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# Is it possible to pick your Ed25519 public key?

Is it possible to generate an Ed25519 keypair that has a very similar public key as another keypair (fooling a casual visual comparison) or is this as hard as solving one of SHA-512 or the discrete logarithm problem?

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If you can store the private key with some pre-computed work, then you can pick almost any public key you want. So in a way, it depends on the implementation.

Here's a diagram of how Ed25519 works, note how keys are generated:

(Image source.)

A more detailed description (that is simpler than the actual paper) of the process is in these slides (slides 9 - 11).

To recap the image and slides: The secret is hashed and public key is derived from the "left half" of the hash. The left half is manipulated to be value $a$, which is a bit string with a few conditions. Then $a$ is applied to a curve point $B$ to produce the public key $A$.

The critical thing to note is that the seed's hash output is used in key generation. Apparently implementations differ on what they store as the private key, but the algorithm itself does not need the original secret, only the hash from it. This means that if you control the implementation you use, you can just make up any hash that suits your purposes and you don't need to care what the corresponding secret would be. (This prevents the need to finding any sort of SHA-512 pre-image.) We can use this by working backwards from the desired public key value and deciding what hash value we want.

The goal of this attack is to choose the value of $A$, or to at least choose something very similar to it. Call the new target public key we're trying to build $A'$, then if $A'$ is a valid public key there is an $a'$ such that $a'B = A'$. This reduces to finding the inverse on the elliptic curve, $a' = AB^{-1}$, which is actually the Discrete Logarithm problem and certainly non-trivial to calculate. But if you could find a valid $a'$, you would have everything you need.

As an important side-note, not all values of $a$ are valid in the scheme. See the Ed25519 paper for all restrictions for $a$ (I believe they are all listed on pg 6). There aren't very many restrictions, I think the most restrictive one is that $a$ be divisible by 8 (ie, the low three bits are 0). Assuming you can can calculate the $a'$ from a chosen $A'$, you could just pick random $A'$s until you find one that produces a valid $a'$. This step would prevent you from explicitly choosing your public key, because you have no guarantee that the desired value for $A'$ is valid. You can probably find a valid $A'$ after only a few tries, so just flip a bit here and there in the desired $A'$ and you should be able to find one that works and has a small Hamming distance from the original desired $A'$, which is probably good enough to produce keys that look alike.

If you have $a'$ then you have the public key and you can create a corresponding private key. You can make up a random "right-side" of the SHA-512 hash value and incorporate the new $a'$ to construct the private key.

This all assumes you can control the implementation and solve the necessary discrete logarithm. Regarding the implementation, the NaCl project has a high-speed performance-oriented implementation of Ed25519, and as such it stores as much pre-computed data as possible in the private key. It would allow you to perform this sort of manual crafting of the keys. But in general, with other implementations, you may have no such luck. I would think that for most attacks, the attacker should be assumed to be able to choose the implementation they use.

(Hopefully this is all correct, I hadn't looked at Ed25519 before this.)

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Doesn't your algorithm assume that the attacker can easily solve the discrete logarithm problem on this elliptic curve? Elliptic curves are essentially logarithmic, so your "division" is actually discrete logarithm. The equivalent of A=aB on an elliptic curve is A=g^a using the multiplicative group modulo a prime. – CodesInChaos Aug 23 '12 at 6:37
Can you fix the division part? The rest of the answer contains a nice explanation why the attacker doesn't care about k, but making it sound like DL is easy is highly misleading. – CodesInChaos Aug 25 '12 at 10:02
@Codes: Yep, you're right. Let me see if there's a better way to go about it, or if I just need to re-word that entire section. – B-Con Aug 25 '12 at 15:37
Your talking about "multiplicate inverse" is still weird. AFAIK there is no operation that multiplies two points on a curve. There is only an operation that multiplies a scalar with a curve point. If you look at the analogous operation in a finite field $B^{a'}=A'$, you can't find an inverse element for $B$ either. – CodesInChaos Sep 6 '12 at 13:38
Thanks, I missed that somehow in the previous edit. I'm not sure why I thought "multiplicative inverse" was the right way to word that, originally. – B-Con Sep 7 '12 at 22:36

The core of the problem is finding a near first pre-image on the function $A = aB$ on an elliptic curve, where $A$ is the public key, and $a$ the private key¹.

For a normal hash function you $2^m$ operations to fix $m$ specific bits.² In particular a full pre-image takes $2^n$ hash function calls.

A full pre-image on $A = aB$ is equivalent to solving the discrete logarithm problem on that curve. This problem only needs $2^{n/2}$ operations, which is something around $2^{126}$ for Ed25519³. i.e. much faster than for a hash function, but still prohibitively slow.

Now the interesting question is, if you want to find only $m < 126$ matching bits, how much work do you need? And I can't answer that, so this doesn't really answer your question. I know of no way to do this faster than brute-force, but I'm certainly no expert on this matter.

¹ The standard implementation of Ed25519 does use a different private key $k$ from which $a$ is derived via hashing, but an attacker would simply skip that step, so we can ignore it. It does not affect security.

² If you don't care about which bits you fix, then the problem becomes a bit easier than $2^m$ but it doesn't matter that much.

³ I believe Curve25519 has a small subgroup, reducing security by a few bits, so I assumed a 126 bit security level. A few bits more or less don't matter, especially since one "operation" can be much more expensive for some primitives than for others.

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The subgroup size is 251 bits, so your estimate is correct give or take half a bit. – Daira Hopwood Oct 18 '15 at 12:37

Looking at the Ed25519 paper, it seems that a key pair consists of a private key $k$ (just a $k$-bit string) and a public key $A$, where $A = a · B$ and $a$ is derived from the first half of the hash $H(k)$ ($B$ is the base point of an elliptic curve (or actually a twisted Edwards curve equivalent to an elliptic curve) and $·$ is the scalar multiplication in this curve).

A different private key $k'$ (even if just different by one bit) ought to create a quite different hash $H(k')$, and thus quite likely also different $a'$ and $A'$.

The only way to succeed here would be to brute-force different values for $k'$ and see if the difference between $A$ and $A'$ is below some limit, but that will take quite long (depending on how high you set the difference limit).

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You can ignore the hashing step, it doesn't affect security. An attacker wins if he finds $a$, he doesn't care about finding a $k$. – CodesInChaos Aug 23 '12 at 8:03
@CodesInChaos: The question was about creating a key pair which looks similar, and this contains a $k$. Of course, the $a$ itself is sufficient for signing, and depending on the actual goal it might be enough to find such one. – Paŭlo Ebermann Aug 24 '12 at 17:21
I believe the root question is: "Can an attacker impersonate a known public key by creating a keypair with a similar looking public key, for which he can execute private key operations such as signing or diffie-hellman." The actual representation of the private key is irrelevant. – CodesInChaos Sep 6 '12 at 13:47

I assume from the "fooling a casual visual comparison" in the question that it is actually talking about visual comparison of the fingerprint of the public key. So, absent a full break of Ed25519, even if we had some fancy efficient way to generate public keys with particular structure, an attack still requires a near-second-preimage for the hash used for the fingerprint. (Not a near-collision attack, because it does not benefit the attacker just to generate two similar public key fingerprints; they have to generate a fingerprint that is similar to an existing one for a victim server or servers.)

An ssh fingerprint is an MD5 hash of the Base64-encoded public key. MD5 is a relatively old hash algorithm and there are known attacks better than brute force for collision-resistance and near-collision-resistance. The status of MD5 for near-second-preimage resistance is less clear.

Note that in principle, an attacker might not need a secure or valid public key. So we might think of generating invalid key encodings, or encodings that vary in redundant information that doesn't require us to do a point multiplication per trial, and combining that with a near-second-preimage attack on MD5. In practice I think this is unlikely to be feasible given that Ed25519 key encodings are designed to always represent elements of a group in which discrete log is hard (even if not the intended elliptic curve subgroup), and AFAIK they do not contain redundant information.

That leaves brute force multi-target attacks. Suppose we are satisfied with finding a second preimage for a fingerprint that agrees in k specified bits with any of n target fingerprints. The expected cost of finding one by brute force is 2k / n hashes. (Note: we don't have to do a full point multiplication for each trial, because it's sufficient to iterate through valid keys by incrementing the secret key and adding the group base to the public key -- so we only need a point addition per trial, and the Base64 encoding and hash, at most.)

If you don't care where the matched bits are, then there are L-choose-k times as many possibilities where L is the hash output size (128 in this case), so the cost is then 2k / (L-choose-k * n). However, what we're actually interested in is matched hex digits (i.e. groups of 4 bits) rather than bits. To match h such digits in a 16-digit hash, the cost is 16h / (16-choose-h * n).

