Chinese Remainder Theorem CRT programming

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Chinese Remainder Theorem (CRT) programming

• Jonas Tan
• jonastan_at_cs.tamu.edu

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Chinese Remainder Theorem

• Used for speeding up modulo computations
• Reconstruct integers in a certain range from
  their residues modulo a set of pairwise
  relatively prime number.
• The 10 integers in Z10(0,1,,9) can be
  constructed from their two residues modulo 2 and
  5 (the relatively prime factors of 10).
• x mod 2 0 x mod 5 3 gt The unique solution is
  8

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Example
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Multiplicative Inverse

• If gcd(d, f) 1, then d has a multiplicative
  inverse modulo f.
• For positive integer dltf, there exists a
  such that
• In addition to finding gcd(d, f), if gcd(d ,f)1,
  Euclidean algorithm can return the multiplicative
  inverse of d.

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Program Exercise

• Add your codes into Crt.cs
• BigInteger CRT(BigInteger r1, BigInteger r2,
  BigInteger m1, BigInteger m2)
• BigInteger x
• by Euclids algorithm, compute inv(m1) and
  inv(m2)
• Calculate x by equation
• return x