For 50 years, lots of aging men and a few women kept lots of secrets – or what they thought were secrets – about really sensitive math. This was the math related to cipher making and cipher breaking.
We can now dumb down the fancy math, since 50 years has passed.
Put yourself in the shoes of a 16 year old Turing, brimming with anticipation about the way the new math represents the latest 1920s-ish breakthroughs in physics. So excited is he about the possibilities, he teaches himself the advanced math – bringing down the pseudo-ish-complexities of academics and intellectuals to the level a slight immature 14 year old (as 16) can grasp. In doing so, he has to earn the annoyance of the drill-teaching scholastic system, prepping the next generation of oppressors to play their role in subjugating the empire’s peon class (and sub-humans, in English human rights lexicon).
It turns out that all the fancy crypto math in the world doesn’t really need much more than a 14 year old (today) gets taught. One just has to be taught right (which starts with realizing WHY one is not taught right, since its not hard).
So to “phase space”. Boy does “phase space” sound complex. Well actually it is…typically since its uses complex numbers. But only for that reason is it complex. Since complex numbers are just a list of 2 real numbers. Wow. Millions of poor little kids were taught drivel about imaginary numbers just to to hide the crypto fact that a complex number is two numbers in a list. It’s that simple (unless you are TURING to make to complex, which is a pun).
So what is phase space? It’s just a pack of cards, or the shuffled pack rather. each “state” in phase space is one of the shuffles you can make. It’s the result of the shuffle. Its not the shuffling process itself, though that’s a bit ambiguous, since each step in a shuffle process is… of course, an intermediate shuffle.
So we arrived at “group theory”. Or shuffling, intermediate shuffles, and an sequence of intermediate shuffles that make a shuffle. Confused? Get a pack of cards, start shuffling, and call each shuffle a “state” (of the pack of cards). yes, in a geometric sense the “space” is the very volume of the cards stack in your paw.
So what is the concept of phase doing there (and why not just have state)? Well that’s a fun question. Phase is like phase of the moon, since its cyclic. And phase in phase space is no different. Since crypto is all about periodicities (just like the moon wandering across a 12 years bedroom window, each night), we have phase of the shuffle – much as there are full moons and half moons, etc.
Of course, the 13 year know that the moons phases are due to the dynamics of the earth, the sun and moons orbits – which cast different earth shadow upon the moon, changing the visible area – known as phase, since the phase “changes” each night … as the 3 parties move around and induce the next shadow in the sequence.
And so it is in shuffling, where phase 1 requires picking a card, phase 2 requires placing a card, and phase 3 requires “giving a state” name to the result. For example, pick a card from the middle, place it on top, and designate this as the state whose name is, forevermore: “top”. “top” varies from another state, formed by picking the second from top card, putting it in the top, and call that state “top top”. The second formulation of a phase space is more predicable, because if you do the required shuffling “action” a third time, the first two cards merely transpose. Do it a fourth, the same thing happens, as it does for the fifth, etc. In this “state” known as top-top all that happens ever is that the top two cards swap over. The rest of the pack is “not involved” – because of the way that the dynamics were set up in “phase space”. The periodicity is two (for this phase space “configuration” known as top-top).
Now to vector spaces- since we have conquered phase (moon shadows!). Boy do many kids hate vectors – being so poorly taught. Which is a shame because they are just list of scalars. So what is a scalar (another frightening word)? Its a scale with a strange r on the end, dummy. (Remember, cryptopolicy was all making simple math seem hard, so you look the other way while you are spied on by the exceptionals, so inducing buyin to the social deception planned by the military psychologists who are part of the program). If boy kids at 12ish are measured by the school doctor as (below/above 4 feet, balls not/dropped), the latter is the “scale” (of two elements there). I seem to recall being measured that way; and presumably a girl’s puberty gets a similar score – on the female puberty “scale”. Quite what the scale is for transgendered kids, I don’t know.
So now that we know that a vector is just a list (of numbers, or our puberty scale points or “points” in general), we can ask: what is a vector *space*? Well its just the space occupied in your paw by a list of 52 numbers making up a single vector, which is a list of points (or a list of measurement results for 28 kids in a medical line…). Or it that’s too abstract, it’s the ordered list of the cards in your actual shuffle (still sitting in your paw, having contemplated phase space). Or it’s the line of 28 boys waiting to get their balls observed by the doctor doing the puberty test, each with a school id number attached to them that orders them. contrary to the 1960s TV show, you are a number in America, remember. And that’s all you are (Vietnam Vet, no vet, or otherwise) when reduced to dead state.