To reduce the cost further it may also be beneficial to consider visually similar characters, e.g. 3 and B, as matches, and similar tricks such as allowing transposition.

Also, note that the attacker doesn't necessarily have to choose the same key exchange algorithm. The OpenSSH command-line client displays the key type by default, but other clients may not, and the user might not know what the key type should be. So, if it were somehow easier to generate visual matches for a different algorithm than Ed25519 (either by a constant factor due to less overhead per trial, or to facilitate a faster-than-brute-force near-second-preimage attack on MD5), then they would probably do that, even if targeting the fingerprint(s) of an Ed25519 key or keys.

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B - L

The difference between the baryon number and the lepton number${\ displaystyle B}$${\ displaystyle L}$ , in short , is a many-particle quantum number in particle physics . Each particle is assigned a baryon number and a lepton number ; At the level of the elementary particles, each lepton receives the lepton number and each quark the baryon number , since a baryon is always made up of three quarks. Antiparticles are given the respective negative quantum number. Both quantities are additive and are independent conservation quantities in the standard model of elementary particle physics described in perturbation theory . In a large unified theory (GUT) as an extension of the Standard Model, neither the number of baryons nor the number of leptons are preserved independently of one another, but their difference is preserved. This enables a baryogenesis with simultaneous leptogenesis to fulfill the Sakharov criteria for the emergence of the asymmetry between matter and antimatter. ${\ displaystyle BL}$${\ displaystyle L = + 1}$${\ displaystyle B = 1/3}$

B - L as the conservation size

The property of the difference between the number of baryons and the number of leptons as a conserved quantity results from the absence of anomalies in a quantum field theory and is related to the weak hypercharge of the particles. The weak hypercharge is the charge of the symmetry group, under which the equations of motion are invariant. With the introduction of an additional symmetry group as an extension of the standard model, only the assignment of the quantum numbers for quarks and for leptons leads to a consistent description (apart from a general normalization factor) . ${\ displaystyle U (1) _ {Y}}$${\ displaystyle U (1)}$${\ displaystyle 1/3}$${\ displaystyle -1}$

If the symmetry represents a calibration symmetry , a corresponding calibration boson must exist. These hypothetical particles are called leptoquarks because they couple to both leptons and quarks in a vertex. ${\ displaystyle U (1) _ {BL}}$

Example: proton decay

The proton decay is a prediction of many GUT variants, so that the final state is a neutral pion and a positron (anti-electron): ${\ displaystyle \ pi ^ {0}}$

${\ displaystyle p ^ {+} \ to \ pi ^ {0} e ^ {+}}$

The proton is a baryon , the pion is a meson and therefore has and the positron is an antilepton . Therefore neither baryons nor leptons are preserved in this process, but the difference . In the standard model, however, the proton is the lightest baryon and is stable. Since the lower bound on the experimental life is very high proton (about 10 33 years, compared to the universe is "only" 10 10 years old), has the mass of any -Eichbosons in the order of 10 16 GeV are . ${\ displaystyle B = 1}$${\ displaystyle B = L = 0}$${\ displaystyle L = -1}$${\ displaystyle BL}$${\ displaystyle U (1) _ {BL}}$

• Sphaleron , a non-perturbative process in the Standard Model that receives neither nor , but .${\ displaystyle B}$${\ displaystyle L}$${\ displaystyle BL}$

format: plain


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# Installing With Cmake Configuration Options¶

This documents discusses the various Cmake build options that can be used when compiling Basilisk. These build options can be set via the GUI Cmake program or specified via the command line.

## Optional Cmake Flags¶

The Basilisk CMake file contains a series of flags that can be turned on to include additional features within Basilisk or to build in a particular configuration. The flags are listed in the above image. Note that tuning on some flags will invoke the conan package manager to download and install additional dependencies. Basilisk modules that require particular dependencies will not be compiled unless their support software is installed. Associated BSK module unit tests are skipped if the required software is not provided through these Cmake flags.

The following table summarizes the possible flags:

cmake Flag

Default

Description

USE_PYTHON3

ON

enables Basilisk to be compiled for Python 3

USE_PROTOBUFFERS

ON

installs the Google protobuffers package via conan

USE_ZMQ

ON

installs the 0MQ distributed messaging system via conan

USE_OPENCV

OFF

installs the OpenCV package via conan

USE_COVERAGE

OFF

turns on code coverage when compiling with GCC on a Linux platform

This table summarizes what cmake flags are required to get particular Basilisk features:

Feature

Flags

Python 3

USE_PYTHON3

Vizard Support

USE_PROTOBUFFERS, USE_ZMQ

VisNav Modules

USE_OPENCV

## Simplest Basilisk Build¶

The bare-bones version of Basilisk is invoked by not turning on any Cmake flags as shown. In the GUI Cmake all the optional compiler flags shown be de-selected as shown in the above figure. To use the command line cmake command use the regular Basilisk set as discussed in the platform specific Basilisk installation files.

## Building for Python 2 or 3¶

To make cmake setup a build process to compile Basilisk for Phython 3, then use:

-DUSE_PYTHON3=ON

If this flag is set to false it is built for Python 2.

## Building to Support Vizard Basilisk Playback¶

To include the vizInterface module that enables saving off and communicating with the Vizard Unity based visualization, the two flags USE_PROTOBUFFERS and USE_ZMQ must be turned on as shown in the following CMake GUI screen show.

To use the command line cmake command use the regular Basilisk compile tool with the additional arguments:

-DUSE_ZMQ=ON -DUSE_PROTOBUFFERS=ON


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## On equilibration and coarsening in the quantum O(N) model at infinite $N$    [PDF]

Anushya Chandran, Arun Nanduri, S. S. Gubser, S. L. Sondhi
The quantum O(N) model in the infinite $N$ limit is a paradigm for symmetry-breaking. Qualitatively, its phase diagram is an excellent guide to the equilibrium physics for more realistic values of $N$ in varying spatial dimensions ($d>1$). Here we investigate the physics of this model out of equilibrium, specifically its response to global quenches starting in the disordered phase. If the model were to exhibit equilibration, the late time state could be inferred from the finite temperature phase diagram. In the infinite $N$ limit, we show that not only does the model not lead to equilibration on account of an infinite number of conserved quantities, it also does \emph{not} relax to a generalized Gibbs ensemble consistent with these conserved quantities. Nevertheless, we \emph{still} find that the late time states following quenches bear strong signatures of the equilibrium phase diagram. Notably, we find that the model exhibits coarsening to a non-equilibrium critical state only in dimensions $d>2$, that is, if the equilibrium phase diagram contains an ordered phase at non-zero temperatures.
View original: http://arxiv.org/abs/1304.2402


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lilypond-user
[Top][All Lists]

## Stanza Numbers

 From: Jan Kohnert Subject: Stanza Numbers Date: Wed, 21 May 2003 17:51:43 +0200 User-agent: KMail/1.5

I have some trouble with the stanza numbers:
If the beginning line of the melody has a partial as first bar, there are
displayed stanza numers, but if I don't wan't to use \partial (or just cannot
if I have "before-play" (sorry, I don't know the right word) ) there are no
stanza numbers displayed. I send two pieces for you to see what I mean.
What am I doing wrong and how do I get this numbers even when there is a rest
at first?
I even tried \property LyricsVoice.stz = "1." but there are the first numbers
displayed in the second line and I want them to beprinted in the first.
I use Lilypond-1.6.9

Regards Jan Kohnert

with.ly
Description: Text document

without.ly
Description: Text document


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# Bounded chain complexes and the bounded derived category

Let $$\mathcal{A}$$ be an abelian category and consider the following categories:

• $$\mathbf{Ch} (\mathcal{A})$$, the category of cochain complexes in $$\mathcal{A}$$.
• The full subcategories $$\mathbf{Ch}^\mathrm{b} (\mathcal{A})$$ (cochains bounded), $$\mathbf{Ch}^- (\mathcal{A})$$ (cochains bounded above), and $$\mathbf{Ch}^+ (\mathcal{A})$$ (cochains bounded below).
• The derived category $$\mathbf{D} (\mathcal{A})$$, i.e. the category of cochain complexes in $$\mathcal{A}$$ localised with respect to quasi-isomorphisms.
• The full subcategories $$\mathbf{D}^\mathrm{b} (\mathcal{A})$$ (cohomology bounded), $$\mathbf{D}^- (\mathcal{A})$$ (cohomology bounded above), and $$\mathbf{D}^+ (\mathcal{A})$$ (cohomology bounded below).
• $$\mathbf{Q}^{-1} \mathbf{Ch}^\mathrm{b} (\mathcal{A})$$, the category of bounded cochain complexes localised with respect to quasi-isomorphisms, and analogously $$\mathbf{Q}^{-1} \mathbf{Ch}^- (\mathcal{A})$$ and $$\mathbf{Q}^{-1} \mathbf{Ch}^+ (\mathcal{A})$$.

