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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. 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

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  1. Public-Key Cryptography The convergence of prime numbers, the history of math, inverse functions, and a contemporary application

  2. 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.

  3. Basic Example Plaintext- MATH RULES Encryption rule - Write the plaintext backwards Ciphertext - SELUR HTAM

  4. 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.

  5. Function Example -Decrypt • Find the inverse of the encryption function. This is the decryption function. • Evaluate the decryption function at each received block.

  6. 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.

  7. Egyptian Multiplication 1 26 2 52 4 104 8 208 16 416 32 > 23

  8. Egyptian Multiplication 1 26 2 52 4 104 8 208 16 416

  9. New Egyptian Multiplication

  10. Egyptian Exponentiation

  11. Modular Arithmetic

  12. 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

  13. 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

  14. Modular Exponentiation Because the exponent 25 = 1 + 8 + 16, the product of the nonzero elements is

  15. 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.

  16. 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.

  17. An Encryption Example Encrypt M A T H R U L E S 2210 2917 3627 3021 1428

  18. Decryption Function The decryption function is m = c1255 mod 7387.

  19. 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.

  20. 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

  21. Contact Information • galoisgroup@mac.com • http://public.me.com/galoisgroup

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