Section 2.9 The Hill Cipher; Matrices

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# Section 2.9 The Hill Cipher; Matrices - PowerPoint PPT Presentation

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.

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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…
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
• 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.
• 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
• 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 nxnmatrix…
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…
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…!