By the universal property of localisation, the inclusion $$\mathbf{Ch}^\mathrm{b} (\mathcal{A}) \hookrightarrow \mathbf{Ch} (\mathcal{A})$$ induces a functor $$\mathbf{Q}^{-1} \mathbf{Ch}^\mathrm{b} (\mathcal{A}) \to \mathbf{D} (\mathcal{A})$$; similarly we have functors $$\mathbf{Q}^{-1} \mathbf{Ch}^- (\mathcal{A}) \to \mathbf{D} (\mathcal{A})$$ and $$\mathbf{Q}^{-1} \mathbf{Ch}^+ (\mathcal{A}) \to \mathbf{D} (\mathcal{A})$$. It is easy to see that a cochain complex whose cohomology is bounded (resp. bounded above, bounded below) is quasi-isomorphic to a cochain complex that is itself bounded (resp. bounded above, bounded below). Thus, these functors factor as essentially surjective functors $$\mathbf{Q}^{-1} \mathbf{Ch}^\mathrm{b} (\mathcal{A}) \to \mathbf{D}^\mathrm{b} (\mathcal{A})$$, $$\mathbf{Q}^{-1} \mathbf{Ch}^- (\mathcal{A}) \to \mathbf{D}^- (\mathcal{A})$$ and $$\mathbf{Q}^{-1} \mathbf{Ch}^+ (\mathcal{A}) \to \mathbf{D}^+ (\mathcal{A})$$.

Question. When are these functors full and/or faithful?

My impression is that the correct definition of bounded derived category is the one denoted by $$\mathbf{D}^\mathrm{b} (\mathcal{A})$$ above, but it did not occur to me until just now that this might be different from $$\mathbf{Q}^{-1} \mathbf{Ch}^\mathrm{b} (\mathcal{A})$$. I imagine that if $$\mathcal{A}$$ has enough projective objects then $$\mathbf{Q}^{-1} \mathbf{Ch}^- (\mathcal{A}) \to \mathbf{D}^- (\mathcal{A})$$ is an equivalence of categories, and dually if $$\mathcal{A}$$ has enough injective objects then $$\mathbf{Q}^{-1} \mathbf{Ch}^+ (\mathcal{A}) \to \mathbf{D}^+ (\mathcal{A})$$ is an equivalence of categories – it should be one of those standard arguments about reducing zigzags using resolutions – but I have not checked, and even if this is correct, it does not answer the question about $$\mathbf{D}^\mathrm{b} (\mathcal{A})$$. It also leaves open the possibility that there is some impoverished $$\mathcal{A}$$ where these categories are genuinely different.

They are always fully faithful, and this doesn't require enough projectives or injectives.

A map in $$\mathbf{D}(\mathcal{A})$$ from $$X$$ to $$Y$$ is represented by a diagram $$X\stackrel{s}{\leftarrow}Z\stackrel{f}{\to}Y$$ where $$s$$ is a quasi-isomorphism (call this map $$fs^{-1}$$), and if $$t:Z'\to Z$$ is a quasi-isomorphism then $$fs^{-1}=(ft)(st)^{-1}$$.

But if $$X$$ (and hence $$Z$$) has cohomology bounded above, then there is always a quasi-isomorphism $$Z'\to Z$$ from a bounded above complex: for sufficiently large $$n$$, let $$Z'$$ be the "good" truncation $$\tau_{\leq n}Z:=\dots\to Z^{n-2}\to Z^{n-1}\to\ker(d^n)\to0\to0\to\dots$$ of $$Z$$, and let $$t$$ be the inclusion map.

So both ways of defining $$\mathbf{D}^-(\mathcal{A})$$ are "the same".

A dual argument works for $$\mathbf{D}^+(\mathcal{A})$$.

For $$\mathbf{D}^b(\mathcal{A})$$ use the first argument followed by the second.

• Ah yes, it's coming back to me now! Prima facie this argument only shows the functor is full, but this argument plus the calculus of fractions shows faithfulness as well. Very good. – Zhen Lin Jul 30 '20 at 4:30


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## Banach Journal of Mathematical Analysis

### On a J-polar decomposition of a bounded operator and matrices of J-symmetric and J-skew-symmetric operators

Sergey M. Zagorodnyuk

#### Abstract

In this paper we study a possibility of a decomposition of a bounded operator in a Hilbert space $H$ as a product of a J-unitary and a J-self-adjoint operators, where J is a conjugation (an antilinear involution). This decomposition shows an inner structure of a bounded operator in a Hilbert space. Some decompositions of J-unitary and unitary operators which generalize decompositions in the finite-dimensional case are also obtained. Matrix representations for J-symmetric and J-skew-symmetric operators are studied. Simple basic properties of J-symmetric, J-skew-symmetric and J-isometric operators are obtained.

#### Article information

Source
Banach J. Math. Anal. Volume 4, Number 2 (2010), 11-36.

Dates
First available in Project Euclid: 7 February 2011

https://projecteuclid.org/euclid.bjma/1297117238

Digital Object Identifier
doi:10.15352/bjma/1297117238

Mathematical Reviews number (MathSciNet)
MR2606479

Zentralblatt MATH identifier
1200.47050

Subjects
Primary: 47B99: None of the above, but in this section
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$

#### Citation

Zagorodnyuk, Sergey M. On a J-polar decomposition of a bounded operator and matrices of J-symmetric and J-skew-symmetric operators. Banach J. Math. Anal. 4 (2010), no. 2, 11--36. doi:10.15352/bjma/1297117238. https://projecteuclid.org/euclid.bjma/1297117238

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• S.M. Zagorodnyuk, Direct and inverse spectral problems for (2N+1)-diagonal, complex, symmetric non-Hermitian matrices, Serdica Math. J. 30 (2004), 471–482.
• S.M. Zagorodnyuk, The direct and inverse spectral problems for (2N+1)-diagonal complex transposition-antisymmetric matrices, Methods Funct. Anal. Topology 14 (2008), 124–131.
• S.M. Zagorodnyuk, Integral representations for spectral functions of some non-selfadjoint Jacobi matrices, Methods Funct. Anal. Topology 15 (2009), 91–100.


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# Contents

## Idea

Classical Galois theory classifies field extensions. It is a special case of a classification of locally constant sheaves in a topos by permutation representations of the fundamental groupoid/fundamental group.

Even more generally one can define a Galois group associated to a presentable symmetric monoidal stable (infinity,1)-category. There is an analogue of the Galois correspondence in this setting, see Mathew 14.

## Classical Galois theory for fields

We discuss the classical/traditional case of Galois theory, which concerns the classification of field extensions. Below in Galois theory for schemes and then in Galois theory in a topos we discuss how this is a special case of a more general concept of Galois theory in a topos.

### Galois theory of fields

###### Definition

We call a field extension $K \subset L$ a Galois extension if $K \subset L$ is algebraic and there exists a subgroup $G \subset Aut(L)$ of the automorphism group such that $K \simeq L^G$ is the field of elements that are invariant under $G$.

###### Definition

If $K \subset L$ is a Galois extension, we define the Galois group to be

$Gal(L/K) := Aut_K(L) \,.$

This means that we have

$K \simeq L^{Gal(L/K)} \,.$
###### Definition

Let $\bar K$ be a fixed algebraic closure of $K$. If $F \subset K[X] - \{0\}$ is any collection of non-zero polynomials, the splitting field of $F$ over $K$ is the subfield of $\bar K$ generated by $K$ and the zeros of the polynomials in $F$.

We call $f \in K[X]- \{0\}$ separable if it has no multiple zero in $\bar K$.

We call $\alpha \in \bar K$ separable over $K$ if the irreducible polynomial $f^\alpha_K$ of $\alpha$ over $K$ is separable.

A subextension $L \subset \bar K$ is called separable over $K$ if each $\alpha \in L$ is separable over $K$.

###### Definition

We call $L$ normal over $K$ if for each $\alpha \in L$ the polynomial $f^\alpha_K$ splits completely in linear factors in $L[X]$.

###### Theorem

Let $K$ be a field and $L$ a subfield of $\bar K$

$K \subset L \subset \bar K \,.$

Denote by $I$ the set of subfields $E$ of $L$ for which $E$ is a finite Galois extension of $K$. Then $I$, when partially ordered by inclusion is a directed poset.

The following assertions are equivalent:

1. $L$ is a Galois extension of $K$.

2. $L$ is normal and separable over $K$.

