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…
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…
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…
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…
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.
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Example 6:
Example 7:
Example 8:
In general, matrix multiplication is not commutative: AB ≠BA…
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)…
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…
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…
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.
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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…
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…
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…
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…
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) = 456976 different matrices.
– The way to make it more difficult is to increase the size of the
key matrix…
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…!