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CSC2110 Discrete Mathematics Tutorial 6 Chinese Remainder Theorem, RSA and Primality Test

CSC2110 Discrete Mathematics Tutorial 6 Chinese Remainder Theorem, RSA and Primality Test. Hackson Leung. Announcement. Homework Set 2 is released! Deadline 30 Oct 17:00 Sharp No late submission is accepted Submit at the drop box near SHB 924 Project

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CSC2110 Discrete Mathematics Tutorial 6 Chinese Remainder Theorem, RSA and Primality Test

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  1. CSC2110 Discrete MathematicsTutorial 6Chinese Remainder Theorem, RSA and Primality Test Hackson Leung

  2. Announcement • Homework Set 2 is released! • Deadline • 30 Oct 17:00 Sharp • No late submission is accepted • Submit at the drop box near SHB 924 • Project • Those who have not registered, we assigned for you, please check CUHK email

  3. Agenda • Chinese Remainder Theorem • RSA • Primality Test

  4. Chinese Remainder Theorem • Example 1 • Solve for • Since • Then 3-1 exists and • Therefore,

  5. Chinese Remainder Theorem • Example 2 • Solve for • Since • We reduce it to • Same as example 1 • What if ? • Contradiction!

  6. Chinese Remainder Theorem • Solve the following

  7. Chinese Remainder Theorem • Consider so that • Step 1: Let • Step 2: Construct

  8. Chinese Remainder Theorem • Step 3: Find the multiplicative inverse of • Remember how to find multiplicative inverse? • Extended Euclid’s Algorithm! • Step 4: • Step 5: Adjust to meet the requirement

  9. Chinese Remainder Theorem • Example 1 • Solve for largest such that

  10. Chinese Remainder Theorem • Step 1: • Step 2: • Step 3: • Step 4: • Step 5:

  11. Chinese Remainder Theorem • What if ? • We can always reduce them • Example 2 • Solve the largest such that

  12. Chinese Remainder Theorem • Analyze first • Thus, we have

  13. Chinese Remainder Theorem • Take a look at • So • Same as example 1 • We want s to be relatively prime only!

  14. RSA • Step 1: , and very large prime • Step 2: • Step 3: Choose • Step 4: Find • Public key: • Private key:

  15. RSA • Example 1 • Let • Give the public and private keys in RSA cryptosystem

  16. RSA • Step 1: • Step 2: • Step 3: , the choice is ok • Step 4:

  17. RSA • Public key: • Private key: • Example 2: Encrypt 5 • Example 3: Decrypt

  18. RSA • Example 3

  19. Primality Test • Step 1: Pick a random number , set • Step 2: Calculate • Step 3: If not 1 (and not -1), composite, done • Step 4: If -1, “probably” prime, done • Step 5: If 1 and k is odd, “probably” prime, done • Step 6: , go back to step 2 Check when k < n - 1

  20. Primality Test • Example: Test if 221 is prime • Pick 174 to test • Under this test, 221 is “probably” prime • Pick 137 to test • We are sure 221 is composite! • 174: strong liar, 137: witness

  21. The End

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