3. There is a set $F \subset K[X] - \{0\}$ of separable polynomials such that $L$ is the splitting field of $F$ over $K$.

4. $\coprod_{E \in I} E \simeq L$

If these conditions are satisfied, then there is a group isomorphism

$Gal(L/K) \simeq {\lim_{\leftarrow}}_{E \in I} Gal(E/K) \,,$

where on the right we have the limit over the poset of subfield of the contravariant functor $E \mapsto Gal(E/K)$.

Since each group $Gal(E/K)$ is finite, the above isomorphism can be used to equip $Gal(L/K)$ with a profinite topology (i.e. take the limit in the category of topological groups, where each $Gal(E/K)$ has the discrete topology), making it into a profinite group. We henceforth consider $Gal(L/K)$ as a profinite group in this way.

###### Theorem

Let $K \subset L$ be a Galois extension of fields with Galois group $G$. Then the intermediate fields of $K \subset L$ correspond bijectively to the closed subgroups of $G$.

More precisely, the maps

$\{E | E\;is\;a\;subfield\;of\;L\;containing\;K\} \stackrel{\overset{\phi}{\to}}{\underset{\psi}{\leftarrow}} \{H|H\;is\;a\;closed\;subgroup\;of\;G\}$

defined by

$\phi(E) = Aut_E(L)$

and

$\psi(H) = L^H$

are bijective and inverse to each other. This correspondence reverses the inclusion relation: $K$ corresponds to $G$ and $L$ to $\{id_L\}$.

If $E$ corresponds to $H$, then we have

1. $K \subset E$ is finite precisely if $H$ is open (in the profinite topology on $G$)

$[E:K] \simeq index[G:H]$ if $H$ is open;

2. $E \subset L$ is Galois with $Gal(L/E) \simeq H$ (as topological groups);

3. for every $\sigma \in G$ we have that $\sigma[E]$ corresponds to $\sigma H \sigma^{-1}$;

4. $K \subset E$ is Galois precisely if $H$ is a normal subgroup of $G$;

$Gal(E/K) \simeq G/H$ (as topological groups) if $K \subset E$ is Galois.

This appears for instance as Lenstra, theorem 2.3.

This suggests that more fundamental than the subgroups of a Galois group $G$ are its quotients by subgroups, which can be identified with transitive $G$-sets. This naturally raises the question of what corresponds to non-transitive $G$-sets.

### In terms of separable algebras

#### Free modules

Let $A$ be a commutative ring and $N$ a module over $A$.

A collection of elements $(w_i)_{i \in I}$ of $N$ is called a basis of $N$ (over $A$) if for every $x \in N$ there is a unique collection $(a_i)_{i \in I}$ of elements of $A$ such that $a_i = 0$ for all but finitely many $i \in I$ and $x = \sum_{i \in I} a_i w_i$.

If $N$ has a basis it is called free (over $A$). If $N$ is free with basis a finite set of cardinality $n$, then we say that $N$ is free with rank $n$ (over $A$). In this case, $N$ is a finitely generated free module.

Let $N$ be a finitely generated free $A$-module with basis $w_1, w_2, \cdots, w_n$ and let $f\colon N \to N$ be $A$-linear. Then

$f(w_i) = \sum_{j = 1}^n a_{i j} w_j \;\;\; (1 \leq i \leq n)$

for certain $a_{i j} \in A$, and the trace $Tr(f)$ of $f$ is defined by

$Tr(f) = \sum_{i = 1}^n a_{i i} \,.$

This is an element of $A$ that only depends on $f$, and not on the choice of basis. It is easily checked that the map $Tr : Hom_A(N,N) \to A$ is $A$-linear.

#### Separable algebras

Let $A$ be a ring, $B$ an $A$-algebra, and suppose that $B$ is free with finite rank $n$ as an $A$-module. For every $b \in B$ the map $mult_b\colon B \to B$ defined by $mult_b\colon x \mapsto b x$ is $A$-linear, and the trace $Tr(b)$ or $Tr_{B/A}(b)$ is defined to be $Tr(mult_b)$. The map $Tr\colon B \to A$ is easily seen to be $A$-linear and to satisfy $Tr(a) = n a$ for $a \in A$.

The $A$-module $Hom_A(B,A)$ (underlying which is the hom-set in the category of modules) is clearly free over $A$ with the same rank as $B$. Define the $A$-linear map $\phi\colon B \to Hom_A(B,A)$ by

$\phi(x) : y \mapsto Tr(x y) \,,$

for $x, y \in B$.

###### Definition

If for an $A$-algebra $B$ the the morphism $\phi$ is an isomorphism we call $B$ separable over $A$, or a free separable $A$-algebra if we wish to stress the condition that $B$ is finitely generated and free as an $A$-module.

#### Separable closure

Recall the notion of separable elements

###### Definition

Let $K$ be a field and $\bar K$ an algebraic closure of $K$. The separable closure $K_S$ of $K$ is defined by

$K_S \simeq \{x \in \bar K | x \; is \; separable \; over \; K\} \,.$
###### Remark

We have that $K_S$ is a subfield of $\bar K$ and that $K_S \simeq \bar K$ precisely if $K$ is a perfect field, in particular if the characteristic of $K$ is 0.

From xyz it follows that the inclusion $K \subset K_S$ is Galois.

###### Definition

The Galois group $Gal(K_S/K)$ is called the absolute Galois group of $K$.

#### Galois theory for separable algebras

###### Theorem

Let $K$ be a field and $\pi_1(Spec K)$ its absolute Galois group. Then there is an equivalence of categories

$SAlg_K^{op} \simeq \pi_1(Spec K) Set \,.$

## Galois theory for schemes

The classical Galois theory for fields is a special case of a general geometric/topos theoretic statement about locally constant sheaves and the action of the fundamental group on their fibers.

### Statement of the main theorem

###### Definition

A morphism $f : Y \to X$ of schemes is a finite étale morphism if there exists a covering of $X$ by affine open subsets $U_i = Spec A_i$, such that

• for each $i$ the open subschemes $f^{-1}(U_i)$ of $Y$ is affine,

• and equal to $Spec B_i$, where $B_i$ is a free separable $A_i$-algebra.

In this situation we also say that $f : Y \to X$ is a finite étale covering of $X$.

A morphism from a finite étale covering $f : Y \to X$ to a finite étale covering $g : Z \to X$ is a morphism of schemes $h : Y \to Z$ such that $f = g \circ h$.

This defines the category $FEt_X$ of finite étale covers of $X$.

###### Theorem

Let $X$ be a connected scheme. Then there exists a profinite group $\pi_1(X)$ – the fundamental group of $X$ – uniquely determined up to isomorphism, such that the category of finite étale coverings $FEex$ is equivalent to the category $Fin \pi_1(X) Set$ of finite permutation representations of $\pi_1(X)$ (finite sets, with the discrete topology, on which $\pi_1(X)$ acts continuously).

This appears for instance as Lenstra, main theorem 1.11. It is fully discussed in SGA1.

The profinite group, $\pi_1(X)$, is often called the étale fundamental group of the connected scheme $X$. In SGA1, Grothendieck also considers coverings with profinite fibres, and a profinitely enriched fundamental groupoid. In the above the actual group $\pi_1(X)$ depends on the choice of a fibre functor given by a geometric point of $X$. Different choices of fibre functor produce isomorphic groups. Taking two such fibre functors yields a $\pi_1(X)$-torsor for either version of $\pi_1(X)$. This is important in attacks on Grothendieck's section conjecture.

###### Examples
• The disjoint union of $n$ copies of $X$ corresponds, under this theorem, to a finite set of $n$ elements on which $\pi_1(X)$ acts trivially.

• The fact that for $X = Spec \mathbb{Z}$ there are no other finite étale coverings of $X$ is thus expressed by the group $\pi_1(Spec \mathbb{Z})$ being trivial .

• The same is true for $\pi_1(Spec K)$, where $K$ is an algebraically closed field.

• If $K$ is an arbitrary field, then $\pi_1(Spec K)$ is the absolute Galois group of $K$; i.e. the Galois group of the separable closure $K_S$ over $K$. In this case theorem 4 is a reformulation of classical Galois theory.

• In particular, if $K$ is a finite field, then $\pi_1(Spec K) \simeq \hat \mathbb{Z}$.

• Let $X = Spec A$, where $A$ is the ring of integers in an algebraic number field $K$. Let $N$ be the maximal algebraic extension of $N$ that is unramified at all non-zero prime ideals of $A$. Then $\pi_1(X)$ is the Galois group of $N$ over $K$.

