Comments about the article in Nature: The ultimate physical limits of privacy

Following is a discussion about this article in Nature Vol 507 27 March 2014, by Artur Ekert & Renato Renner in the section: Perspective
In the last paragraph I explain my own opinion.


Introduction

The article starts with the following text:
The best that cryptography can offer are security reductions, telling us, that breaking RSA, is at least as hard as hard as factoring large integers. But is factoring really hard? Not with quantum technology.
The large integer we call N. Factoring means to calculate the two prime numbers pm1 and pm2 which product is N. That means pm1*pm2 = N. To do that Shor's algorithm is used. For more detail read: Shor's Algorithm - 11 questions . IMO the whole strategy is like a dream, which will never be fullfilled.
Indeed, RSA, and many other public key cryptosystems, will become insecure once a quantum computer is built.
That day will never come when a Quantum Computer will defeat a classical computer.
Admittedly, that day is probably decades away, but can anyone prove or give any reliable assurance that it is?
A Perpetuum mobile is a device which moves in a closed room, without any external form of energy, indefinitly.
Is it possible to prove that such a machine does not exist? No.
The only way to prove that such a machine exist is by building one. Until today noone has been capable of doing this.
The same is true for a Quantum Computer. Anyway what means reliable assurance.
This said, the requirements for perfectly secure communication are well understood.
The article mentions not what they are but they also should be practical. Practical means that the communication system involved (which requires secrecy) should be used between ordinary people with out any technical training.

The power of free choice"

Technologically superior Eve could have manufactured an additional coin, magically linked to the coins held by Alice and Bob. The three coins tally and Eve knows all the bits in the string.
What happens if Eve coin only is magically linked to the coin held by Alice?
In that case IMO Eve knows also all the bits in the string.
Clearly, to achieve secrecy we must let Alice and Bob do something that is beyond Eve's control.
I would say: Clearly to improve secrecy.

Key distribution

To establish a cryptographic key etc Alice is choosing between A1 and A2 and Bob between B1 and B2.
The resulting setup coulde be something like:

Select A   12122121212211222
Select B   11221212111221212

Special    ..x..x.x....x....

Output A   01001101001110110 
Output B   01101000001100110

The top line shows the coin that Alice has selected: A1,A2,A1,A2 etc.
The next line shows what Bob has selected:B1,B1,B2,B2 etc
The line indicated "Special" indicates with the letter "x" that Alice has selected A1 and Bob B1. Only in that case the outcomes of Alice and Bob are complementary. This is shown in the lines Output A and Output B. That means in all these cases Bob has to flip his bit.
except when A1 and B2 are tossed in which case either Bob or Alice must flip his or her bit.
You have to decide that either Bob or Alice in that case always has to flip bits.
If Eve is capable to monitor both the two coins A1 and A2 of Alice and does not flip bits than in 50% of the cases the the decoded message by Eve is wrong, because cryptographic key is wrong. To solve this in the cryptographic key of Alice (where Alice selected A1 and Bob B1) the bits should be flipped. When this new cryptographic key is used the decoded message becomes correct.
If Alice and Bob notice a deviation from the magic correlations they abort the key distribution and try again with another set of coins.
Who? Alice only? Bob only? There should be in procedure in place to test a cheque which set of coins is wrong. To be more specific which coin is wrong.

The quantum of solace

A number of quantum optical techniques can be employed to generate pairs of polarization-entangled photons.
What are polarization-entangled photons?
As far as I know that when one photon is polarised in the direction +X,+Y and +Z than the other entangled photon should be in the direction -X,-Y and -Z.
The practical issue is how do you demonstrate this in practice, because you need 3 measurements.

Less reality, more security

The impossibility of assigning numerical values to certain physical quantities, for example the different polarizations of a photon, has been baffling physicists for almost a century. After all, most of us grew up holding it self-evident that there is an objective reality in which physical objects have properties that can be quantified and whose values exist regardless of whether we measure them or not.
The problem is that we can not quantify the polarization of a photon exactly at any moment. We can not know the position of an electron at any moment. We know the electron is somewhere but we do not know exactly.
Shocking as it may be our world is not of this kind.
Then how is our world?

Should we trust ourselves?

We have already stressed the power of free choice.
IMO a good encrypting system should operate independent upon any human influence


Reflection

Understanding our world and performing science starts by performing simple experiments. In order to understand the results of what is measured you need a model. Such a model is first of all a physical model, which describes the objects (subprocesses) and the chemical reactions involved. Secondly the model involves mathematical equations, input variables and parameters.
In order to test the model different input variables and parameters can be used and by using the mathematical equations the output of the experiments can be predicted. When the predicted outcomes match which the observed (measured) outcomes the model is correct.
In a second phase the first simple experiment can be modified and enlarged to improve our understanding.

What important is, in this scenario, is that you do not start with mathematical equations which are a rather abstract description of an imaginary process and secondly that you search for a real process which behavior should (or should not) match with the predicted behavior. Bell's inequality mathematical statement about correlations falls in this cathegory.

