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

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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 MathematicsTutorial 6Chinese Remainder Theorem, RSA and Primality Test

Hackson Leung

Announcement
• Homework Set 2 is released!
• 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
Agenda
• Chinese Remainder Theorem
• RSA
• Primality Test
Chinese Remainder Theorem
• Example 1
• Solve for
• Since
• Then 3-1 exists and
• Therefore,
Chinese Remainder Theorem
• Example 2
• Solve for
• Since
• We reduce it to
• Same as example 1
• What if ?
Chinese Remainder Theorem
• Solve the following
Chinese Remainder Theorem
• Consider so that
• Step 1: Let
• Step 2: Construct
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
Chinese Remainder Theorem
• Example 1
• Solve for largest such that
Chinese Remainder Theorem
• Step 1:
• Step 2:
• Step 3:
• Step 4:
• Step 5:
Chinese Remainder Theorem
• What if ?
• We can always reduce them
• Example 2
• Solve the largest such that
Chinese Remainder Theorem
• Analyze first
• Thus, we have
Chinese Remainder Theorem
• Take a look at
• So
• Same as example 1
• We want s to be relatively prime only!
RSA
• Step 1: , and very large prime
• Step 2:
• Step 3: Choose
• Step 4: Find
• Public key:
• Private key:
RSA
• Example 1
• Let
• Give the public and private keys in RSA cryptosystem
RSA
• Step 1:
• Step 2:
• Step 3: , the choice is ok
• Step 4:
RSA
• Public key:
• Private key:
• Example 2: Encrypt 5
• Example 3: Decrypt
RSA
• Example 3
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

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