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

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

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### CSC2110 Discrete MathematicsTutorial 6Chinese 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
- 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 ?
- Contradiction!

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

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