chapter 2 section 9 the hill cipher and matrices.pptx

8/14/2019 Chapter 2 Section 9 The Hill Cipher and Matrices.pptx

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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


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Introduction to Matrices

A matrix is a rectangular array of numbers madeup of rows and columns.

The sizeof 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 entryin a matrix A, we use aijwhere i = row and j = column.

The general form of a mxn matrix has the formindicated here.

A square m x n matrix is a matrix where m = n. That isthe number of rows equals the number of columns
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Equality of Matrices

Two matrices A and B are equalif

They have the same size and

There corresponding entries are equal.

Special types of MatricesVectors

A row vectoris a matrix with one row.

A column vectoris a matrix with one column
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Matrix Addition and Subtraction Two matrices can be added and subtracted only if they have the

same size.

Example 1: A + B and AB

Example 2: A + B and AB

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 thematrix by the given scalar.

Example 3: 3A

Example 4: 5A2B

Addition and Scalar Multiplication Properties of Matrices
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Introduction To Matrices

Matrix Multiplication

Multiplying two matrices requires how you multiplya row vector timesa 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 jthcolumn of B.

Example 6:

Example 7:

Example 8:

In general, matrix multiplication is not commutative: AB BA
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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)


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Addition Identity Matrices

The additive identity has all entries of zero. It is called thezero 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 identityhas 1s on the main diagonal(row number = column number) and 0s everywhere else.

Example 9: AI and IA
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Determinants

The determinant of a matrix is a real number.

The determinantof 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
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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)

Notationof the inverse.

The inverse for the 2x2 matrixis fairly simple to find.

The B is the inverse of A the AB = BA = I. (I call it B here becausethis stupid program doesnt 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
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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: aijMOD m.

Example 15: Compute matrix MOD 26.

Example 16: Find A + B and AB MOD 5

Example 17: 3A MOD 13 Example 18: Product AB MOD 26
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Finding the inverseof 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
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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 aninverse. The procedure requires breaking the code up into small

segments. If the matrix is nxn, then each segment consists of nletters.

If A is the matrix and x is the n letter segment code, then theciphertext is found by calculating Ax = y. Y is the ciphertextsegment.

To decipher the text we use the inverse of the matrix A. If wecall this inverse B, then By deciphers the code returning x.

Note: It is required that the plaintext message have n letters. Ifit does not have some multiple of n letters, we pad the messagewith extra characters until it does.

Example 22: Encrypt a Message.

Example 23: Decrypt a Message
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Cryptanalysis of the Hill System

Having just the ciphertext when trying to crypto-analyze aHill cipher is more difficult then a monoalphabetic cipher.

The character frequencies are obscured (because we areencrypting each letter according to a sequence of letters).

When using a 2x2 matrix, we are in effect creating a 26^2 = 676character alphabet. That is, there are 676 different two lettercombinations.

If you in fact knew that the ciphertext was created using a 2x2matrix, then a crypto-analyst could break the code with bruteforce, 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


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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!
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chapter 2 section 9 the hill cipher and matrices.pptx

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