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Therefore, <math>a*X + b*\bar{X} \to a*Y + b*\bar{Y}</math> where <math>a, b \in \mathbb{C}</math> | Therefore, <math>a*X + b*\bar{X} \to a*Y + b*\bar{Y}</math> where <math>a, b \in \mathbb{C}</math> | ||
| − | '''3)'''First, one has to find the matrix that was used in encoding. To find this, we take input '''<big><math>(1,0,4,0,1,0,1,0,1)</math></big>''' and turn it into matrix <math>\begin{Bmatrix}1 & 0 & 4\\ 0 & 1 & 0\\ 1 & 0 & 1\end{Bmatrix}</math>. | + | '''3)'''First, one has to find the matrix that was used in encoding. To find this, we take input '''<big><math>(1,0,4,0,1,0,1,0,1)</math></big>''' and turn it into matrix <math>\begin{Bmatrix}1 & 0 & 4\\ 0 & 1 & 0\\ 1 & 0 & 1\end{Bmatrix}</math>. Then find the inverse of the output matrix <math>\begin{Bmatrix}2 &0&0\\0&1&0\\0&0&3\end{Bmatrix}</math> which is <math>\begin{Bmatrix}\frac{1}{2}&0&2\\0&1&0\\ \frac{1}{3}&0&\frac{1}{3}\end{Bmatrix}</math> |
<math>\begin{Bmatrix}a * b* c\end{Bmatrix} * \begin{Bmatrix}1 & 0 & 4\\ 0 & 1 & 0\\ 1 & 0 & 1\end{Bmatrix} \to \begin{Bmatrix}2 &0&0\\0&1&0\\0&0&3\end{Bmatrix}</math> which yields three equations with three unknowns. | <math>\begin{Bmatrix}a * b* c\end{Bmatrix} * \begin{Bmatrix}1 & 0 & 4\\ 0 & 1 & 0\\ 1 & 0 & 1\end{Bmatrix} \to \begin{Bmatrix}2 &0&0\\0&1&0\\0&0&3\end{Bmatrix}</math> which yields three equations with three unknowns. | ||
Solving for coefficients '''<big>a, b, c</big>''' | Solving for coefficients '''<big>a, b, c</big>''' | ||
Revision as of 17:45, 19 September 2008
1) Bob can decrypt the message by multiplying the encrypted message by the inverse of the 3x3 matrix that his friend gave him, if it exists.
2)Yes she can! Reason being that this is a linear system in which input $ X $ yields output $ Y $ and input $ \bar{X} $ yields output $ \bar{Y} $.
Therefore, $ a*X + b*\bar{X} \to a*Y + b*\bar{Y} $ where $ a, b \in \mathbb{C} $
3)First, one has to find the matrix that was used in encoding. To find this, we take input $ (1,0,4,0,1,0,1,0,1) $ and turn it into matrix $ \begin{Bmatrix}1 & 0 & 4\\ 0 & 1 & 0\\ 1 & 0 & 1\end{Bmatrix} $. Then find the inverse of the output matrix $ \begin{Bmatrix}2 &0&0\\0&1&0\\0&0&3\end{Bmatrix} $ which is $ \begin{Bmatrix}\frac{1}{2}&0&2\\0&1&0\\ \frac{1}{3}&0&\frac{1}{3}\end{Bmatrix} $
$ \begin{Bmatrix}a * b* c\end{Bmatrix} * \begin{Bmatrix}1 & 0 & 4\\ 0 & 1 & 0\\ 1 & 0 & 1\end{Bmatrix} \to \begin{Bmatrix}2 &0&0\\0&1&0\\0&0&3\end{Bmatrix} $ which yields three equations with three unknowns.
Solving for coefficients a, b, c