• More generally, if $a \in A$, $a \neq 0$, then $\pi_1(Spec A[\frac{1}{a}])$ is the Galois group, over $K$, of the maximal algebraic extension of $K$ that is unramified at all non-zero prime ideals of $A$ not dividing $a$.

### Reproducing classical Galois theory of field extensions

In this section we explain the connection between the main theorem of Galois theory for schemes, theorem 4, and classical Galois theory.

We denote by $k$ a field. It is our purpose to show that the opposite category of the category of free separable $K$-algebras is equivalent to the category of finite $\pi_1(X)$-sets, for a certain profinite group $\pi_1(X)$. This is a special case of the main theorem 4, with $X = Spec K$. In the general proof we shall use the contents of this section only for algebraically closed $K$. In that case, which is much simpler, the group $\pi_1(X)$ is trivial, so that the category of finite $\pi_1(X)$-sets is just the category of finite sets.

(…)

## Grothendieck’s Galois Theory

In SGA1, Grothendieck introduced an abstract formulation of the above theory in terms of Galois categories. A Galois category is a category, $\mathcal{C}$, satisfying a small number of properties together with a fibre functor $F: \mathcal{C}\to FinSet$, preserving those properties. The theory is more fully described in the entry on Grothendieck's Galois theory.

## Galois theory in topos theory

One notices that classical Galois theory has an equivalent reformulation in topos theory. That puts it into a wider general abstract context and leads to a topos-theoretic general Galois theory.

### Reformulation of classical Galois theory

###### Observation

The étale morphisms $f : Y \to X$ corresponds precisely to the locally constant sheaves on $X$ with respect to the etale topology, in that it is equivalently a morphism for which there is an etale cover $\{U_i \to X\}$ such that $f$ is a constant sheaf on each $U_i$.

For $K$ a field let $Et(K)$ be its small étale site. And

$\mathcal{E} := Sh(Et(K))$

the sheaf topos over it. This topos is a

Then Galois extensions of $K$ correspond precisely to the locally constant objects in $\mathcal{E}$. The full subcategory on them is the Galois topos $Gal(\mathcal{E}) \hookrightarrow \mathcal{E}$.

The Galois group is the fundamental group of the topos.

### Topos-theoretic Galois theory

Accordingly in topos theory Galois theory is generally about the classification of locally constant sheaves. The Galois group corresponds to the fundamental group of the topos .

(…)

### Higher topos theoretic Galois theory

In the context of higher topos theory, there are accordingly higher analogs of Galois theory. According to shape theory, any (∞,1)-topos $\mathbf{H}$ has an associated fundamental ∞-groupoid $\Pi(\mathbf{H})$, which in general is a pro-space whose $1$-truncation $\Pi_1(\mathbf{H})$ is the ordinary fundamental groupoid of the underlying 1-topos. Classical topos-theoretic Galois theory states that locally constant sheaves (of sets) on a locally connected topos are equivalent to representations of $\Pi_1(\mathbf{H})$, i.e., functors $\Pi_1(\mathbf{H})\to Set$. This generalizes to higher topoi as follows:

###### Theorem

Let $\mathbf{H}$ be a locally n-connected (n+1,1)-topos, $-1\leq n\leq \infty$. Then there is an equivalence of categories

$\mathbf{H}^lc \simeq \Fun(\Pi_{n+1}(\mathbf{H}), n Grpd),$

where $\mathbf{H}^{lc}\subset\mathbf{H}$ is the subcategory of locally constant objects.

This generalization of Galois theory is discussed in (Grothendieck 75, Hoyois 13, Hoyois 15). For further discussion in the case $n=\infty$, see

## ‘La Longue Marche à travers la Théorie de Galois’

Between January and June 1981, Grothendieck wrote about 1600 manuscript pages of a work with the above title. The subject is the absolute Galois group, $Gal(\overline{\mathbb{Q}},\mathbb{Q})$ of the rational numbers and its geometric action on moduli spaces of Riemann surfaces. This will (one day be) discussed at Long March. Other entries that relate to this include anabelian geometry, children's drawings (in other words Dessins d’enfants, which is the study of graphs embedded on surfaces, their classification and the link between this and Riemann surfaces) and the Grothendieck-Teichmuller group.

The anabelian question is: how much information about the isomorphism class of an algebraic variety, $X$ is contained in the étale fundamental group of $X$? Grothendieck calls varieties which are completely determined by their étale fundamental group, anabelian varieties. His anabelian dream was to classify the anabelian varieties in all dimensions over all fields. This can be seen to relate to questions of the étale homotopy types of varieties.

Tim: I have a feeling that this anabelian question should have a form that generalises to higher dimensions. (Not that I can shed much light on progress in dimension one.) Perhaps there is an anabelian version of the homotopy hypothesis or something of that nature.

(…)

## References

Lecture notes on the Galois theory for schemes are in

• Hendrik Lenstra, Galois theory for schemes , Mathematisch Instituut Universiteit van Amsterdam (1985) (pdf)

Some of the material above is taken from this.

A comprehensive textbook is

a review of which can be found at Galois Theories.

The locally simply connected case is discussed for instance in

• Marco Robalo, Galois Theory towards Dessins d’Enfants (pdf)

Galois theory in a presentable symmetric monoidal stable (infinity,1)-category is developed in


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# Simplicial structures over the 3-sphere and generalized higher order Hochschild homology

Document Type : Research Paper

Authors

1 Department of Mathematics, Ohio Northern University, Ohio, United States of America

2 Department of Mathematics, St. Norbert College, Wisconsin, United States of America

Abstract

In this paper, we investigate the simplicial structure of a chain complex associated to the higher order Hochschild homology over the $3$-sphere. We also introduce the tertiary Hochschild homology corresponding to a quintuple $(A,B,C,\varepsilon,\theta)$, which becomes natural after we organize the elements in a convenient manner. We establish these results by way of a bar-like resolution in the context of simplicial modules. Finally, we generalize the higher order Hochschild homology over a trio of simplicial sets, which also grants natural geometric realizations.

Keywords

#### References

[1] Anderson, D.W., Chain functors and homology theories, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, Wash., 1971), pages 1-12. Lecture Notes in Math. 249, 1971.
[2] Carolus, S., Properties of higher order Hochschild cohomology, OhioLINK Electronic Theses and Dissertations Center, Columbus, Ohio, 2019. Dissertation (Ph.D.)– Bowling Green State University.
[3] Carolus, S., Hokamp, S.A. and Laubacher, J., Deformation theories controlled by Hochschild cohomologies, to appear in São Paulo J. Math. Sci., 14 (2020), 481-495.
[4] Carolus, S., Laubacher, J., and Staic, M.D., A simplicial construction for noncommutative settings, to appear in Homology Homotopy Appl., 23(1) (2021), 49-60.
[5] Carolus S. and Staic, M.D., G-algebra structure on the higher order Hochschild cohomology H S2 (A;A), to appear in Algebra Colloq. arXiv:1804.05096, 2020.
[6] Cohen, R.L., Hess, K., and Voronov, A.A., String topology and cyclic homology. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2006. Lectures from the Summer School held in Almería, September 16-20, 2003.
[7] Corrigan-Salter, B.R. Coefficients for higher order Hochschild cohomology, Homology Homotopy Appl. 17(1) (2015), 111-120.
[8] Corrigan-Salter, B.R. Higher order Hochschild (co)homology of noncommutative algebras, Bull. Belg. Math. Soc. Simon Stevin 25(5) (2018), 741-754.
[9] Corrigan-Salter, B.R. and Staic, M.D., Higher-order and secondary Hochschild cohomology, C. R. Math. Acad. Sci. Paris 354(11) (2016), 1049-1054.
[10] Gerstenhaber, M., The cohomology structure of an associative ring, Ann. of Math. 78(2) (1963), 267-288.
[11] Gerstenhaber, M., On the deformation of rings and algebras, Ann. of Math. 79(2) (1964), 59-103.
[12] Gerstenhaber, M. and Schack, S.D., Algebraic cohomology and deformation theory, In: Deformation theory of algebras and structures and applications (Il Ciocco, 1986), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 247, 11-264, Kluwer Acad. Publ., Dordrecht, 1988.
[13] Ginot G., Higher order Hochschild cohomology, C. R. Math. Acad. Sci. Paris 346(1-2) (2008), 5-10.
[14] Ginot, G., Tradler, T., and Zeinalian, M., Higher Hochschild homology, topological chiral homology and factorization algebras, Comm. Math. Phys. 326(3) (2014), 635- 686.
[15] Hochschild, G., On the cohomology groups of an associative algebra, Ann. of Math. 46(2) (1945), 58-67.
[16] Laubacher, J., Secondary Hochschild and cyclic (co)homologies, OhioLINK Electronic Theses and Dissertations Center, Columbus, Ohio, 2017. Dissertation (Ph.D.)–Bowling Green State University.
[17] Laubacher, J., Properties of the secondary Hochschild homology, Algebra Colloq. 25(2) (2018), 225-242.
[18] Laubacher, J., Staic, M.D.,and Stancu, A., Bar simplicial modules and secondary cyclic (co)homology, J. Noncommut. Geom. 12(3) (2018), 865-887.
[19] Loday, J-L., Opérations sur l’homologie cyclique des algébres commutatives, Invent. Math. 96(1) (1989), 205-230.
[20] Loday, J-L., “Cyclic homology", Grundlehren der Mathematischen Wissenchaften [Fundamental Principles of Mathematical Sciences] 301, Springer-Verlag, Berlin, second edition, 1998. Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with T. Pirashvili.
[21] Mac Lane, S., “Homology". Classics in Mathematics, Springer-Verlag, 1995. Reprint of the 1975 edition.
[22] Pirashvili, T., Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. 33(2) (2000), 151-179.
[23] Staic, M.D., Secondary Hochschild cohomology, Algebr. Represent. Theory 19(1) (2016), 47-56.
[24] Staic, M.D. and Stancu, A., Operations on the secondary Hochschild cohomology, Homology Homotopy Appl. 17(1) (2015), 129-146.
[25] Weibel, C.A., “An Introduction to Homological Algebra", Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1994.


