Chinese Remainder Theorem

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Chinese Remainder Theorem
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An example

• Find a number x such that have remainders of 1
  when divided by 3, 2 when divided by 5 and 3 when
  divided by 7. i.e.
• x 1 mod 3
• x 2 mod 5
• x 3 mod 7

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Introduction

• used to speed up modulo computations
• Integers can be represented by their residues
  modulo a set of pairwise relatively prime moduli.
• E.g. In Z10, integer 8 can be represented by the
  residues of the 2 relatively prime factors of 10
  (25) as a tuple (0, 3)

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Chinese Remainder Theorem (CRT)

• Let m1, m2, , mk be pairwise relatively prime
  integers. That is, gcd(mi, mj) 1 for 1?iltj ?k.
  Let ai?Zmi for 1?i ?k and set Mm1m2mk. Then
  there exists a unique A ?Zm such that A?ai mod mi
  for i 1k.
• A can be computed as
• Where for 1?i?k.

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Proof

• A is a solution
• Since
• for any j?I
• Therefore,

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Proof (cont.)

• A is unique in Zm
• If A is not the unique answer, there must exist
  another answer A? ai mod mi in Zm.
• Then A ? A mod mi
• ? A-A r1m1 r2m2 rkmk
• ? rimj where i?j (since mis are relatively
  prime)
• ? rimi rim1m2mk rim
• ? mA-A
• ? A ? A mod m, proving uniqueness.

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Properties

• (AB) mod M ?
• ((a1 b1) mod m1, , (ak bk)mod mk)
• (A-B) mod M ?
• ((a1 - b1) mod m1, , (ak - bk)mod mk)
• (A?B) mod M ?
• ((a1 ? b1) mod m1, , (ak ? bk)mod mk)

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Example 1

• Represent 973 in Z1813 as a k-tuple
• Answer
• M 1813 37 49 ? m1 37 m2 49
• A 973
• ? A (A mod m1, A mod m2) (11, 42)

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Example 2

• Give x 11 mod 37 x 42 mod 49, find x.
• Answer
• since M1 49 M1-1 mod m1 34 and M2 37
  M2-1 mod m2 4

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Exercises

• Question 1
• Represent 75 in Z77 using Chinese Remainder
  Theorem.
• Question 2
• Given x 6 mod 13 and x 2 mod 17, find x.