertaining lies to amuse their readership. They are generally not really trying to "pain correct visions" of anything, and you are greatly exaggerating the extent they have success in this endeavor, which again, is generally not something they are even trying to accomplish.