text:
A note on meromorphic $$m$$-valent starlike functions.(English)Zbl 1206.30013

Summary: In the present investigation, we derive sufficient conditions for starlikeness of meromorphic $$m$$-valent functions in the punctured disc.

MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
Full Text:


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# Back to the Future with Smart Clients

Billy Hollis
Elysian Consulting

March 24, 2004

Summary: While Web applications have received the bulk of the interest over the last few years, improvements in the client mean that it's time to re-investigate client-side development. In this new column, .NET in the Real World, authored by Microsoft Regional Directors, Billy Hollis looks at smart clients and how you can use them to build applications today. (6 printed pages)

Related Books

Once upon a time dinosaurs ruled the earth; they were called mainframes. If your hair isn't as gray as mine, you might not remember that world. Let me tell you a bit about how it worked. In that world, the big mainframe system did all the processing. It assembled a page of information and sent that page out to a device called a terminal. The most common type of terminal was called a 3270.

Terminals were rather stupid devices. They had no processing power to speak of. All a user could do with a terminal was fill in data fields and navigate around the screen.

When the user was done with a page of data, he or she pressed a button on the terminal and sent the page's information back to the mainframe. Then the mainframe took in the page's information and processed it, and got ready to send out a new page.

Then PCs came along, and then they got hooked to networks, and finally somebody said, "Hey, we can do better than that mainframe stuff. We can use our PCs' processing power to make the user interface more intelligent and remember lots of things about how the user needs to work. That would help the users do their tasks, and be a lot more productive."

Thus was a new type of system born. The best of these systems did a great job of distributing the work among the users' PCs and the big central machine, which was rechristened a server. These client-server systems allowed both PCs and large central systems to each do what it did best. Client-server displaced many of the mainframes, and people liked it much better. The users were productive and happy.

Then along came the Internet. At first it was just used to browse static, hyperlinked pages. But then some very bright young fellows said, "Hey, look what we can do! We can have an application run on this Internet. We'll have a large central Web server that will do all the processing. And it will send out pages to users, who will view the page in a browser, which will let them navigate around the page and enter some data. Then the users will press a button and send the page's information back to the big system, which will process it and send back a new page. Won't that be great!"

The browser didn't have much intelligence built into it. It was supposed to be a "thin client," meaning the PC wasn't supposed to do anything except run this browser. Thus PCs with more processing power than the mainframes of twenty years earlier were reduced to being pretty analogs of 3270 terminals.

Why did this happen? Why did we take this giant leap backwards? As with so many questions of the "Why is it done that way?" type, the answer is "money." With browser-based systems, we could for the first time allow users all over the world to access our systems, and that was a big step forward. Allowing a new user to access our system had essentially zero cost.

The alternative of a "rich client" or "smart client" required us to install some special software out on the user's system. This used to be easy, back in the days of DOS. We just copied it out there and it ran. But then COM came along—about the same time as the browser, in fact—and we learned the term DLL hell. So we gave up on getting our software to run on the user's machine, and just used the browser, because we couldn't afford to do anything else. After all, something beats nothing. An inferior alternative we can afford beats a superior alternative we can't afford.

But let's shift an underlying assumption. Let's assume deployment of software to a user's system becomes cheaper—perhaps very close to the zero cost of using a browser. What happens then?

Billy's First Law of Software Development states: Users outnumber programmers and tech support reps. Users are the reason our systems exist. If they learn they can get application software interfaces that are intelligent, and help them do their job better, how long do you think they are going to put up with clunky interfaces? How long will they be happy with pages built with a protocol that was not even designed for application interfaces, but for browsing hyperlinked pages? I believe that the answer is, "not very long."

Consider the possibilities. If a system has one thousand operators, and smart client software makes them just five percent more effective, then the cost savings can be huge. Assuming a loaded employee cost of $50,000, a five-percent productivity enhancement for one thousand operators is fifty full-time equivalents, which comes out to$2.5 million. And that doesn't even take into account lower training costs, lower error rates, less user frustration and stress, and other potential by-products of smart client software. (By the way, that five percent assumption above is quite conservative. I've seen productivity improvements in the ten- to twenty-percent range, depending on the task.)

If this analysis is correct, then how does that affect you decision makers in the world of software development and information technology? I think it leads to an interesting choice: You can be reactive or proactive about adapting smart client systems. And you can pay the price or reap the rewards of the strategy you choose.

On one hand, you can lay back and keep doing things in 1990's fashion, depending heavily on browser-based software. Depending on what industry you are in and how fast your organization moves on new technology, that strategy might not get you in trouble. But in any fast-moving industry, in which technology can confer a competitive advantage, such a strategy carries significant, career-affecting risk. Someday, a person who really calls the shots on technology in your organization may come to you and say, "Hey, this browser-based stuff is junk. The users hate it, and it slows them down. I've seen other systems that have smart user interfaces that let them do their jobs better and faster, and save their companies tons on money. I want some of that. Get it for me—and we need it yesterday." Or worse, they might start the conversation with, "We've decided to make a change in the management of your department..."

If this doesn't sound very attractive, you can adopt a proactive strategy. You can start by deciding whether that new application you need to write for 500 users should have an old-fashioned browser interface, or a hot, sexy, new intelligent user interface. You can be the hero that creates software that improves the productivity of hundreds or thousands of users, maybe saving millions of dollars. You can have your users go, "Wow! I didn't know you could have Internet applications this nice! You're a genius!"

## Getting Started

If you like the "genius" choice, then you need to learn the ins and outs of smart client development. There are four major pieces of technology needed to make a smart client architecture work:

• A forms package to create the intelligent client software.
• Data transport technology to communicate data between the smart client and the central server, either on the local network or across the Internet.
• A means to deploy smart client software to the client machines, in many cases across the Internet.
• Security technology to protect the system from malicious access.

Right now, the Microsoft? .NET Framework is the clear choice of platform to attain all of these capabilities. It is not the only choice as a platform for distributed smart client systems, but it obviously has a large lead over alternatives such as Java. Let's look at the capabilities of .NET in each of the areas above to see why.

## Forms on the Client

The .NET Framework includes one of the most advanced forms engines available. It's called Microsoft? Windows? Forms. It's completely object oriented, has a wide variety of visual controls (both from Microsoft and third parties) available for it, and the Microsoft? Visual Studio? development environment includes a great visual designer to quickly create Windows Forms interfaces.

Using the event-driven paradigm of Windows Forms, plus the ability to store as much state information locally as necessary, user interfaces can be much more responsive. Heads-down data entry tasks can be done with interfaces specially designed for that purpose. Sophisticated data validation can help users get the data right the first time, instead of needing to reload pages just to see if the data is right. (Yes, yes, I know. Browsers can do some of that too. But it takes so much programming to make them do it right that the effort is seldom worth it.)

Windows Forms interfaces can include tutorial windows, tooltips, translucent help forms, dynamic controls that adapt their behavior to the user, and much more. And they are still typically faster to develop than equivalent Web pages.

Microsoft is investing heavily in this area of technology. The next generation of UI technology has already been announced. The code name for the project is "Avalon," and it's part of the next release of Microsoft? Windows?, code-named "Longhorn," currently under development. Avalon adds even more capabilities to make user interfaces responsive, including some new paradigms for user interaction that would be virtually impossible in a browser.