So, can we add vectors? Well hats just asking can we put two lists of numbers back to back. And that’s all it means. Its just strange that the result of the addition means we have to think most about the join, versus the ends. And it really helps if you now “complete the parallelogram” thinking about the 2 anti-lists that complement the first lists. When I was 14, I just imagined each list to be 2 of the sides of soccer field, starting at a corner (which represented the “join” of the 2 vectors, in my mind). The two other sides (joined at the diagonally opposite corner) were the anti field – that took a long time to run to – even if you ran along the middle of the parallelogram. That area was reputed to exist though (being so far away, when you have little legs of a 12 year old, and you don’t have your glasses on while playing soccer). I didn’t really need to ever go there, since it was just like the local corner and its two sides (just reversed..) Later, I learned to call it the “adjoint”.
Now, part of school sports for any 12 year old requires running around a football field, especially if you are being class punished (for looking at the girls perhaps, in England). Since we didn’t have any girls (till 16), we all felt we were being false punished so I for one punished the impunity of the punishers (later to be associated with the exceptionals) by doing math, instead. starting at a near-corner, I’d run OUT along vector A (the long side of the pitch). Then I’d run IN along short vector B (of the adjoint side of the pitch, connected to the initially far-corner). Then id run OUT along vector A (of the adjoint side, still connected to the initially-far corner). Then on the home run, Id run IN along long vector B (or the home corner). Then I do what MOST 14 year old, after 4 years of English math teaching, could NEVER do. I could associated the “addition of” the vectors as that line connecting the far ends of the IN and the OUT spoke of either the near or the far corner.
Now, that helps but the real help came when then considering the nature of a commutator. A commutator is what’s left over when you take away BA from AB. Or in football field, when you have run around sides A, B’, A’ and B and despite having take the same number of steps on AB and B’A’, you are still not at the home corner. It’s a couple of feet short (or you had to go passed, a couple of feet long), since the parallelogram ain’t regular. Said in math terms (to confuse you and make you think crypto is hard), its what happens when two terms don’t commute, so AB ain’t BA. Of course, real numbers do (5 + 4 = 4 +5). But if you remember your matrices from being 14, THERE you learned that the order mattered (since matrices being multiplied con’t commute, just like a funky football field where the linesman drawing the while lines got drunk…and failed to square things up). Or . in America, the typical road lines (which are really crappily painted, by “the exceptionals” who have to do it all 4 times, making a horrid visual mess of endless corrections).
So, now we know what a space is, a group is, an operation is (adding list..), a phase is, and even what a commutator is (the result of drunk exceptional in charge of a painting machine for football fields).
So what is a quantum mechanical wave function, when its element in the group, and stuff like vector spaces, vector fields and other university math stuff are thrown in to make it all complex?
Well, I have to admit that one got me for years. I could not explain that to 12 year olds for 2 decades, until crypto saved the say. Now its easy. Thanks Turing!
Any 12 year knows that an enigma wheel is a disk, with electrical wires connecting a metal pad on one side to the pin on the other. And, you can put several disks in a row (pads to pins) making a circuit through them, when you connect up the battery and lamps. And, some know that the last wheel is a reflector (just like on the back of your bike) – in which 13 of the pads are just wired to one of the other 13 pads (rather than pins). Thus the electricity goes through the 3 wheels one way, gets reflected back, and comes back through the 3 wheels, coming our on some pad – that is not the same as the we stuffed the electricity in, on. Well, we just did functions, in phase space, where each function is a “point” – just like the ink dot that results when you press your ball-POINT pen into page. Looking just at the first wheel, we put power in (on point/pin A, say), and got power out (on point/pin B, say)
Lets see how how we get to make things hard, and fall things “functions” now. Its really easy (to make things hard). Just think like a 12 year old who is taught math in a US school. Or don’t… so we stand a change of understanding crypto. I.e. be UN-exceptional (i.e. not an American ).
Lets get back to vectors, that are sides of the football pitch (which is that parallelogram drawn by the drunk exceptional, high on weed, having got exceptional standards to meet, with a little cheating help…and with which to mask the true lack of exceptionalism!). Now, Ive never smoked anything (cough! splutter! at just the idea); let alone weed. But I’m led to believe that it makes you VERY interested in the few “commutator” feet at the end of the parallelogram that “don’t make sense” – but do.. (when you are high). Having had too many vicodin once too quickly (since the tooth ache was just unbearable), I get the idea. I seem to remember incredible focus (like I was at the dentist’s door; knocking at 8am and 1 mS). And that 1mS seems endless, enough time to commute with the gnat whose natural lifetime was 1% over and also ponder both the missing 2 feet at the end of the parallelogram run and the nature of functions (that are points). Hmm (reaching for the empty bottle of opiates).
