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L06: The RSA Algorithm

L06: The RSA Algorithm. Objective: Present the RSA Cryptosystem Prove its correctness Discuss related issues. The RSA Algorithm. Exponentiation mod n The RSA Cryptosystem Correctness Fermat’s Little Theorem Decipherability of RSA Security of RSA

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L06: The RSA Algorithm

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  1. L06: The RSA Algorithm • Objective: • Present the RSA Cryptosystem • Prove its correctness • Discuss related issues

  2. The RSA Algorithm • Exponentiation mod n • The RSA Cryptosystem • Correctness • Fermat’s Little Theorem • Decipherability of RSA • Security of RSA • Calculating exponentiation mod n efficiently • The Chinese Remainder Theorem

  3. Exponentiation mod n • Encryption with addition and multiplication mod n • Easy to find the way to decrypt • RSA: use exponentiation mod n

  4. Exponentiation mod n

  5. Corollary of Lemma 2.19

  6. Page 8 The RSA Algorithm • Exponentiation mod n • The RSA Cryptosystem • Correctness • Fermat’s Little Theorem • Decipherability of RSA • Security of RSA • Exponentiation mod n efficiently • The Chinese Remainder Theorem

  7. Public-Key Cryptography

  8. RSA Algorithm • Questions to answer

  9. One-Way Function

  10. RSA Algorithm • Builds a one-way function using • Exponentiation mod n • Prime numbers • gcd • Multiplicative inverse

  11. RSA Algorithm

  12. RSA Algorithm

  13. RSA Example • Key generation

  14. RSA Example • Encryption and decryption • Try: http://cisnet.baruch.cuny.edu/holowczak/classes/9444/rsademo/

  15. Page 17 The RSA Algorithm • Exponentiation mod n • The RSA Cryptosystem • Correctness • Fermat’s Little Theorem • Decipherability of RSA • Security of RSA • Exponentiation mod n efficiently • The Chinese Remainder Theorem

  16. A Lemma

  17. Fermat’s Little Theorem

  18. What is a is a multiple of p?

  19. Simplifies computation

  20. Page 25 The RSA Algorithm • Exponentiation mod n • The RSA Cryptosystem • Correctness • Fermat’s Little Theorem • Decipherability of RSA • Security of RSA • Exponentiation mod n efficiently • The Chinese Remainder Theorem

  21. Decipherability

  22. Decipherability

  23. Decipherability Proved!

  24. Page 35 The RSA Algorithm • Exponentiation mod n • The RSA Cryptosystem • Correctness • Fermat’s Little Theorem • Decipherability of RSA • Security of RSA • Exponentiation mod n efficiently • The Chinese Remainder Theorem

  25. Page 38 The RSA Algorithm • Exponentiation mod n • The RSA Cryptosystem • Correctness • Fermat’s Little Theorem • Decipherability of RSA • Security of RSA • Exponentiation mod n efficiently • The Chinese Remainder Theorem

  26. Exponentiation mod n efficiently

  27. Exponentiation mod n efficiently

  28. Exponentiation mod n efficiently

  29. Exponentiation mod n efficiently

  30. Complexity of Repeated Squaring

  31. Page 44 The RSA Algorithm • Exponentiation mod n • The RSA Cryptosystem • Correctness • Fermat’s Little Theorem • Decipherability of RSA • Security of RSA • Exponentiation mod n efficiently • The Chinese Remainder Theorem

  32. The Chinese Remainder Theorem

  33. The Chinese Remainder Theorem

  34. The Chinese Remainder Theorem

  35. The Chinese Remainder Theorem

  36. The Chinese Remainder Theorem

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