CSC2110%20Discrete%20Mathematics%20Tutorial%206%20Chinese%20Remainder%20Theorem%20and%20RSA

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CSCI2110 Discrete MathematicsTutorial 6Chinese
Remainder Theorem and RSA
Hackson Leung
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Announcement

• Project
• Those who have not registered, we assigned for
  you, please check CUHK email
• Midterm
• Next Monday 230pm to 400pm
• Bring your own calculator

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Agenda

• Chinese Remainder Theorem
• RSA

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Chinese Remainder Theorem

• Example 1
• Solve for
• Since
• Then 3-1 exists and
• Therefore,

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Chinese Remainder Theorem

• Example 2
• Solve for
• Since
• We reduce it to
• Same as example 1
• What if ?
• Contradiction!

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Chinese Remainder Theorem

• Example 3
• Solve the followings, or show no solution if it
  has no solution

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Chinese Remainder Theorem

• Example 4
• Solve the following, or show no solution if it
  has no solution

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Chinese Remainder Theorem

• Solve the following

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Chinese Remainder Theorem

• Consider so that
• Step 1 Let
• Step 2 Construct

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Chinese Remainder Theorem

• Step 3 Find the multiplicative inverse of
• Remember how to find multiplicative inverse?
• Extended Euclids Algorithm!
• Step 4
• Step 5 Adjust to meet the requirement

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Chinese Remainder Theorem

• Example 1
• Solve for largest such that

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Chinese Remainder Theorem

• Step 1
• Step 2
• Step 3
• Step 4
• Step 5

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Chinese Remainder Theorem

• Example 2
• Solve for such that

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Chinese Remainder Theorem

• Example 3
• Solve for smallest such that

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Chinese Remainder Theorem

• What if
  ?
• We can always reduce them
• Example 4
• Solve the largest such that

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Chinese Remainder Theorem

• Analyze first
• Thus, we have

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Chinese Remainder Theorem

• Take a look at
• So
• Same as example 1
• We want s to be relatively prime only!

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Chinese Remainder Theorem

• Example 5
• Solve for smallest such that

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Chinese Remainder Theorem

• Example 6
• Solve for largest such that

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RSA

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

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RSA

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

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RSA

• Step 1
• Step 2
• Step 3 , the choice is
  ok
• Step 4

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RSA

• Public key
• Private key
• Example 2 Encrypt 5
• Example 3 Decrypt

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RSA

• Example 3

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The End