Sep 7

## Chess Boards, Exponential Growth and the Ice Bucket Challenge

Yesterday I was finally nominated for the Ice Bucket Challenge, I had actually thought this inevitable for the reasons in this post, but then it all kind of passed by. As it happens, it was my Daughter Aimee's fault in the end. .

So I thought I would use the opportunity to bore people about the arithmetic behind it, in the end, I suspect that wasn't too coherent since I was being watered at the time by my older Daughter Aimee, while my younger Daughter Matilda generally screamed at both of us.

So here is what I was trying to say. I like to talk to my students about a new version of an old problem. There's an old story about a chess game where the winner will take away a certain amount of rice. The amount is calculated by having one grain on the first square, two on the next, four on the next, and so on, doubling all the way to the 64th square. My modified version is to consider coins, a UK 10 pence piece to be precise, piling up on the squares as we go along. The question is, how high is the pile of coins on the last square? I encourage you to guess, approximately what sort of size that is.

We double each time, so the number of coins on the first eight squares are 1, 2, 4, 8, 16, 32, 64, 128. Another way to write that is 20, 21, 22, 23, 24, 25, 26, 27. So, we are going along 64 squares, and start at 0, so the number of coins on the last square will be 263. This is a big number, but just how big? Do we change our guess on this information?

Now 263 ≈ 9,223,000,000,000,000,000 = 9.223 × 1018 and a ten pence piece is 1.85 mm thick, or 0.00185 m, so when we multiply these we get

17,063,000,000,000,000 m = 1.7063 × 1016 m. Wow. That seems like a lot. Just how big is that number as a distance?

To put it in perspective...
• by the 19th square, the coins are higher than the radio mast on the Empire State building;
• by the 29th square, the pile of coins would have reached the Moon;
• by the 48th square, the pile of coins would have reached the Sun;
• by the 63rd square, one from the end, the coins are a light year high.
This is an example of the staggering power of exponential growth, and just how unintuitive it is. So what has this got to do with the Ice Bucket Challenge? Well suppose you start with a single individual, who then nominates three people, each of whom nominate three people. Now on the Chess Board we have 1, 3, 9, 27 and so on. This is, again, exponential growth with powers of three rather than two (actually somewhat higher growth).

Mathematically these are often called Geometric Progressions or just G.P.s for short. These are sequences of the type

a, ar, ar2, ar3, ar4, ...

There is a formula that can be derived (it's not hard, the derivation is on the above Wikipedia link) for the Sum of the first terms. (In Mathematics, contrary to popular opinion, a Sum specifically means the result of an addition process).

(Yuck, LaTeX rendering on my Blog is horribly broken at the moment). In the above case we can see that a = 1 because that's the first number in 1, 3, 9, ... and r = 3 because that is the number we are multiplying by each time. So here

In other words, every "generation" each person nominates a further three people, so the number of people added each generation ramps up exponentially, and the total number involved increased rapidly too.

GenerationNew People Cumulative Total
1 1 1
2 3 4
3 9 13
4 27 40
5 81 121
6 243 364
7 729 1,093
8 2,187 3,280
9 6,561 9,841
10 19,683 29,524
11 59,049 88,573
12 177,147 265,720
13 531,441 797,161
14 1,594,323 2,391,484
15 4,782,969 7,174,453
16 14,348,907 21,523,360
17 43,046,721 64,570,081
18 129,140,163 193,710,244
19 387,420,489 581,130,733
20 1,162,261,467 1,743,392,200
21 3,486,784,401 5,230,176,601
22 10,460,353,203 15,690,529,804

So you can see that after just 22 steps in the nomination process, you have literally more people than you have on the planet. Pyramid selling schemes become untenable for this reason too... you quickly run out of suckers on any given land mass.

So if this really ran its course unimpeded (and I know that the original challenge was money or dowsing, with a requirement to undertake the challenge in 48 hours (some say 24) then the whole planet would have done the challenge in at most six weeks. If each person had used say 4 litres of water then at the time of writing the 7,259,289,122 people alive today would have collectively used 29 billion litres of water. Since an Olympic swimming pool has 2.5 million litres of water, that represents over 11,614 swimming pools filled with water. While it is easy to criticise those who have been uneasy about the water wastage here, to be honest they have a good point. Also one organisation would have billions of charity income rather than it being spread.

Clearly this has not happened[citation needed] so people have run out on enthusiasm here and there. Anyway, have fun, and give some money to something you love, if you want, when you want to.

