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The Rise of Bitcoin

Bitcoin is in the news again as it hits $10,000 a coin – the online crypto-currency has seen huge growth over the past 1 1/2 years, and there are now reports that hedge funds are now investing part of their portfolios in the currency.   So let’s have a look at some regression techniques to predict the future price of the currency.

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Here the graph has been inserted into Desmos and the scales aligned.  1 on the y axis corresponds to $1000 and 1 on the x axis corresponds to 6 months.  2013 is aligned with  (0,0).

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Next, I plot some points to fit the curve through.

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Next, we use Desmos’ regression for y = aebx+d. This gives the line above with equation:

y = 5.10 x 10-7 e1.67x + 0.432.

I included the vertical translation (d) because without it the graph didn’t fit the early data points well.

So, If I want to predict what the price will be in December 2019, I use x = 12

y = 5.10 x 10-7 e1.67(12) + 0.432 = 258

and as my scale has 1 unit on the y axis equal to $1000, this is equal to $258,000.

So what does this show?  Well it shows that Bitcoin is currently in a very steep exponential growth curve – which if sustained even over the next 12 months would result in astronomical returns.  However we also know that exponential growth models are very poor at predicting long term trends – as they become unfeasibly large very quickly.   The two most likely scenarios are:

  1. continued growth following a polynomial rather than exponential model
  2. a price crash

Predicting which of these 2 outcomes are most likely is probably best left to the experts!  If you do choose to buy bitcoins you should be prepared for significant price fluctuations – which could be down as well as up.  I’ll revisit this post in a few months and see what has happened.

If you are interested in some more of the maths behind Bitcoin, you can read about the method that is used to encrypt these currencies (a method called elliptical curve cryptography).

 

Elliptical Curve Cryptography

This post builds on some of the ideas in the previous post on elliptical curves. This blog originally appeared in a Plus Maths article I wrote here.  The excellent Numberphile video above expands on some of the ideas below.

On a (slightly simplified) level elliptical curves they can be regarded as curves of the form:

y² = x³ +ax + b

So for example the curve below is an elliptical curve.  This curve also has an added point at infinity though we don’t need to worry about that here.  Elliptical curve cryptography is based on the difficulty in solving arithmetic problems on these curves.  If you remember from the last post, we have a special way of defining the addition of 2 points.

elliptical1

Let’s say take 2 points A and B on y² = x³ -4x + 1. In the example we have A = (2,1) and B = (-2,1). We now want to find an answer for A + B which also is on the elliptical curve. If we add them as we might vectors we get (0,2) – but unfortunately this is not on the curve. So, we define the addition A + B through the following geometric steps.

elliptical3

We join up the points A and B. This line intersects the curve in one more place, C.

elliptical4

We then reflect the point C in the x axis. We then define this new point C’ = A + B. In this case this means that (2,1) + (-2,1) = (1/4, -1/8).

Trying another example, y² = x³ -5x + 4 (below), with A = (1,0) and B = (0,2) we have C = (3,-4) and C’ = (3,4). Therefore (1,0) + (0,2) = (3,4).

elliptical5

We also need to have a definition when A and B define the same point on the curve. This will give us the definition of 2A.  In this case we take the tangent to the curve at that point, and then as before find the intersection of this line and the curve, before reflecting the point.  This probably is easier to understand with another graph:

elliptical10

Here we used the graph y² = x³ -5x + 4 again.  This time point A = B = (-1.2, 2.88) and we have drawn the tangent to the curve at this point, which gives point D, which is then reflected in the x axis to give D’. D’ = (2.41, -2.43).  Therefore we can say 2A = D’, or 2(-1.2, 2.88) = (2.41, -2.43).

Now addition of points is defined we can see how elliptical curve cryptography works.  The basic idea is that given 2 points on the curve, say A and B, it takes a huge amount of computing power to work out the value a such that aA = B.  For example, say I use the curve y² = x³ -25x to encrypt, and the 2 points on the curve are A = (-4,6) and B = (1681/144 , -62279/1728).  Someone who wanted to break my encryption would need to find the value a such that a(-4,6) = (1681/144 , -62279/1728).   The actual answer is a =2 which we can show graphically.  As we want to show that 2(-4,6) = (1681/144 , -62279/1728) , we can use the previous method of finding the tangent at the point (-4,6):

elliptical12

We can then check with Geogebra which shows that B’ is indeed (1681/144 , -62279/1728).  When a is chosen so that it is very large, this calculation becomes very difficult to attack using brute force methods – which would require checking 2(4,-6), 3(4,-6), 4(4,-6)… until the solution (1681/144 , -62279/1728) was found.

elliptical14

NSA and hacking data

Elliptical curve cryptography has some advantages over RSA cryptography – which is based on the difficulty of factorising large primes – as less digits are required to create a problem of equal difficulty.  Therefore data can be encoded more efficiently (and thus more rapidly) than using RSA encryption.  Currently the digital currency Bitcoin uses elliptical curve cryptography, and it is likely that its use will become more widespread as more and more data is digitalised.  However, it’s worth noting that as yet no-one has proved that it has to be difficult to crack elliptical curves – there may be a novel approach which is able to solve the problem in a much shorter time.  Indeed many mathematicians and computer scientists are working in this field.

Government digital spy agencies like the NSA and GCHQ are also very interested in such encryption techniques.  If there was a method of solving this problem quickly then overnight large amounts of encrypted data would be accessible – and for example Bitcoin currency exchange  would no longer be secure.  It also recently transpired that the NSA has built “backdoor” entries into some elliptical curve cryptography algorithms which have allowed them to access data that the people sending it thought was secure.   Mathematics is at the heart of this new digital arms race.

If you enjoyed this post you might also like:

RSA Encryption – the encryption system which secures the internet.

Circular inversion – learn about some other geometrical transformations used in university level mathematics.

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