Linear representation theory of quaternion group – Groupprops

Center and zero
More great Turing

View as CPU design, instruction set for rational – i.e. Probability densities seen in cryptanalysis

View in terms of early comp sci, searching for a computable group (suited to improbability calculus)

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Subgroup structure of quaternion group – Groupprops

Relate center to quotient as Klein.

We saw this in tunny

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subgroups of quaternion group – Google Search

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Quaternion group picture

Contrast with fano plane which holds for octionions

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More Turing simila

Turings on permutation argument

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Eigenvalues and eigenkets in turings on permutations

Looking at turnings overall argument form, I now see that he does address pure states. When he talks about those group element with particular discriminators,these are the eigenkets for his rotational (pre-measurement) operator.

His system is still constrained to be looking at lambda2 (max difference from constants etc).

Interesting also that the inner product norm is to him just a (easy sum of squares) measure of the squiggle path one takes, with respect to another vector (including his eigenket vectors, k).

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Metric and Topological Spaces index
excellent uk teaching.

as turing was taught it.

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Fano plane – WikiVisually

Back to the future

Gives a nice notion of distance for measurable probability densities such as those found in Markova ciphers resisting 1943 era differential cryptanalysis

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The Turing era teaching on doubly transitive and a sequence of h1….hi, playing the role of n des rounds making a markov cipher

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Doubly transitive like Turing P22

Simple groups

Now we see why Turing left unstated the relationship to differential cryptanalysis.

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We must defend and protect federal networks and data,” Trump said during a meeting on cybersecurity. “We operate these networks on behalf of the American people and they are very important and very sacred.” The executive order had been scheduled for signing after the meeting. It was unclear when it would be signed.

Link needed.

Because NSA objected and , just as with the dhs attempt to usurp NSA role earlier, was able to state why things must remain with NSA. ;and that’s not even discounting the alternatives lack of deployed capability).

The emotional reason doesn’t take a lot of understanding (and even trump – t for torture and p for pussy, recall – understands that without control over the metadata, you cannot spy. And that means spying as much on congress as vlad the impaler.

At this point NSA will be pitching trump on how they validate and protect him (oh king, imperatur) as the preatorians. At his level of paranoia it will be doubly effective.

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​Let C be a conjugacy class of the symmetric group Sn . Recall that C consists of all of the permutations of a given cycle type. Hence we may define the support of C, denoted by supp(C), to be the number of nonfixed digits under the action of a permutation in C. If C has “large” support, then each element of C“moves” a lot of letters. If the permutations in C are all even, then we say that C is an even conjugacy class. If the permutations in C are all odd, we say that C is an odd conjugacy class. For n≥5, the subgroup generated by C, denoted by C, is Sn if C is odd, and An if C is even. 

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tunny run and quantum coins

think of each tunny run as a particular biased quantum coin space.

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building now see on each side of e (the changing slit pattern)a torus. 2 donuts in a box

ie the screen is the torus surface
we want the circle packing to  measure the embedding into spectral uniformity
we want gap minimzed when number of tangents on bith torus are equal

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Now we have broken through on des by applying turings quantum walk (on graphs whose eigenvalues are functions of the characters) we can turn to his 2d lattice, coded way of expressing his cipher knowhow.

What’s is interesting will be to relate light cones and Padic labeling to the n particle case of the des e functions “slit”, as subkey expansion evolve the positions of the slits, and thereby imposes a discriminator (attacking des)

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Shifting coined operators

Turing describes a quantum walk, using only the apparatus of a rotor machine.

What is cute is the very simple way the evolution shifts (the distances between the transformed state).

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Lovely orientation material for understanding math of colossus cryptographic attack on Tunney”>

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S box nibble sized graph evolution (with quantum coin from support between pairs of plaintext chars).

Distinguished from e(), that de-localized flows, simulating quantum coherence

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CS359G Lecture 4: Spectral Partitioning | in theory”>

We even get (after 3 years) why the tunny alphabet was ordered the way it was, when applied to cryptanalysis. See spectral partitioning and the way in which nodes were ordered.

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CS359G Lecture 4: Spectral Partitioning | in theory


As we will see in a later lecture, there is a nearly linear time algorithm that finds a vector {x} for which the expression {\delta} in the lemma is very close to {1-\lambda_2}, so, overall, for any graph {G} we can find a cut of expansion {O(\sqrt {h(G)})} in nearly linear time.

…for any tunny “run” known to exhibit a stats measurable bias, through just counting (on colossus).

Think of each std deviation worth of amplification as refining the cutset And tuning to the ciphertext, to best decide between improbably candidates (wheel s that in x or still in the edge expansion set)

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Des spectral gap theory

So the 7point geometry is special since it embodies cut sets (those nodes on the circle vs the “outer” triangle

U also look at the area of each as probability (of the event space). Noting obviously circle is less than triangle.

This is highly pertinent to quantum walks.

Now in terms of resistance to differential cryptanalysis one has to think of tunny attack: which wanted to chose just that cutest (between circle and triangle) that allowed a decision on two hypothesis tied to the improbability of each.

To resist D.C., the weight difference between the two h must be within the spectral gap.

Do I get the two sides of the des design argument. We are requiring the quantum random walk but also needing it amplifications to evolve the density to within the spectral gap do that h cannot be weighted in an attack.