  1. When you toss a coin a 100 of times and when the outcome is not 50 50 (or close) than you know that the coin is tampered.
  2. When you measure the polarization of two entangled photons a 100 times in the same direction and the correlation is always 1 (100%) than this becomes a physical fact. This type of behaviour allows you (at A) to predict the state of the other photon (at B) with 100% accuracy. This physical fact can be used to create a cryptographic key assuming that the key is "random". The same when the correlation is -1
  3. When you measure the polarization of two entangled photons a 100 times in the same direction first undisturbed as explained before and next with one of the photons tampered which results that the correlation becomes zero than this also is a physical fact. This means that you can detect eavesdropping.
  4. When you measure the polarization of two entangled photons a 100 times in the same direction and the correlation on average is constant than this also a physical fact. In fact this means that direction is not the same. However this type of behaviour does not allow you (at A) to predict the state of the other photon (at B) at each instant with 100% accuracy. This physical fact can not be used to create a cryptographic key To solve this problem you must change the direction of one of the boxes until the correlation becomes 1 (or -1) but that is not simple.


Evaluation paper "Less Reality, More security"

This is a discussion of the paper Less Reality, More security by Artur Ekart.
In this paper at page 2 we read:
This line of argument hinges on the interpretation of phrases such as "the value does exist". The EPR paper offered a carefully worded definition:
If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.
IMO the following example explains what is meant:
Suppose a device generates identical photons all with the same frequency (A laser). I measure 100 and they all have the same frequency f1. Then the physical reality of the device is that it generates photons with frequency f1.
The next line at page 2 explains in more detail:
The paper then went on to demonstrate that there are cases where one can establish the existence of the "element of reality" of physical quantities, such as position and momentum, so that their values exist, and yet when these quantities are measured the results are random.
The fact that we (humans) can not measure these quantities is no reason to assume that these quantities do not exist.
Next we read at page 2:
Here we rephrase the original argument in terms of polarizations. Think about two photons, labeled A and B. One can prepare a pair of photons in such a way that the measurement of polarization on B provides precise information about the value of a corresponding polarization of A.
The importance is the definition of precise. To establish that you have to prepare 100 pairs of photons in this way and observe the correlation between A and B. When this correlation is 1 than when B is measured you also know the polarization of A precise. With any other value it is not precise. Most probably in reality by performing an experiment this is always the case.
Moreover, because the two photons can be far apart from each other the measurement on B cannot disturb A.
Again you have to perform 100 pairs of photons and observe if this correlation between individual photons, when both are measured, holds or decreases. I expect the correlation will decrease as a function of distance.
Next we Read:
This is the EPR locality requirement, which, in its original form, reads:
"The real factual situation of the system A is independent of what is done with the system B, which is spatially separated from the former".
Generally speaking this should be always the case. A measurement overthere which could change the state overthere (at B) should not influence the state here (at A).
This polarization of A has a certain value and, it follows from the locality requirement, this value must exist even if the measurement on B is not performed.
This value does not come from the locality requirement. This value comes from performing 100 experiments on both A and B. And if there is a correlation this can only be established when both A and B are measured.
Still, the best quantum theory can do is to make statistical predictions whenever the polarization of A is measured directly.
Quantum theory can do "nothing". What we can do is by performing 100 experiments is to collect the statistical data when both A and B are measured.
The world, he (Einstein) firmly believed, might be inordinately complicated, but at the bottom of it there should be order and predictability.
Are this Einsteins exact word? I also think that at "the bottom" there is order. The problem is we can not predict what is happen in the future because we cannot measure what this order actual is (ie what the positions and velocities of the electrons actual are)

2. For whom the Bells Tolls

At page 3 we read:
Let us label the two photons in each pair as A and B respectively and let us assume that both A and B have well defined values of their polarizations.
In reality this is may be not the case. It is possible that during one measurement both A and B measure someting, A only, B only or both nothing. Those tries should be monitored and taken into account in the final calculations.
Next at page 3:
Let us define a new random variable S,
(1) S = A1(B1 + B2) + A2(B1 - B2):
Why this equation ?
Next
It is easy to see that one of the terms (B1 + B2) or (B1 - B2) must be equal to zero and the other to + 2 or - 2, hence S = +2 or -2.
This is not clear.
When A select A1 and B select B1 then S = A1B1 can be calculated in four different ways:
1*1 = 1, 1*-1=-1, -1*1 = -1 and -1*-1 = 1
All the other values are zero in one test.
The average value of S must lie somewhere in-between, i.e.
(2) -2 < or = S < or = 2:
That's it! Such a simple mathematical statement about correlations, to which we refer simply as Bell's inequality, and yet so profound.
The question is how is this value of "S" calculated in an actual experiment.
I expect that the average value of S = 0, but this is not necessary for each experiment, specific if there are correlations involved.
In fact, instead of photons and polarization analyzers Alice and Bob may be given sealed, impregnable boxes each. etc.
Please read the full text in the paper at page 3.
Technical details of the hardware are irrelevant.
I expect that the technical details of the experiment are extremely important.
The focus is on correlations alone and, surprisingly enough, there are correlations that violate Bell's inequality.
Why surprisingly?

