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Public-Key Cryptography. The convergence of prime numbers, the history of math, inverse functions, and a contemporary application. Introduction to Cryptography. Cryptography is the study of ways of writing a message that hides its meaning from everyone except the intended recipient.

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public key cryptography
Public-Key Cryptography

The convergence of prime numbers, the history of math, inverse functions, and a contemporary application

introduction to cryptography
Introduction to Cryptography
  • Cryptography is the study of ways of writing a message that hides its meaning from everyone except the intended recipient.
  • Encryption is a method of changing plaintext, the message to be hidden, to ciphertext, the message in its hidden form.
  • Decryption is the procedure that changes ciphertext back to plaintext.
basic example
Basic Example

Plaintext- MATH RULES

Encryption rule - Write the plaintext backwards

Ciphertext - SELUR HTAM

function example encrypt
Function Example - Encrypt
  • Choose a function that has an inverse.
  • Rewrite the message as blocks of numbers. 2210 2917 3627 3021 1428

M A T H R U L E S

  • Evaluate the function at each block. This becomes the encrypted message.
function example decrypt
Function Example -Decrypt
  • Find the inverse of the encryption function. This is the decryption function.
  • Evaluate the decryption function at each received block.
path to a public function
Path to a Public Function
  • Create a function whose inverse is extremely difficult to determine without precise details of how the function was created.
  • Publish this function in a data base of public functions.
  • Use the inverse function only you can determine to decrypt messages intended for you.
egyptian multiplication
Egyptian Multiplication

1 26

2 52

4 104

8 208

16 416

32 > 23

egyptian multiplication1
Egyptian Multiplication

1 26

2 52

4 104

8 208

16 416

modular exponentiation
Modular Exponentiation

23325 mod 537

1 233 mod 537

2 2332 mod 537 = 52 mod 537

4 2334 mod 537 = 522 mod 537 = 19 mod 537

8 2338 mod 537 = 192 mod 537 = 361 mod 537

16 23316 mod 537 = 3612 mod 537 = 367 mod 537

modular exponentiation1
Modular Exponentiation

23325 mod 537

1 233 mod 537

  • 2332 mod 537 = 52 mod 537

4 2334 mod 537 = 522 mod 537 = 19 mod 537

8 2338 mod 537 = 192 mod 537 = 361 mod 537

16 23316 mod 537 = 3612 mod 537 = 367 mod 537

modular exponentiation2
Modular Exponentiation

Because the exponent 25 = 1 + 8 + 16, the product of the nonzero elements is

public key cryptography1
Public-Key Cryptography
  • Choose two large prime numbers, p and q, and form their product n = pq.
  • Calculate
  • Randomly choose e such that

and

  • The values of e and n are the public key.
  • The ciphertext, c, is c = me mod n, where m is the message being encrypted.
an encryption example
An Encryption Example
  • Let p = 83 and q = 89. Then n = 7387.

= (83 – 1)(89 – 1) = 7216

  • Randomly choose e = 23. Verify
  • The encryption function is c = m23mod 7387, where m is a plaintext message block and c is a cipher block.
an encryption example1
An Encryption Example

Encrypt M A T H R U L E S

2210 2917 3627 3021 1428

decryption function
Decryption Function

The decryption function is m = c1255 mod 7387.

public key cryptography2
Public-Key Cryptography
  • Theorem: The decryption function is given by m = cdmod n, where d is the solution of
  • Basically, we have to prove that

cd = (me mod n)d = med mod n = m.

other applications
Other Applications
  • Digital signatures
    • Olivia encrypts a message using her private key. Henry decrypts the message using her public key.
    • Better: Olivia first encrypts her message using Henry’s public key. Then uses her private key to encrypt that message.
  • HTTPS
contact information
Contact Information
  • galoisgroup@mac.com
  • http://public.me.com/galoisgroup