To take advantage of these technologies, developers will need to learn more about user interface design principles. If they've only done browser-based programming, they may not realize how much there is to learn about UI design. They can't make your systems include responsive, intelligent interfaces, until they know how to write responsive, intelligent interfaces.

There is one limitation to note. Windows Forms is a part of the .NET Framework, and is therefore available only on Windows platforms. However, if your organization has standardized on the Microsoft? Windows Server System? for client workstations, as many have, then this is not a concern for you.

### Applications Need Data

The vast majority of corporate and commercial applications manipulate data, which is usually stored in some central server. For a smart client application to be viable, it must be practical to get the data from the server to the client, allow the client to make changes to the data, and then send the data back again.

Microsoft's newest data access technology, Microsoft? ADO.NET, is designed for just such a scenario. Unlike previous data access models, it was created for distributed use. XML-based containers of data can be created on the server and transported to the client. The client can use the container to work with the data without maintaining a connection to the server, making changes and additions, and then send the container back to the server.

There are two primary technologies than can be used to do the actual transport of data between client and server machines. One is Web services. The advantages of Web services include ease of implementation and configuration, and cross-compatibility to many kinds of servers. Web services can even talk to non-Microsoft servers, though you'll have to do a bit more work to create an appropriate data container.

If all the systems involved in your application can run Microsoft .NET, then another alternative is called .NET remoting. This has some performance and security advantages, but is more difficult to configure. Large-scale smart client systems for internal users often depend on remoting, while systems with users outside the organization are more likely to use Web services.

In either case, it's quite feasible for a smart client application to handle data much more intelligently than a browser-based system could hope to do. For example, if the smart client loses its link to the server, it can maintain a local copy of the data until the server is available again. The smart client application can dynamically adapt its behavior for online and offline scenarios.

### Low Cost Deployment

Remember that we started this discussion by asking about the effect of making deployment of smart client systems as cheap or nearly as cheap as browser-based systems. Now, it's hard to get much cheaper in deployment costs than a browser-based application, which has essentially zero deployment costs for each additional user. But it's possible to reduce smart client deployment costs enough to be competitive. And remember, with the potential large savings of a smart client through productivity improvements, taking on some additional deployment cost can make good economic sense.

The key technology allowing cheap deployment is the copy-and-run capability of the .NET Framework. There's no need to perform complex registration of system components as in COM. Just copy the components onto the disk and the Framework does the rest. And with side-by-side execution of multiple DLL versions, DLL incompatibilities are banished.

A limited form of automatic Internet deployment is already included in the .NET Framework, and a more advanced version, code-named "ClickOnce," is slated for the next version. In the meantime, I've found it easy to create custom deployment systems that take advantage of copy-and-run to easily and cheaply produce self-updating applications.

Just as browsers must be installed on the client systems to allow browser-based applications to run, the .NET Framework must be installed on client machines for .NET-based smart clients to run. At present, this does mean taking extra responsibility, because it's not automatically installed for the operating systems (Windows XP, Windows 2000, and Windows 98/Me) used by most client machines. However, installation of the .NET Framework is free for these systems. Over time, as systems turn over, we'll see the .NET Framework become ubiquitous because Microsoft intends to include it with future operating system products. In fact, they've already started by including it in Microsoft? Windows Server? 2003.

### Making It All Secure

We've learned in the past few years just how malicious some people on the Internet can be, and we've responded by building more robust security into our systems. But it's clear that new security capabilities are needed for a distributed smart client system.

Fortunately, the design of the .NET Framework includes robust security principles. Besides eliminating vulnerabilities such as buffer overruns, there is a new form of security called code access security. It awards security privileges to pieces of code, based on information about the code, such as where it came from or who wrote it. This security sits on top of the normal user-based security, allowing another layer of protection.

I would argue that a properly designed smart client system is more secure than a typical browser-based system, because the interface between the client and server systems can be more carefully controlled, and because the executable parts of the system can receive only the privileges they need. But it's true that designing such advanced security means learning some new techniques and concepts.

## Your Action Plan

You don't have to rush into this smart client world. Any time in the next two years or so will probably be fine—as long as your competitors don't get there first, of course. The users aren't clued in yet. And deployment, while economically reasonable in cost, still requires you to take responsibility for getting the .NET Framework on the client machines.

But these are temporary conditions. There is no doubt in my mind that smart client applications will displace a lot of browser-based applications over the next few years. The reason I have no doubt is that I've already had five clients decide to do it, and they are all thrilled with the results.

That doesn't mean the shift was painless for them. It required developers to learn new distributed architectures and new technologies. In many cases, they had to get better at object-oriented development and user-interface design.

Developers who have invested the last five years in learning browser-based development may not want to change. They are accustomed to being the leading edge developers of their day. But all technologies peak, and then decline, and are replaced by newer technologies. While we will continue to see browser-based applications used in certain scenarios for many years, I believe browser-based development has now passed its peak. The decline from the peak may take a while, but the direction is clear. Get ready for the return of the smart client!

## Related Books

VB.NET Deployment Handbook

Windows Forms Programming in C#

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## Wednesday, June 19, 2019

### Second Bell's theorem and the irrational cult around it

Off-topic, climate bragging: Steve Koonin is an NYU physics professor and a high-tier DOE official under Obama who proposed to introduce "the red team, the blue team" to the climate debate. I successfully urged him to respond to Gavin Schmidt's attack against a Koonin's talk at Purdue University – attack posted at RealClimate.ORG – and made sure he knew the address to submit it to the Anthony Watts' blog where it became the most successful text in quite some time.
A stupid 1976 paper by Bell did more than anything else for the deluded anti-quantum zealots to explode

Five days ago, I reviewed Bell's theorem. Two spins that are entangled – in a singlet state – produce a simple correlation $$\langle \sigma_a \sigma'_b\rangle=-\cos\vartheta_{a,b}$$ between the measurements of projections of the two spins along axes $$\hat a,\hat b$$, respectively. This nice and simple quantum mechanical prediction cannot be reproduced by a local classical theory (even with hidden variables) because such a local classical theory implies inequalities for correlations involving three axes that the (experimentally verified) cosines do not obey.

A voodoo doll. Anti-quantum zealots believe that quantum mechanics must enable some magic action-at-a-distance (and break Einstein's relativity, at least in principle) but it doesn't.

My blog post was a polished, more effective version of a 1964 paper by Bell. That paper made some sense, Bell has proven an actual result that was somewhat analogous to no-go theorems banning hidden variables. In fact, it was technically a more correct paper than John von Neumann's theorem on the same topic from the 1930s.

Although he wasn't terribly clear, it was implicit in the 1964 paper that the class of theories that was ruled out was composed of theories that were (1) classical theories with hidden variables that simultaneously (2) interacted locally.

But John Bell continued to write... and drift towards the confusion and ideologically motivated stupidity. So the misinterpreters of quantum mechanics love to say that he's proven not one Bell's theorem but two Bell's theorems. The second theorem was described in a 1976 paper "The Theory of Local Beables" – although my CERN version of that preprint was already written in 1975.

In this later paper, Bell coined the new terms "local beables" and "local causality" which have turned his writing into complete mess combining outright wrong statements with totally illogical definitions. He also tried to retroactively rewrite what he did in 1964. While in 1964, as I said, he was rather clear that he made two main assumptions about the theory, namely that it is local and it is classical (although he wasn't a sufficiently clearly thinking physicist to actually use the word "classical"), in the 1976 paper, he already tried to claim that he had only made one assumption, "local causality".

Eric Cavalcanti was trying to repeatedly push the slogan "local causality is important, violated by quantum mechanics, and it's a big deal" so I had to blacklist him according to the rules as an obnoxious troll who wants to spread wrong views by permanent repetition. But it wasn't just four or five comments where he writes all this junk. If you open Google Scholar, you will see that he has actually written tons of papers about this "local causality" stuff. All the papers are wrong and they're being referred to by very similar papers by other writers who are totally clueless about the actual rules of quantum mechanics.

In many previous blog posts, I explained that the entanglement experiments don't show any nonlocality and don't depend on any nonlocality. In a non-relativistic formalism, the faraway subsystems are typically described by tensor factors of the Hilbert space – so the full Hilbert space is a tensor product of the subsystems' spaces – and the Hamiltonian is close to the sum of the Hamiltonian for two parts which means that the subsystems evolve independently and locally.

In relativistic quantum field theory, the full Hilbert space isn't constructed as a tensor product but it is approximately a tensor product if only excitations in two separated regions A,B are allowed. More generally, the operators in the region A have vanishing (graded) commutators with the operators in a spacelike-separated region B. The word "graded" means that you should use anticommutators if both fields are Grassmannian i.e. fermionic. The vanishing of the commutators is what guarantees the independence of the evolution of objects in faraway regions.