3. Hello world, are you there ?

If we take this unorthodox approach then the expression
(3) S = A1B1 + A1B2 + A2B1 - A2B2:
admits +4 or -4 as its two extreme values.
That is correct when you perform 4 tests and add the results.
The paper does not give enough details what is involved in practice
Next at page 4
However, generating correlations of this kind involves either instant communication between distant objects or inherent randomness, or both. Instant communication is hard to swallow.
More information about the actual experiment is a must
For example, Bell's inequality is maximally violated when A1, A2 and B1 are fixed at +1 and B2 takes value +1 when measured together with A1 and -1 when measured together with A2.
How is it possible that in each test of an experiment A1, A2 and B1 are fixed at +1 ?
Next at page 5
The atom subsequently decays, emitting two "polarization-entangled" photons, so that if the polarization analysers A and B are set "theta" degrees apart then the results agree (AB = 1) with probability sin2(theta) and hence differ (AB = -1) with probability cos2(theta)
There is nothing surprinsingly about this result.
Correlations of this kind cannot be used to send instantaneous messages but they do violate Bell's inequality.
The only thing that is important that the correlation between the different photons is a function of "theta"
Choose angles 0, etc and you obtain S = -2 sqr(2)
This, by the way, is the maximal violation that quantum correlations can offer.
There is nothing violated !!!!
Experiments teach you what physics is. Different experiments teach you different things. In order to understand you need a model of the physical reality. This model can be based on mathematics.

4. Less reality, More security

The secrecy of communication depends entirely on the secrecy of the key.
This is important to know. In many cases the secrecy depends about the decryption method.
However, this implies that new keys must be repeatedly generated and distributed.
This can be very unpractical and also unsafe.
They keep a detailed record of the settings and the corresponding results. Then they communicate in public the settings etc
Next at page 6
If the violation of Bell's inequality is observed then the remaining results, which were recorded but not communicated in public, remain secret. etc They can be turned into a secret key.
The details of how this is done are missing.

5. Yes We can

Last but not least - even if one day quantum physics is refuted and superseded by a new theory, even then, as long as the new theory does not admit any instant communication, we can use Bell's inequality as an indicator of secrecy.
First of all you have to define what instant communication means and secondly you have to demonstrate that instant communication is possible. If quantum theory allows for instant communication (Yes or No) is unimportant.
At the end of the paragraph:
The sheer fact that Bell's inequality allows us to make sensible statements about security of devices operating according some yet to be discovered laws of physics is amazing.
Pure mathematical statement and the results of experiments have nothing in common.

6. Epiloque

For one thing, Einstein, such a smart guy, and yet you are telling me that when it comes to quantum theory he got it all wrong?'
You must describe exactly what Einstein's had to say related to the Quantum Mechanical experiments.
The notion of "reality", as understood by Einstein and his colleagues, was probably too simplistic.'
You must describe in detail what Einstein understood by "reality". To call his opion too simplistic is too simplistic.
Then, with some trepidation, I added. `If the formalism of quantum theory is anything to go by then one particular interpretation, proposed by Hugh Everett in 1956, indicates that "reality" is more complex, that everything that can possibly happen actually happens, and all possible outcomes of measurements have the same status
All of this text is very unclear.
In this multiplicity of "universes", often called the multiverse, Bell's argument, which requires a predetermined single outcome of a quantum experiment, simply does not make sense.
In general the outcome of an experiment which involves single photons or single electron's is not predetermined implying that you can predict with 100% accuracy what the outcome of a single test is.


Reflection paper "Less Reality, More security"

What is missing in the above paper is a detailed description how by using the "sealed boxes of unknown provenance" Alice and Bob generate their own private keys. This description should specific describe that the private keys are 100% the same.
For example:
  1. The process starts by means of an beta-barium-borate crystal atom, at a central location, which receives a photon, decays, and in return emits two "polarization-entangled" photons, These photons are both send to Alice and Bob, a distance away, were they are both "measured" in a sealed box. Before this is done Alice sets her switch in position A1 or A2 to select a direction. Bob does the same but he selects B1 or B2.
  2. Next Alice informs which position A1 or A2 she has selected using her "free will". Bob does the same. What each one has measured stays secret.
  3. The following step is to calculate a bit of the cryptographic key. How this is done is not explained. How the decoded message is transfered to Bob can be done in public.
To do this process and to produce a key of 100 bits does not seem very practical, which is also prone to errors. Specific if you have to repeat this process each time when a new key is required. To do this procedure by means of a robot and a random number generator (for example also based on photon decay) is not allowed because a robot has no free will, which (apparently) is an important part of the experiment. If you want that more people decode the same message is also not explained. Apparently in each case a different cryptographic key has to be generated.

IMO to use a scheme based on very long prime numbers is much more practical.


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Created: 4 April 2014

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