I am donating to Since people I care about have been troubled by both, and since it can't be denied I live in extraordinary luxury to be able to waste perfectly good water on this kind of thing.

Posted by Colin Turner

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Jul 14

## Aikido in Dead Straight Lines

There's been an “elephant in the room” on my blog for quite a while now, and it has prevented me completing a number of articles that I have had in draft for some time.

A bit over a year ago, my friend and mentor, Alan Ruddock died. I've been trying to articulate what that meant for me, and what I thought about Alan, but I have repeatedly failed. This doesn't fully resolve that issue, but at least I can put down some thoughts about Alan here now, or at least about Aikido

Disclaimer: this post probably badly needs some photos, and I'll try to retro-fit that at some point.

I was kindly asked to present an hour at this year's Galway Aikido Summer School, where Alan Ruddock and Henry Kono traditionally taught together for many years. This year Henry continued with his excellent classes in the morning and other instructors that knew Alan took an hour each in the afternoon. I confess I was a bit daunted by some of the others teaching in these slots, particularly the inestimable Lorcan Gogan of PSAC, with whom I shared a session. When I said to him I would have to follow that, he simply replied "Hey, I had to follow Henry!". A fair point. But I did find myself more reticent than usual in my teaching style.

I chose to try and present some thoughts that have arisen from Alan talking about his “dead straight line”. That is that Aikido is often thought of as being circular in nature but Alan was keen to stress it was not at its heart.

In Aikido, we often see uke (the attacker) whirled around nage (the defender) in circles. There are beautiful diagrams about this, and allusion to circles everywhere. Our own club is called the “Belfast Aikido Circle”. So it's impossible to deny circles don't appear. Indeed, there are semi-physical, semi-mystical links to squares and triangles too.

But what Alan meant, in my opinion, (all disclaimers apply) is that you always behaved as if you were operating in a dead straight line.

Aristotle believed that objects, in a perfect (celestial) environment travel in perfect circles, but many centuries later, Newton thought otherwise.

“A body will continue in its state of rest, or uniform motion in a straight line, unless acted upon by a resultant force.”

is better known as Newton's First Law of Motion. In Aikido terms its consequences are simple, if an attacker comes along a straight line, and is subsequently diverted off that straight line, Force, and Energy has been added by someone. Not in some kind of mystical sense of the use of these words you might see in other places, but in their elementary definitions in Physics. So the Force has been added, the big question then is by whom, followed up by why, and an analysis of the consequences.

Let's look at the question of “whom” first. In a previous article, I wrote at some length about the spectator problem in Aikido. Sitting at the side lines you can never know for sure just who is doing what to whom. You can see the nage's hands rise, move or turn, but you cannot know, from outside, whether this is nage initiating these things, or a reaction to uke's movements and attack.

In fact, we can take it a step further than this and say that at best only the two involved can fully know, since it is entirely possible that neither of them will fully know either. In other words, the problem of “whom” is a really knotty one – it's altogether possible that no-one knows.

In theory, at the beginning of a “technique” the nage first moves to a position of safety from the immediate attack and then “blends” with the attack. Even this is an over simplification since nage can be more proactive, but let's set that side for a moment. This moment of initial blending is pivotal. Alan used to tell a story about O-Sensei coming to watch a class of aikido at the Hombu dojo, and after watching, smiling, for some time, he announced “you are all doing a wonderful job, after having your heads cut off.” The analysis of this could be that, especially when facing an armed attack, even if the blend if a “bit out”, there may be more than a “bit” of you missing by the time you start your technique.

Aiki means the harmony of ki, or “energy”, so your blend is the moment where you, and your attacked end in a position of aiki, where both your energies are pointed in the same direction. If the uke's energy was directed at you in a dead straight line, as it often is, then in theory, you should both be pointing in the same dead straight line.

But this is often not the case. You can immediately see deviation from the straight line. Why is this? From Newton, the answer is obvious, one or both person(s) have put extra force into the situation.

This may have been uke, who immediately realising things are not going as planned, starts to react and often turns towards nage to strike them; this being the case nage has to move off the original straight line too to provide space for uke, and to keep the “aiki” principle, continuing to move with the uke. If that's all it is then this is fine. The force and energy being contributed by nage are minimal, it is necessary and sufficient.

