That's a good question. I thought they are only using PQC for key exchange (which is referred to as Level 2) but they are not.
In the article, Apple explains why they choose to use Level 3:
> At Level 2, the application of post-quantum cryptography is limited to the initial key establishment, providing quantum security only if the conversation key material is never compromised. But today’s sophisticated adversaries already have incentives to compromise encryption keys, because doing so gives them the ability to decrypt messages protected by those keys for as long as the keys don’t change. To best protect end-to-end encrypted messaging, the post-quantum keys need to change on an ongoing basis to place an upper bound on how much of a conversation can be exposed by any single, point-in-time key compromise — both now and with future quantum computers. Therefore, we believe messaging protocols should go even further and attain Level 3 security, where post-quantum cryptography is used to secure both the initial key establishment and the ongoing message exchange, with the ability to rapidly and automatically restore the cryptographic security of a conversation even if a given key becomes compromised.
Those networks are named as discrete neural networks. There are already research on those, mainly for (homomorphic) encryption purposes (because it’s much easier to homomorphically encrypt NNs of just 0’s and 1’s than normal NNs).
I grew up in China and came to the same conclusion as yours! I never expect such a similarity. I've always thought that education in the US must be much better.
After graduating from college, I realized that the problem I was facing was a systematic one of the whole society, rather than one limited to particular teachers, middle schools, or even the entire education system.
Many people say Chinese maths education is better than the US but I can hardly agree. But based on what I have seen, there are problems on both sides. Chinese education is focusing too much on memorizing existing pieces of knowledge, but too less on teaching the young how to create new ones. The knowledge which our ancestors had struggled for thousands of years to find was taught to us in a spendthrift manner. Aside from lacking training on how to find/create new knowledge, Chinese education does not encourage students to learn advanced topics since it could have negative effects on the students' grades. But there is nothing you can do to change it, because too many things are correlated: fair distribution of teaching resources, less demand for highly educated people in the job market, and the overall not-so-innovation-appealing social vibe. I cannot foresee any possibility of a true, self-driven, systematic reform.
Education in the US, especially math education, on the other hand, is somehow too frivolous. I have no learning experience in any US middle school, so my opinions can be biased. But it seems that US education is more like elite education. The average/universal maths education level should be a little higher in such a highly modernized society.
These different (or even opposite) problems surprisingly show some similarity. Shall I say the problems actually reflect some real problems in the two societies?
I had the experience of going to US schools among competitive immigrants from Taiwan and HK(in the late 90's, i.e. they left while it was still under the British), and a little bit of mainland China as well.
Reflecting on it, it produced an odd dichotomy in classroom expectations where nobody was really on the same page: I'm fourth-generation American to a mixed European background - my mom insisted on me attempting advanced math, but in a distinctly Eastern European sense, with emphasis on learning theory, which wasn't anything like what I was confronted with at school, which was primarily computational drills that I didn't know how to prepare for and which my parents tried to pretend I could just power through, as my older brother did(he had more of a direct interest, and later confessed that he probably got through it all just with short-term memory, because he was diagnosed with ADHD as an adult and started medicating, and thought I should too). My classmates, meanwhile, had clearly normalized strict study habits but could not usefully communicate what they were to me, or maybe did not want to give up their secrets. And the teachers were just pleased that the class behaved so well and could withstand being assigned piles of homework, but they didn't have particularly advanced backgrounds themselves and often couldn't hold their own when challenged by the best students in the city.
And then I went off to college and the student body was now mostly white. I realized that this was a completely different vibe and I didn't understand that, either.
