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Types of Ciphers

  • Classical Ciphers
  • Transposition Ciphers
  • Substitution Ciphers
  • Monoalphabetic/Polyalphabetic
  • Steganography
  • Modern Ciphers
  • Computation-intensive mathematical ciphers

Practical Applications

  • Digital authentication
  • Time stamping
  • Exchanging digital currency (cryptocurrency)
  • Secure network communication
  • Disk encryption

Cryptography and Linear Algebra

Cryptography Background

The Information Age

  • Math-based cryptography > Linguistic cryptograpy
  • Increase in sophistication of both cryptography and cryptanalysis
  • Deciphering requires much more effort than enciphering

Alan Turing

  • Goal: secure communication
  • First used in Egypt
  • Message confidentiality
  • Linguistic and lexicographic patterns
  • Data Encryption
  • Mathematics

Definitions

  • Cryptography = the study of encoding and decoding messages
  • Ciphers = Codes
  • Plaintext = Uncoded Messages
  • Ciphertext = Coded Messages

Bobby McGrane

Michael Tang

Enciphering with Hill Ciphers

  • Lester S. Hill
  • Assign Numbers to Characters

  • Matrix A
  • Pair the Plaintext
  • Make the pairs into vectors
  • Make the vectors into a matrix
  • C=AP
  • Make C into vectors
  • Convert to letters

Example

Given the start of the plaintext matrix provided by Bobby, decipher the following ciphertext matrix.

Deciphering With Hill Ciphers

  • P = CA^-1
  • Set up Augmented Matrix
  • RREF
  • Transpose

Example

Use the 2 x 2 matrix A to encipher the message "Linear Algebra"

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