But all too often, the honest truth is that as nage, we anticipate the move from uke, or worse don't even think about it, and just start whirling them around us. It may not be immediately clear why this is a Bad Thing. There are two reasons; one is that the aiki principle has been immediately broken, you are no longer in harmony with the attacker but attempting to direct them. The second is why the aiki principle itself matters; when you inject force and energy into the situation a skilled opponent can make use of it. In fact that's practically the central principle of aikido. Probably 90% of the time, particularly in training, your opponent may not even notice, so we all get away with it, and we probably never learn.

So here are some thoughts on training to enforce honesty on this.

Try and do some techniques along a straight line on the mats. You'll need your uke to be initially very well behaved. Don't even think of it as an attack, imagine that you meet a friend at the gates of a park. You see them as they approach and you walk backwards, then sideways, then alongside them as you extend your hand to shake hands with them. They key here is not to make it a conflict. This is astoundingly easy to do when you aren't worrying about having your head punched – just practice the naturalness of this first.

Then, as you walk along your straight line, just allow your hand to rise across the line, the analogy I used as if you are pointing at a squirrel in a tree. It's a simple, gentle irimi-nage that requires a compliant sensible uke who just has to keep walking straight, even into the “throw”.

It's a silly exercise, and you can play with greatly shortening it, but the big deal is, practising doing the throw along a straight line.

The next phase is to allow your uke to, once every so often (and without prior warning), lifting their free hand and turning to – well - lovingly caress your face shall we say? This changes the dynamic, the uke wants to move off the straight line, and so you have to as well. But the key of this exercise is:

1. do you actually pull them off the line on occasions where they do not turn;
2. can you really convince yourself you don't move off the line with a little extra force?
It may be a useful exercise.

Another example worthy of note is Shomen Uchi, Ikkyo (tenkan). Or – that when your opponent tries to strike your forehead, you turn, contact their arm and bring it, and the uke to the ground.

Traditionally this is often done by grabbing the arm as it descends and whirling your uke around you. There can be real consequences to this because their other fist is being whirled around you too. There are other problems too – you can have a tendency to pull the arm so closely into your space that uke can merely step behind you and topple you over their leg. It isn't a nice fall. It also provides , ironically, a slow descent of uke into the ground, since you provide them with a lot of implicit support.

So try, from the moment of “Shomen Uchi contact” where your hands meet, to step to the side, and without gripping contact the arm with both hands (the other hand at the elbow). Let your uke continue on their straight line straight forward and downwards. Potentially this is a hard fall, you are providing no force or breaking but just allowing them to sail majestically (potentially teeth first) into the mat. Some care needs be taken with this initially! It is most important that you do not push them down, just act as a ratchet so as they descend they cannot rise.

Again, in real life uke will often start heading towards you early on, trying to recover their balance, and also because in aikido, the tendency for uke and nage to start running around in circles is ever present, and once again, that's fine, the issue is not to interfere with this, but also not to seek to amplify it by hauling them around.

At an early stage you may find it useful not to grab their wrist – this can, in any case, cause all sorts of postural problems – and once you grab something the urge to pull it in is not far behind.

A suitable training for this can be found at your local Supermarket. Find yourself an empty trolley, and as you move it around, you will probably grip it with both hands and pull in one while you push out the other. If you are really mindful to your body you may notice muscles in your abdomen and spine taking the strain. But then fill the trolley to the brim, you will rapidly discover that this trick is not so easy. It's also not so wise, your muscles will shriek in protest at you, and if you do this kind of thing with a large opponent you will likely both be unable to move them, and injure yourself trying.

These are some simple thoughts on the Dead Straight Line that Alan used to talk about. There should always be a delicate positive pressure in that straight line in front of you, so that as things move and the gap appears there you will be immediately slot yourself into it. It is the spike in O-sensei's “Ki” calligraphy; it is the imaginary sword held in front of you that tells you always where you want to go.

There is a nice, and relatively elementary parallel from Einstein's General Theory of Relativity. The curved paths that objects like planets make, as commented on by Aristotle appear circular and curved, but actually they are Dead Straight Lines (geodesics) through curved space-time. When the Moon travels around the Earth, it does so in a series of Dead Straight Lines so that it keeps missing the Earth and it perpetually falls towards it. If you do decide to have uke orbit around you, then this is the same principle you need to prevent them spiralling into you, complete with their body weaponry.

But anyway, try playing with Dead Straight Line. I hope you will find it rewarding.

Posted by Colin Turner

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Jun 28

## Tau versus Pi

Today, two of my friends independently sent me a story about Tau Day which I had hitherto never heard of. One of them asked for me comment about whether this had any point to it. At first I thought the article was just mathematical trolling, thought about it a bit more, thought there might be a real point to it, thought some more and concluded it seemed rather silly.

