Question

Consider Hill Cipher with block length b = 3. That is, the secret key is a 3 × 3 matrix over Z26. The definition of Hill Cipher is given in the lecture notes. In the data file a8.mat posted on Canvas you will find five plaintexts x1,x2,x3,x4,x5 as well as five ciphertexts y1,y2,y3,y4,y5. It is given that yi is the encryption of xi under the same secret key K, for i=1,2,3,4,5. Perform the known plaintext attack in order to find the secret key, using the given data. Hint: Let X be a 3 × 3 matrix whose rows are some three plaintexts, and let Y be the 3 × 3 matrix whose rows are the corresponding ciphertexts, in the same order. Let K be the secret key. Determine a relation among the matrices X, Y and K. Once you write down the relation among X, Y and K, solve this relation (equation) for K, which is still just linear algebra using pen and paper. Once you have a general expression for K in terms of X and Y , only then start working on the numerical data with Matlab. Note that some matrices X can not be used in this method (specify the required condition on X).

Answer 1

In the known plaintext attack in Hill Cipher, the relation between the X, Y, and K matrices is Y = XK. We can find the secret key by solving the equation K = X^(-1)Y using the inverse of X.

Step-by-step explanation:

The objective of the known plaintext attack in Hill Cipher is to determine the secret key by analyzing the relation between the plaintext matrix (X), ciphertext matrix (Y), and the secret key (K). In this case, we are given five plaintexts and their corresponding ciphertexts. We can form a matrix X using the plaintexts and a matrix Y using the ciphertexts. The relation between X, Y, and K can be expressed as Y = XK, where X, Y, and K are 3x3 matrices. To find the secret key, we solve the equation by finding the inverse of X and multiplying it with Y: K = X^(-1)Y.