Editing Geometric algebra (section)

==How we used it?==
As mentioned, the dominant method used for our project was rotations with GA, and fortunately enough, the formula for doing said rotations in GA is a very general one. It's shown below and is applicable to rotations of any dimension, whether it be 3D, 4D 5D or even 20D. It's important to note that DeMoivre's theorm applies.

[[Image:GARotations.png|center|450px]]


So as mentioned several times throughout the wiki, the original plan for the project was to examine a particular dimension thoroughly, beginging with 3D and either prove or disprove the question of whether or not rotations can be used to achieve secure communication in that dimensions according to the KS cipher. Once this was done, we would move on to a higher dimension and keep doing so until we found a solution or until project closeout- whichever came first.

[[Image:RotationPicture.png|center|700px]]

So in 3D Rotations for an initial message vector '''m''' in the form shown below, we apply a '''Rotation Operator''' or '''Rotor''' R to the message in the shown form, wit a corresponding '''Reversion Operator''' applied to opposite side of the message vector as shown. 

[[Image:proof1a.png|center|450px]]


[[Image:proof1b.png|center|240px]]

This gives us the final message vector of the form:

[[Image:proof1c.png|center|450px]]


Now as mentioned, we started to explore the 3D rotations to circumvent the vulnerability of 2-dimensional rotations, as given the initial and final rotated message vector in 3D it would not be possible for an eavesdropper to simultaneously deduce the rotation axis and rotation angle, as only the plane of possible rotation axes can be found, and hence it appears to be secure against and eavesdropper. Which as depicted in the image above, the '''''v''''' in the rotation operator '''R''' represents the rotation axis around which the rotation occurs and '''Theta''' the clockwise rotation angle.

So given 2 rotation operators selected independently by Alice and Bob, let’s say '''Ra''' and '''Rb''', each with a unique rotation axis and rotation angle, we have the encryption process shown on the screen to give our '''final message vector'''. Where the most inner 2 '''Rotors''' around our initial message vector define the original encryption by Alice, the next two the encryption of Bob, and then the 2 on the far left and far right act as the decryption process of both Alice and Bob. However, as we know from the Final Report in order for this process to work we require '''Ra''' and '''Rb''' to commute, that is we needed '''RaRb to equal RbRa'''.

[[Image:AliceBob1.png|center|450px]]


[[Image:AliceBob2.png|center|590px]]


However as we can see through the application of Demoivres theorem, this breaks down to the form shown below, which implies that the cross product of the 2 rotation axes must be 0, and hence the 2 rotation axes of Alice and Bob must be parallel. However in order for Alice and Bob to use parallel Rotation Axes, a preferred direction would need to be communicated beforehand, however this would make the process vulnerable and goes against the premise of the double padlock protocol.


[[Image:AliceBob3.png|center|710px]]


For a much more detailed explanation of the proof for 3D and the difficulties that arose in 4D, please read the [https://www.eleceng.adelaide.edu.au/personal/dabbott/wiki/index.php/Semester_B_Final_Report_2013_-_Secure_communications_without_key_exchange%3F_2013#2D_.26_3D_Rotations Final Report].