The argument is about whether the mathematical constant pi, would be better being replaced throughout mathematics with an alternative tau, which is just twice pi (in other words, replacing pi everywhere with a half of this tau). It's suggested that formulae with tau will be more simple.

Basic Geometry

So this is all about the fact that pi was defined historically as the ratio of the circumference to the diameter of the circle, a very old classical reference stemming back to Greek geometry (incidentally pi is also known as Archimedes' constant since he attempted to calculate an approximation to it). Once upon a time, the formula used in schools would have been:

related the circumference C to the diameter D. But generally now, we use the radius r rather than the diameter. And so that gives us (for circumference and area):

The argument for tau begins by observing the extra 2 in the first formula, and wouldn't be nicer if we just defined tau to be twice pi so that these formula would be so much nicer. Would they?

Set aside for the moment the fact that pi is probably the most recognisable Greek letter in the world that speaks languages based on the Latin alphabet (aside from those that are, or appear to be the same). Set aside the fact that tau is used for other specific purposes in much of modern Mathematics, and in particular in the discipline of Topology. The first formula might be nicer, but the second one is probably worse, and by enough to make the improvement of the first rather parlous. OK. But the article talks about this being the problem behind radians, so maybe that's where we get the big gain. Let's explore that.

There's nothing particularly clever about using degrees. It's an arbitrary choice (360 degrees in a circle) that probably owes a lot to do with historical factors in one civilisation. It is true that when you start to do some significant mathematics with degrees, it starts to look quite unwieldy. The classic two formulae to consider are the length of an arc and area of a sector.

Suppose we have a circle of radius r and we want to work out the length of an arc (a part of the circumference) where the angle subtending this arc is theta degrees (don't panic, no more Greek to come). Then in degrees the formula will be:

The reason why is that the fraction on the left is the fraction of the relevant angle out of all the angle available, multiplied by the total arc length available (the whole circumference). The formula is not beautiful, and the similar formula for sector area is also a big ugly.

You will note that in both cases there is a 360 on the bottom of the fraction and a 2 pi on the top. This looks like nature's way of trying to tell us something. What would happen if we used an unit of angle so that, instead of having 360 of them in a circle, we had 2 pi of them in a circle (proponents of tau will just say tau of them in a circle)? The formula, derived using the same logic, become much nicer.

So we get

Now these are beautiful, elegant formulae, and the underpinning of why radians (the unit of angle we are talking about here) are used instead of degrees in much of higher mathematics, the formula are much simpler (particularly true when using calculus). Also, look at that first formula, it has all the resonance of F = ma. 1 unit of arc length is found in a circle of radius 1 unit with an angle of 1 radian. So beautiful is this that it used as the definition of the radian in many books. So far, so good. Did we really need tau to produce these? Does it matter that it it tau and not pi that cancels out? I can't see why.

Fourier Series

Another example owes to the work of Fourier, who showed that repeating patterns can be broken into sums of the most basic repeating functions, the ones that are most simple are the sine and cosine functions. These are used to model waves of any sort which are of course ubiquitous in nature. It turns out you can build up more odd shapes like triangular and square waves out of these sinusoidal ones. Being able to do this is important in many aspects of Science and Engineering, and eventually this theory leads to all sorts of cool stuff like the way data is compressed in photos and more.

The graphs of Sine and Cosine (sin t and cos t)

This graph is labelled in degrees, so you can imagine if we change it to radians as we should then the 360 would become 2 pi. This is where again, proponents of tau will argue that replacing the 360 with a simple tau makes things easier. And it does, if all you want to do is to label that diagram. But the foundation of Fourier theory is building functions up in combinations of these:

The formulae you need to be able to deal with to do this are (among others):

which I grant, strike fear into the hearts of many. But they don't look nicer with tau (and are a little more awkward):

Remember, the people arguing for tau are claiming it simplifies formulae, not making them look worse.

Euler's Identity

Finally, I cannot leave this without talking about Euler's identity considered by most mathematicians (including myself) to be one of the most beautiful results in Mathematics.

This result can be written in a few ways, but this way is very commonly used. This is because in this form you can see how this identity connects the five most important numbers of Mathematics: 0, 1, pi, i and e. With tau, it just doesn't have the same beauty:

so I will stick with pi. Thanks all the same.

Posted by Colin Turner

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Nov 15

## More about LHC black holes

Last week I wrote a little about the size of black holes, and incidentally discussed very primitive calculations I did on the lifespan of any black hole created by the LHC.

