Maths Behind Credit Card Transactions

[James...]

I am doing a qualification which requires me to chooses a subject for which I will research & produce a presentation & a 5000 word essay in December.

As I plan to study maths at Degree Level I have decided to do a maths based project.

My two basic ideas are...

How does Maths keep our Money safe?

Here I would look into the Mathematics behind credit card transactions & online money trnasfers & the encryptions etc used to ensure it cannot be obtained & understood.

Or,

How the golden ratio is found in nature - something about fibonachis.


I know nothing about the above subjects (which is the point of the qualification) so before I decide I was wondering if anyone can help me out with the first, the credit card one. I have 5 books about divine proportion but very little on encryption.

Is there enough maths behind it to do 6 months of research & understand it, I only have AS Maths in college so my theory will be minimal I assume. I do have a contact who does this for a living but cannot get in touch with him for a while so I am looking for other sources of information.

So if anyone knows any of the basics behind the transactions or knows where I can read up on it I would appreciate it. Got a few websites saved to have a read through tomorrow.

Cheers
James

 

[mgb_phys]

There isn't really encryption behind credit cards as such.
The two best writers on credit card security are probably Ross Anderson Bruce Schneier (both from a technical and a real world point) both have good books on crypto and security.
If you are doing this as a maths project RSA is an interesting example - showing how the function can be easy to do one way but hard to reverse shoudl be within A level maths.

 

[Castilla]

I know that there are, at least, 2 different mathematical basis
for asimetric cryptography:

1. RSA (Rivest-Shamir-Adlemann - the inventors) method.

2. Elliptic curves method.

To understand the first one you only need some easy theorems of elementary number theory.

TO understand the second you need very complicated math.

 

[James...]

Thank you for the reply, just looked at those authors, quite expensive books so will ask my local libery to get some copies, or request anyway!

Just looked up RSA & I presume you mean the algorithm. Looks fairly complicated but I try going over it tonight & see if I can make sense of what is going on.

Cheers
james

 

[James...]

To understand the first one you only need some easy theorems of elementary number theory.

TO understand the second you need very complicated math.

I will have a quick scan over the elliptical curves, do you know anywhere online where I can read up on some of the number theory theorems as I have ne knowlage of it as of yet.

Cheers
james

 

[CRGreathouse]

I will have a quick scan over the elliptical curves, do you know anywhere online where I can read up on some of the number theory theorems as I have ne knowlage of it as of yet.

Have you taken a number theory course? If not, you probably won't be able to pick it up just from a book, at least not quickly.

 

[James...]

Not that I know of, never heard it mentioned anyway.

This is what I am doing...

http://www.ocr.org.uk/Data/publications/key_documents/L_GCE_Maths_Spec.pdf

On page 4, I have done 5.1, 5.2 & 5.16

Currently studying - 5.3

And going on to do 5.4, 5.5, 5.6, 5.7 & 5.8 next year if it makes any difference.

It gives a brief overview of what my course covers.

james

 

[mgb_phys]

Elliptical curves is grad school level maths - I don't know if there is an easy level you can look at.
RSA is much simpler (in principle at least) it only uses exponents and modulus - the maths is quite interesting and should be accessible at your level, there is also a lot written about it. Check out any of schneier's books

 

[James...]

I'm having a go at the RSA at the moment, just sticking some values through it to see if I can get it to work. I will try & check out Schneier's books too.

Just wondering if anyone knows, when you first introduce (mod x) into it, does the remainder (R) have to be a set amount?

eg,

I have 13*j = R (mod 60) up to now & I need to work out J.

On the example I am using online they have

7*j = 1 (mod 20)

which they can use j = 3 to get the remainder of 1

but if I was to use a remainder of 1 I would need 13j = 61 giving me j = 4.6923...

which would be hard to work with. Does the remainder have to be 1 or could I use 5 so I could have...

13j = 5 (mod 60)
13*5 = 5 (mod 60)

which would give me a nice value for j to work with.

Cheers
James

Edit - having a go with a remainder of 1 after realising I can leave j as a fraction!

 

[Petek]

You might want to look at Koblitz's https://www.amazon.com/dp/0387942939/?tag=pfamazon01-20, which is at the undergraduate level. It covers just enough number theory to study cryptography. It also covers elliptic curves, omitting some of the more difficult proofs.

 

[Dragonfall]

Fun Fact: If you take any credit card number and swap any two consecutive numbers, it is not a valid number.

 

[camilus]

^pretty cool where'd you learn that?

 

[mgb_phys]

Credit cards use the luhn checksum http://en.wikipedia.org/wiki/Luhn_algorithm