There only exists one physically meaningful notion of locality or local causality – and it's what holds whenever the special theory of relativity (or its Lorentz invariance) is correct. The probabilities of measurements done purely in region A are determined by the prehistory of A – and don't depend on further data in the region B although it may contain objects previously in contact with the objects in A. There may exist correlations between measurements in A and B but all of those are calculable from the conditions in the past when A,B were a part of a single system AB. What's important is that people's decisions (e.g. what to measure), nuclear explosions, and other random events in region B don't cause changes to the system A.

The 1976 paper by Bell – and the cult around this wrong paper that has grown into a whole industry of pseudoscientists who try to turn physics into one of the humanities if not grievance studies – has tried to obscure all these things. So you may see Bell saying that the conventional "locality is not important". It should be downgraded and he renamed it to "signal locality" (clearly, only members of that cult respect this silly renaming). He not only tried to give an awkward name to locality; he also tried to say it's not important and something else, "Bell locality", is more important – and violated in any quantum mechanical theories.

All these statements are absolute rubbish. Every competent particle physicist sees this fact clearly. Every good atomic, molecular, nuclear, or condensed matter physicist or optic sees it, too. What the special theory of relativity guarantees – by its nontrivial Lorentz invariance – is locality or the local version of causality. It's the thing that Bell called "signal locality" - "signal locality" is the only physically meaningful locality-like conditions because signals (influences) are what actually determines the character of causation and causality according to any physicist. Outcomes at point P are only affected by the data in its past light cone, not by the past of the whole Universe. This locality or local causality holds because of relativity – and that's a great insight and the main reason why we actually consider Einstein a very important physicist. Other things that someone may imagine to understand physics may "look" nonlocal (e.g. the sudden collapse of a wave function for 2 particles in a reference frame) but they are not physics. If someone is imagining something while doing a calculation and this something is nonlocal, it doesn't prove that the theory is actually making nonlocal predictions – because the person is adding something unphysical to the theory!

If some "term that has locality in its name" is violated by quantum field theory, then this term is absolutely silly. Confused people often say that entanglement makes quantum mechanics nonlocal – but the only reason is that there exist faraway objects that are correlated. But those existed in local classical theories, too! Needless to say, Bell – a full-blown anti-quantum zealot by 1976 – deliberately incorporated the assumption that the world follows classical physics into all his terms so that his stupid readers – such as all the "researchers" writing Bell-style junk papers even in 2019 – don't even dare to think about thinking outside the classical box!

So a theory is "locally causal" only if it is classical. Needless to say, no viable state-of-the-art theory has been classical since 1925, so it cannot be "locally causal" according to Bell's silly definition. But is this conclusion really logically correct? Let's look at page 3 of the 1976 paper a bit more carefully. Here is his definition of [Bell's] local causality:
Let $$N$$ denote a specification of all the beables, of some theory, belonging to the overlap of the backward light cone of spacelike-separated regions 1 and 2. Let $$\Lambda$$ be a specification of some beables from the remainder of the backward light cone of 1, and $$B$$ of some beables in the region 2. Then in a locally causal theory$\{ A| \Lambda,N,B \} = \{ A| \Lambda,N \}$ whenever both probabilities are given by the theory.
Look at the definition again. Is the Standard Model a "locally causal" theory according to Bell's definition? I previously said that "locally causal" theories should be classical and the Standard Model is not a classical theory, so it is not "locally causal".

But the actual definition above doesn't demand the theory to be classical. Instead, it only requires the independence of the values of beables on some other beables. This independence has to hold for all beables with some properties. Is the condition obeyed by the Standard Model?

It actually is obeyed because the only beables of the Standard Model – or any quantum mechanical theory – are the $$c$$-numbers such as $$\pi\approx 3.14159$$. There are no nontrivial or dynamical beables in a quantum mechanical theory because a beable is a variable that obeys the laws of classical physics – and no dynamical observables in quantum mechanics (a $$q$$-number) obeys the laws of classical physics!

Clearly, $$c$$-numbers such as $$\pi$$ are independent of the beables $$B$$, not only because $$\pi$$ is a universal constant but also because the list of beables $$B$$ is empty. So the definition is vacuous, it demands zero non-trivial identities to be obeyed, and every theory – including every quantum mechanical theory – actually obeys it. Every quantum mechanical theory (including nonlocal quantum mechanical theories!) is "locally causal" according to Bell's definition because the condition in that definition is completely vacuous!

The claim by Bell, Eric Cavalcanti, and this whole stupid cult that quantum mechanical theories cannot be "Bell locally causal" is wrong. But it's wrong for rather stupid, formal reasons. The condition of "local causality" is obeyed but just because it is a completely stupid condition to be demanded from quantum mechanical theories. Quantum mechanical theories don't have any beables, so any talk about "beables in quantum mechanical theories" is simply and unquestionably a hopeless stupidity. This kind of stupidity fills a vast majority of papers by Eric Cavalcanti and lots of other grievance studies researchers pretending to do physics.

Everyone who wants to seriously talk about a particular quantum mechanical theory or quantum mechanical theories as a category must respect the basic postulates of quantum mechanics. And a basic postulate is that the dynamical observables in quantum mechanics aren't $$c$$-numbers, variables that would behave as coordinates and momenta in classical physics. Instead, all observables are expressed by $$q$$-numbers or linear Hermitian operators acting on the Hilbert space. Possible outcomes of their measurements are eigenvalues i.e. elements of the observables' spectra and the probability of each eigenvalue is calculable from Born's rule.

People who talk about "beables" in quantum mechanics are doing an equally silly mistake as an evolutionary biologist or a molecular geneticist who builds his science on the seven days of creation. Life wasn't created by a Creator in seven days according to the evolution theory or genetics; and analogously, observables that can be measured within quantum mechanics aren't expressed by beables or $$c$$-number-valued functions of time. Too bad, the people incapable of getting these complete basics have created a whole would-be field that they market as a branch of physics although it is really a branch of moronic grievance studies.

Incidentally, if you carefully fixed the silly features of Bell's definition of "local causality" – you would erase the requirement that the observables are "beables" etc. – you would get a definition that would be equally obeyed by local classical field theories as well as the local quantum field theories. If you allow the observable $$A$$ to be determined by Bayesian inference, then of course results of measurements done in the faraway region $$B$$ may be relevant for determining the value of $$A$$ once it's measured. That boils down to the fact that outcomes measured in region $$B$$ are also affected by the joint prehistory in $$AB$$, so they may be relevant for Bayesian inference. But this influence of Bayesian inference doesn't mean that the region $$B$$ influences the region $$A$$ now! Instead, all these predictions for the region $$A$$ now may be reduced to the properties of the system $$AB$$ in the past that contained both the predecessors of $$A$$ and the predecessor of $$B$$. By reverse inference, recent observations in $$B$$ determine some properties of the joint system $$AB$$ in the past which means that they also affect our expectations about the recent outcomes we will get in $$A$$.

All these statements are exactly equally true in classical and quantum mechanical theories! So strictly speaking, with the definition that talks about observables and not beables, no theory – neither classical theory nor quantum mechanical theory – would be "locally causal" according to that fixed definition – simply because that definition of "local causality" prohibits Bayesian or logical inference of properties of $$A$$ now from $$B$$ now that is affected by $$AB$$ in the past. There's no real difference between classical and quantum physics here. The psychological would-be difference is that in a classical theory, the measurement done later in $$B$$ may be identified with an observable that already existed in $$AB$$, before $$A$$ and $$B$$ separated. But if you think carefully, this statement is true in quantum mechanics, too! The observables measured later in $$B$$ have also Heisenberg-evolved from some operators that used to act on the joint system $$AB$$ in the past, too! The only difference is that the measurements always matter in quantum mechanics – if that predecessor-of-an-observable were measured in $$AB$$, that would have changed the experiment. But there's no reason to demand that the observable in $$AB$$ had to be measured – it isn't really being measured in the classical situation, either. What matters is that the observable mathematically existed.

If you're careful to avoid these and similar elementary mistakes, you will see that relatively to classical physics, quantum mechanics doesn't add any non-locality in any sense whatsoever. Locality is linked to relativity – but as a condition, it's pretty much orthogonal to the classical-vs-quantum axis. Locality has nothing to do with quantum mechanics. Locality is related to relativity – and relativity's $$c$$ and quantum mechanics' $$\hbar$$ are independent beasts. Whoever conflates these two pillars of modern physics simply fails to understand quantum mechanics as well as relativity – sadly, this includes all or virtually all writers of popular books about the foundations of quantum mechanics.