A few days later, this interesting article showed the results of professional physicists on just how little such little black holes could grow, in some cases even if their lifetime was not restricted. Enjoy.

Posted by Colin Turner

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Nov 9

## Black Hole Radius, or How I came to love the LHC

I allowed myself to become rather distracted by my second years last week as the class was finishing. They were talking about an episode of Horizon that discussed General Relativity and theories of Quantum Gravity. What followed was a free ranging discussion on the nature of infinity, mentioned briefly in the program. But we also talked about the nature of a black hole and its size.

It's surprisingly easy to calculate this with reasonably elementary maths and physics. I first did this when I was about 17 (how very sad) using classical physics equations, and was astounded to discover that even so, the answer was correct (I checked it in the Encyclopedia Britannica in the library at the time).

Here is Newton's universal law of gravitation, between two bodies. It describes the force F between two bodies that are r metres apart. Let's take the one with mass M to be the black hole. G is a small (though mysterious) constant.

You can work out the energy needed to escape the black hole using the old stand by equation that work done is the force times distance traveled against that force, but that only works with a constant force, this force will change as we move, so we need to use the big daddy of multiplication, integration.

Specifically, we will work out the energy needed to escape from the event horizon, the surface at which the escape velocity is the speed of light, which is c (299,792,458 m/s). So the energy will be given by moving my little mass m from the radius of the event horizon, let's call is R to infinity, to show we have broken away.

Now, this should just balance the kinetic energy possessed by my little mass m traveling at the speed of light.

and if we rearrange for R we get that

In other words, the radius of the event horizon, the bit we think of as the "hole" is dependent entirely upon the mass of the object. Please note this is based on a very simple model of a non rotating black hole. Nevertheless we can do some nice calculations from this.

The Sun, would have to be compressed from its diameter of about 700 million kilometres into a radius of just under 3 kilometres. The Earth's mass would need to be compressed so much to form a black hole you would need to squeeze its radius of over 6 thousand kilometres into a radius of around 9 millimetres. That's how dense we're talking here.

We can also consider the radius as described by the contained energy of the black hole, since we know that

and so, replacing our M in our above equation we get

Wow. Remember c is a big number, taking it to the power of four is a lot. So why do this? There's been a lot of speculation about the possibility the Large Hadron Collider (LHC) could create a black hole. This has caused a fair degree of panic, and at least one suicide. It's a physicist's dream that a black hole might be created. I just looked up the "high" energies used by the LHC, and high is a relative term. It plans to bash protons together with 7 TeV (Tera electron volts) of energy each, or lead nucleii with 574 TeV each, let's take the latter. Just how much energy is that in a collision? Well, doubling and converting to good old Joules gives 184 micro Joules. That's really not a lot, 184 millionth's of a Joule. A 100W light bulb uses 100 Joules each and every second. How big would the radius of such a black hole that might form be, from that energy? Check the maths, because so far I haven't but I get.

metres

which is 0.000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,003 metres wide (I hope, I didn't double check the zeros either), which is probably not the planet swallower of people's imagination (but this is fun). But the problem is people think this tiddler will grow very rapidly, but that's because they don't know about Hawking Radiation. This is an interesting quantum effect that means black holes aren't really black, they do emit a little radiation. Large holes would gather surrounding matter faster than their low radiation rate, but small holes have the opposite situation, they radiate more rapidly. The maths for all that is pretty complex, and you need to make lots of assumptions, but the time taken for our little black hole to "evaporate" is (hurriedly calculated by me)

a tiny, tiny fraction of a second. Even allowing for the ambient temperature and some fall in of matter, this little baby is not in equilibrium, it's not getting mass fast enough to accumulate more. It's safe*.

* All disclaimers apply. No liability is assumed for foolish unvalidated experiments done by you or other members of your species. Do not attempt to create black holes in your garage. Any subsequent destruction of your civilization, planet or solar system is at your own risk, and any "EPIC FAIL" signs placed by aliens on the remains is not due to me or my calculation. No calculations have been done on the matter of strange matter either. If you break the planet / system / galaxy or universe you own all the parts.

Posted by Colin Turner

Oct 4

## Derren Brown tells porky pies, get over it

Derren Brown has been back on our screens recently with a series of big events. It's brought him a lot of publicity, and a lot of the public comment has been amusing to say the least. I've been watching everyone comment on the TV and Radio that he's a trickster, and his explanations of how he does things are often bogus.

