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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.

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section 2 9 the hill cipher matrices
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
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 matrices1
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 matrices2
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 matrices3
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 matrices4
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 matrices5
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 matrices6
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 matrices7
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 matrices8
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 matrices9
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 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
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 system1
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…!