I think the places in which the US system manages to work are because sometimes the collision of varied cultures against the institutions produces useful sparks. The institution itself tracks political winds, which vary at the state and local level. Struggling schools have the usual issues of domestic insecurity spilling into the classroom, and being in the public school system, occasionally I would cross paths with those students instead of the "gifted and talented" track that I was on. But "good schools" tend to be "home owners association" schools, whipped into doing whatever the parents ask for, which usually amounts to fairy tale fantasies. When my mom started pressuring the faculty for me to stay in the advanced math track despite my not fitting there, it was, I now see, in this latter mode. Eventually, not getting the desired result, she insisted that I argue my own case, which of course I was terrible at, and left me confused, ashamed and other feelings which took years to work through. I just wanted to withdraw from everything at that point, but I was being hurried along. That is the one quality I would say tends to always be the case throughout, at least in the large schools I went to - nobody has time for anything, because everyone has a deadline to meet. It's mostly an illusion and busywork, but it nevertheless sucks out societal energy.
The elite students, some of whom I ran into in college, tend to have a path carefully paved for them through subtle signalling and tracking - opportunities and experiences that are just not the norm for anyone less wealthy. They aren't getting well-rounded educations either, rather, they are normalized to self-identify as strivers, which when combined with some early connections, is enough for most of the cohort to advance. I had a housemate who was an heir to an major beer company executive. He was an alcoholic and his dad was, too, and he bemoaned the idea that his summer job was being the boss to people ten decades older than him. His goal in getting a CS degree was to prove that he could do something for himself, essentially.
In the end, looking at it, the way the US system is set up is to not know you are in a rat race until it's too late and you're tracked at the bottom for reasons beyond your control.
Agreed. So what you need is the 'complex structure' behind rather than just 'complex numbers'. Any form of representations (numbers, matrices, and so on) should correspond to a unique structure. The question why the complex structure emerges in quantum mechanics is more interesting.
Complex numbers have two roles in mathematics. The first is as a number system based upon SO(2) the group of rotations in 2D, the second is as the algebraic closure of the reals. That these two are the same thing is somewhat of a fluke (it doesn't work in higher dimensions).
Physics uses complex numbers in the first sense. There's really nothing too special about SO(2), there's an SO(n) for all n.
Whereas mathematics uses complex numbers in both senses. There is something rather special about complex numbers as the algebraic closure of the reals and it's what makes a lot of modern math tick.
The complex numbers are the closure regardless of dimension. When I was writing that I was thinking of the Quaternions, which are the 4 dimensional analog of the complex numbers, in 2^N dimensions this is the Cayley Dickson construction.
The fluke is this: Euclidean space of dimension N has N(N-1)/2 rotational dimensions. If you plug 2 into that you get 2x1/2 which is 1 dimension. i.e. the rotations in 2D space look like a circle. If you add an extra dimension (the radius) you get the polar form of complex numbers.
In other dimensions this doesn't always work. In 3 dimensions we have 3x2/2 = 3 rotational dimensions, so we need a space with dimension 4 (the quaternions). In 4 dimensions we need a 6 dimensional rotation space. We just established that Cayley Dickson algebras only come in powers of 2, so it doesn't fit at all.
But couldn't one use two unit quaternions to describe rotations in 4d space? An antisymmetric matrix in 4 dimensions has N(N-1)/2 = 6 independent variables. But each unit quaternion has three independent variables, so two of them would be enough to describe rotations in 4d.
Using modern languages for hardware design seems to be a tendency today. I wonder how does MyHDL compare to Chisel? Chisel is another hardware description language based on Scala, which has already been wildly used in the RISC-V community. Check it out here: https://www.chisel-lang.org
In the article, Apple explains why they choose to use Level 3:
> At Level 2, the application of post-quantum cryptography is limited to the initial key establishment, providing quantum security only if the conversation key material is never compromised. But today’s sophisticated adversaries already have incentives to compromise encryption keys, because doing so gives them the ability to decrypt messages protected by those keys for as long as the keys don’t change. To best protect end-to-end encrypted messaging, the post-quantum keys need to change on an ongoing basis to place an upper bound on how much of a conversation can be exposed by any single, point-in-time key compromise — both now and with future quantum computers. Therefore, we believe messaging protocols should go even further and attain Level 3 security, where post-quantum cryptography is used to secure both the initial key establishment and the ongoing message exchange, with the ability to rapidly and automatically restore the cryptographic security of a conversation even if a given key becomes compromised.
Article link: https://security.apple.com/blog/imessage-pq3/