Yes.... Yes... That's the point. He's a magician, a showman, he tells you right up front at the start of the show. Misdirection, magic, showmanship and more. It's entertainment. He is the first to make it clear that what he does is a trick. He doesn't claim otherwise. If you can't bring yourself to enjoy that, I humbly suggest you watch something else. And while we are on it, I'm relieved people see it's a trick. I think David Blaine's closeup magic is among the most impressive I have ever seen, and I shudder when those around him don't even seem to consider it's just a trick.

Personally, I love Derren, I love his cheeky smug as hell smile as he gets away with it. I enjoy his faked discomfiture. I enjoy it when he fakes near success in tricks to make them more convincing. I enjoy calling him names as I laugh as he lies through his teeth to the audience. I have casually studied a bit of magic in my time, and he is a fine magician. Many big stage magicians rely on assistants, who actually do all the hard work, but he is clearly the architect of most of his own tricks. I'm reading his book and it is informative and very amusing to me, and I very much respect him as a fellow in fight against (rampant) irrationality (a little bit can be a fine thing).

Sadly I've missed a few of the "events", since it clashes with my Iaido class and I wasn't organised enough to record it. I'm slowly catching up on 4OD. But I did see the program on how he stole the lottery. And I'm a mathematician, and I laughed and laughed at it. I didn't believe a word of it, but even the deceit was cleverly convincing and it's not trivial to say exactly what he really did.

He claims he averaged the results of a crowd of people picking lottery numbers. There's a consequence to this. Consider this: how many ways can 50 people pick a number that averages to 1? How many ways can they pick numbers that average to 30? If you understand this point you'll see the possible "answers" from the crowd have an odd distribution. But the cover story was very amusing in its own right. You don't want to know how he really did it, it'll be shockingly dull.

So yes. We know he didn't show the numbers in advance of the draw. Yes, we know his explanation is nonsense. It's a trick. That's the point. Enjoy it and smile along with his insufferable smugness .

Posted by Colin Turner

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Aug 31

## Campaign for an apology for treatment of Alan Turing

The website for Number 10 Downing Street has a new petition calling for an apology for the treatment which led Alan Turing to commit suicide at the age of 41, rather than submit to having his homosexuality "cured" by chemical castration.

Turing was, not only a founding father of Computer Science, but a leading member of the dedicated team at Bletchley Park, who decoded the Nazi Enigma code. This work saved thousands of lives.

As reported on the BBC, the originator of the petition, John Graham-Cumming, is not only suggesting an apology, but a posthumous knighthood for Turing. It would seem the least his country could do for him, after all he did for his country, mathematics and computing.

If you're a UK resident, please consider signing the petition.

Posted by Colin Turner

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Jul 27

## Steganography, prevention before is better that detection after

New Scientist recently ran an article about steganography. If you don't already know, steganography is essentially encryption with a difference. Specifically, encryption is usually obvious. It may be that the data Alice sends via email to Bob with public key cryptography is entirely secure from eves-dropping by Eve (pun intended, sorry), but Eve will know data is being sent that she might be interested in. Steganography, by contrast, seeks to hide the encrypted data so Eve is not aware of its very existence.

It's a very ancient idea, stretching back to ancient Greece. In modern times a common way to perform the trick is to hide data in an image. One of my more gifted undergraduate students did a final year project on this with me. We used a known password as a seed for a pseudo-random number generator to determine which pixels of the image we would embed the data in. By playing with the least significant bit of one colour in randomly spaced pixels, you can very effectively hide data.

The New Scientist article suggests that if you detect the steganography, and if we obtain the computer of the suspect and if they have carelessly wiped the software, there might be traces that tell you this was done. Now let's remember the whole point of steganography is that the first step is improbable, you most likely won't detect it.

The issue is, in today's geopolitical situation, reasonably serious. It has been suggested (see the wikipedia article I linked above), that such techniques were used to exchange data on site like ebay to plan major terrorist attacks. With lots of analysis software only playing with known algorithms, or relying on comparing modified images with the original (where the original may not be available) what can such a major website do to prevent such abuse? Well, I thought an approach would be to essentially employ the same techniques with random data. That is, randomly poking data into bits in pixels here and there will, up to a certain point, not affect image clarity to the naked eye, but unless the encrypted data is loaded with huge amounts of error correcting code, it will destroy the payload. You could easily automatically run such a filter over uploaded data. I'm sure similar approaches would work for digital sound.

Posted by Colin Turner

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