Ok so I was right on my intuition, the other day. Now we have histologucal and pure math support.

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Turing reasoning when search

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How des works – again

So here we are again, explaining
the principles upon which des is designed. Yes it’s the nth time; and the same caveat as before: go anywhere else to learn someone’s description of the algorithm. Here we motivate the design.

So imagine you are studying the infamous quantum mechanics 2 hole screen – separating electron gun on the left from intensity screen. Or the 1 hole. Or the two holes shifted up (or down) a bit on the grate.

Oh and don’t forget that we can flip the grate, should we want the gun on the right (and the screen on the left).

After all like des, quantum mechanics is inevitably reversible, since “information” is conserved.

Now imagine that the e function of des is the grate.

And 2 of the six input bits of the current sbox are the slits in the grates.

The purpose of the way the e function shift inputs left (and right) is to simulate shifting the slits in the grate ip and down.


So that the qm normal density is shifted.

Our goal is to uniformly fill an intensity space, on the screen. Or better, average the left and right intensity screen (per the classical functional form.

Now, even though we shift our normal curve up and down (as the number of supports in the pair wise plaintext’s chats induce a key particle to move 1,2 3… lambada from the center, within the normal curve (and while constructing/destructing as we go) we still need the addition of intensity curves to uniformly fill the output space.

And here is where comes in the particular key schedule. It’s particular sequencing moves the grate around not only so feistels multiplexor covers all subpace but does so in such manner that guarantees that the concentration of intensity at any point (in 2d intensity space) is never more than the second eigenvalue gap.

Now recall that des does not have reversible sboxes. But we don’t need them! After all we have interleaved diffraction grates, since left 2 right we have half s des round and (right 2 left) we have the other half.

Now view the des subkey generatio functions own (highly programmed) bit duplication as a means of subtly (at huge granularity) measuring the quantum effect, thus influencing just how the left and right particle motions occur – giving a characteristic.

And ensuring that there exists no matrix representation of the same graph.

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beetles and support

​Let C be a conjugacy class of the symmetric group Sn . Recall that C consists of all of the permutations of a given cycle type. 
Hence we may define the support of C, denoted by supp(C), to be the number of nonfixed digits under the action of a permutation in C. 
If C has “large” support, then each element of C“moves” a lot of letters.

 If the permutations in C are all even, then we say that C is an even conjugacy class. If the permutations in C are all odd, we say that C is an odd conjugacy class. 

For n≥5, the subgroup generated by C, denoted by C, is Sn if C is odd, and An if C is even. 

now we understand how turing thought about avalanche -and the significance of 4. he used gccs terms, like beetles….

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From Enigma to rods to logs (to a fractional base)

The proper (fractional) base used when cryptanalyzing runny messages was not such a strange idea. After all, napiers original log tables were conceived in term of using a base that was a small offset from 1. E.g. O.9999999

I you were 1920 trained math person, you be trained in such “strange” log tables – and such mechanical devices with vertical wooden “rods” bearing abacus like beads. The latter could compute, using the log basis in question.

So think! Tunny back to rods.

And from rods we get back to enigma/Hebern crytanalysis, and isomorph searching.

Of course, rods in enigma algebra are less about logs and more about relative automorphisms (as one computes conjugates).

But you see the eureka transitions.

Once turing and co figured that a discriminator could exist for a rotor set, now one can leap to its log (and an algebra of bulges).

What unclassified docs don’t say is the parallel analysis going on with quantum mechanics calculations, given the parallel effort going on with the ;mostly compartmentalized) atomic bomb making efforts (circa 1944).

Must have been fun to be thinking about the ” potential” of the electron cloud in a colossus tube/valve and that similar controlled electron (well neuron) flow used to accelerate a uranium chain reaction.

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Des backdoor

Let’s imagine it’s 1980 again and you have a field full of custom des processors (just doing exhaustive search).

The kind trapdoor I’d want is to probably abandon a given key before having to
Complete all rounds, putting that key at the back of the queue. Given the cost of io in a cluster and the accounting, I want every key derived from that candidate by key schedule to similarly get filtered out of the early trials

So now imagine your field had a special precomputation area with electronics dedicated to the subkey scheduling. So what would that look like?

Look at today’s general purpose CPU pipelining for a good clue on how such custom vlsi was built then.

Also? Since io is the main impeder, think of clustering that is centered on the accounting and queue. One thinks of a giant ram cache… so how do multi cores today share gigs of ram? Probably sane back then…

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Hadamard actions

Each row of a Hadamard matrix is perpendicular to the previous one.

And hadamard gates implement conditional qbit operations.

Even in 1940 thinking, multiplying u by r meant figuring what the ciphertect would have been should the rotor have rotated the u.

What we want after n rebounds of des is that if u consider each round output to be ur then ur and ur2 differ as do rows on the hadamard matrix.

Ie half the bit differ.

Which is more tangible than “avalanche” since now we had a limiting condition – when the sequence of conditionals has induced the cipher text itself to be/become an action matrix ;that only produces outputs indistinguishable from uniformity)

In q terms, the data under a long sequence of qbit manipulation has itself evolved to become a hadamard action…

Ok so that means we have a replicating group. We have an expander whose action is to output a (key/plaintext parameyerizrd) expander code that is itself an action…

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