1. Section 2.9 The Hill Cipher; Matrices
The Hill cipher is a block or polygraphic cipher, where groups of plaintext are enciphered as units.
The Hill cipher enciphers data using matrix multiplication.
We will now introduce the concept of a matrix…

2. Introduction to Matrices
A matrix is a rectangular array of numbers made up of rows and columns.
The size of a matrix is given as m x n
m is the number of rows to the matrix.
n is the number of columns to the matrix.
To indicate an individual entry in a matrix A, we use aij where i = row and j = column.
The general form of a mxn matrix has the form indicated here.
A square m x n matrix is a matrix where m = n. That is the number of rows equals the number of columns…

3. Introduction to Matrices
Equality of Matrices
Two matrices A and B are equal if
They have the same size and
There corresponding entries are equal.
Special types of Matrices – Vectors
A row vector is a matrix with one row.
A column vector is a matrix with one column…

4. Introduction to Matrices
Matrix Addition and Subtraction
Two matrices can be added and subtracted only if they have the same size.
Example 1: A + B and A – B
Example 2: A + B and A – B
Scalar Multiplication of Matrices
When working with matrices, numbers are referred as scalars. To multiply a matrix by a scalar, we multiply each entry of the matrix by the given scalar.
Example 3: 3A
Example 4: 5A – 2B
Addition and Scalar Multiplication Properties of Matrices…

5. Introduction To Matrices
Matrix Multiplication
Multiplying two matrices requires how you multiply a row vector times a column vector.
Example 5: Compute AB
For the matrix product AB to exist, the number of columns of A must be equal to the number of rows of B.
If A has size m x n and B has size n x p, then the product AB has size m x p.
The number of row and column vectors that must be multiplied together is mp.
The ijth element of AB is the vector product of the ith row of A and the jth column of B.
Example 6:
Example 7:
Example 8:
In general, matrix multiplication is not commutative: AB ≠BA…

6. Introduction to Matrices
Multiplicative Properties of Matrices
Let A, B, and C be matrices whose sizes are multiplicatively compatible, c a scalar.
(AB)C = A(BC) matrix multiplication is associative
A(B + C) = AB + AC
(A + B)C = AC + BC
c(AB) = (cA)B = A(cB)…

7. Introduction to Matrices
Addition Identity Matrices
The additive identity has all entries of zero. It is called the zero matrix.
If A is mxn then the zero matix is mxn.
The zero matrix is called 0.
A + 0 = 0 + A = A
Multiplicative Identity Matrices
If A is mxn then the multiplicative matrix is nxn.
The multiplicative identity has 1s on the main diagonal (row number = column number) and 0s everywhere else.
Example 9: AI and IA…

8. Introduction to Matrices
Determinants
The determinant of a matrix is a real number.
The determinant of a 2x2 matrix.
Example 10: Find the determinant
Example 11: Find the determinant
Note: the determinant of a 1x1 matrix is just the value of the entry. A =[3] then |A| = 3.
You can calculate the determinant of any nxn matrix…

9. Introduction to Matrices
Matrix Inverses
The additive inverse of a matrix is obvious. You want A + B = 0, where B is the inverse That is B = -A.
The more difficult to find, and not always exists, is the multiplicative inverse.
The matrix A must be nxn (a square matrix)
Notation of the inverse.
The inverse for the 2x2 matrix is fairly simple to find.
The B is the inverse of A the AB = BA = I. (I call it B here because this stupid program doesn’t allow exponents)
Example 12: Find inverse.
Note: For the matrix A, the inverse exists if det(A) ≠ 0.
Example 13: Find Inverse.
Example 14: Find Inverse…

10. Introduction to Matrices
Matrices with Modular Arithmetic
For a matrix A with entries aij we way that A MOD m is the matrix where the MOD operation is applied to each entry: aij MOD m.
Example 15: Compute matrix MOD 26.
Example 16: Find A + B and A – B MOD 5
Example 17: 3A MOD 13
Example 18: Product AB MOD 26…

11. Introduction to Matrices
Finding the inverse of a matrix in modular arithmetic.
Example 19: Find the inverse of a matrix
Example 20: Determine if inverse exists.
Example 21: Solve the system of equations…

12. The Hill System
The Hill Cipher was developed by Lester Hill of Hunter College. It requires the use of a matrix mod 26 that has an inverse.
The procedure requires breaking the code up into small segments. If the matrix is nxn, then each segment consists of n letters.
If A is the matrix and x is the n letter segment code, then the ciphertext is found by calculating Ax = y. Y is the ciphertext segment.
To decipher the text we use the inverse of the matrix A. If we call this inverse B, then By deciphers the code returning x.
Note: It is required that the plaintext message have n letters. If it does not have some multiple of n letters, we pad the message with extra characters until it does.
Example 22: Encrypt a Message.
Example 23: Decrypt a Message…

13. Cryptanalysis of the Hill System
Having just the ciphertext when trying to crypto-analyze a Hill cipher is more difficult then a monoalphabetic cipher.
The character frequencies are obscured (because we are encrypting each letter according to a sequence of letters).
When using a 2x2 matrix, we are in effect creating a 26^2 = 676 character alphabet. That is, there are 676 different two letter combinations.
If you in fact knew that the ciphertext was created using a 2x2 matrix, then a crypto-analyst could break the code with brute force, since there are 26^4 (each entry in the matrix can have 26 different numbers) = different matrices.
The way to make it more difficult is to increase the size of the key matrix…

14. Cryptanalysis of the Hill System
If the adversary has the ciphertext and a small amount of corresponding plaintext, then the Hill Cipher is more vulnerable…!