One can now equate adoint space with a high space; an altered universe, where geometry still holds, but all the lines run the wrong way around (like the sides running out of the opposite corner, so far away…). Or, now to get to functions, with the weird world where you first consider the OUTPUT of the function and think only then about going BACK through the wormhole of the function to the INPUT. This is like the football field run again, in which one runs along a functions input (side A) and then the output (short side B’) of the adjoint’s function (associated with the opposite corner, recall), and then the input (long side A’) of the adjoint – showing in 3 steps we have found – at step 2 – our first case of “start at the output (B’) and get to the input (A’) – backwards thinking-like. Bother to complete the run along side 4 we do see our normal case (input leads to output), which is A from the first side, and B from the fourth side – in “normal order” of input begets output, even!
What’s interesting then is that adjoint function (output begets input) only appeared when stuffed in the middle, so to speak, of a normal “input begets output” function. Consider just those three first sides, we ran A, B’, A’. Considering the last 3 sides, we ran B’, A’, B. Note the pattern. Turing did. B’ is the “input” in the first case conjugated by A-ness – which requires an A (and A’) on EACH SIDE. And similarly, A’ is surrounded by B-ess, by which process B conjugates A’. If this is hard, just run around the football field… If you are an American, pretend exceptionalism by smoking weed, so its easier (by cheating). We will look the other way, for now so long as you will lose a pound or two, too. But the point is that “conjugation” is the class of “function due to football field runs” – unlike boring functions on blackboards like y = f(x). Remember, Turing liked to RUN! He got to write out equations such as:
y(B,A) = A.B.A’ —- or y(A, B’) = B’.A.B
which may = some C – or may = some E (for enigma, perhaps)
which just happen to be the “functional space” in which the enigma machine calculates. Its also where quantum mechanics calculates. So if you can do one, you can do the other! Remember, there is MORE than type of calculator! Neither QM nor crypto is like the $1 calculator. Each is, however, related to other, with the crypto function space being a “stochastic” evolution (vs. a hamiltonian evolution).
So what is stochastic? It’s a long word designed to make you forget that the exceptionals are STILL spying on you, systemically. So what is a system (apart from a long word…)?
Well, thanks to Turing he tells us and leaves us with a simple mental model, that fits12 years olds fascinated by math. Just as each corner of the parallelogram was a “point” and each run between neighboring point was an “edge” to be run, so other “systems” (other than parallelogram group systems) have points and edges. Or vertices and edges, if you prefer. Vertex sound more star trekky, than point. If you imagine the Jewish star (two triangles), it has 5 points (vertices) and the edges between them (that design out the characteristic look). That’s another system (of 5 points, and a certain number of edges you get to count). You can even run around its points/vertices and edges, as a funky run around a funky football pitch. The good news (assuming you are still in your 1mS moment), is that commutators exist here, too. We can be left with a few feet missing in this system, too! Commutators are a system thing. And they are really useful, when making ciphers.
Now, here comes the mind bending part. Each point can be function. And functions can be acted on by edges (much as A ad B got added, when the points were As and Bs.) Lets see how.
Remember the enigma wheel? well it’s a function. why? because it takes 26 inputs and makes 26 outputs (or vice versa, if you put the electricity in reverse … to convention). Call it the innermost wheel (since Turing did). Then call the next wheel in a system of 3 wheels in a cage the Middle wheel, noting it’s a function too. Turing did (and of course, part of the fun of enigma is that any wheel-function could be in any position in the fixed cage, becoming thereby denoted as inner, middle or outer). SO, we have a system of 3 functions, each of which is a point on a VERY funky geometry, which gets hard to visually imagine running around. But one could (and the true exceptionals, American or otherwise, actually can). I cannot (being an untalented unexceptional, fit only to be spied upon).
As we start to finish, we get to where we started; at phase space. phase space is just that geometry where points are functions – including the running electricity through the wirings of enigma wheels.. that then rotate or “step” – so changing the MAP of functions (per step). So, 12 year old, remember that you have to have a map to wander through the complex geometries – which should feel JUST like a easter egg puzzle (another Turing favorite, circa 1920). Just be aware that what stepping of the enigma wheels do is CHANGE the map (so having yesterdays map helps not, today).
Or so those spying on you would have you believe.
There is a coda to the story. Turing taught us to look for the non-exceptional groups. Thus, no American need look (being only concerned with the exceptional). For in the non-exceptional groups there are strong ciphers. When you are ready, go study non-exceptional lie groups. one gets to the notion that the points are all connected to each other (so you can run between one and the other along “just one” side. Within this non-exceptional space (where no Americans need apply), we get strong theorems for stochastic evolutions that expand just how many previous maps you have to have (to have any change of breaking Turings ciphers).
Now you know this, non Americans, who revel in their inherent non-exceptionalism, you can address their spying on you. Go make a cipher! Courtsey Alan Turing.