Lesson 11: Applications of Number Theory

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Lesson 11 Applications of Number Theory

• Objectives
• Calculate bnmod(m) for large b,n,m
• Solve Linear Congruences
• Solve Chinese Remainder Problems
• Represent numbers with pairwise relatively prime
  n-tuples (pg. 187)
• Perform arithmetic with large numbers using a
  limited base.

• Outline
• Modular Exponentiation
• Linear Congruence
• Chinese Remainder Problems
• Integer Representation
• Arithmetic of Large Numbers
• Reading Section 2.6

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Modular Exponentiation

• Recall
• if a b mod(m)
• c d mod(m)
• Then bd mod(m) ac
• Also (bd) mod(m) ac

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Modular Exponentiation

• Goal Find bn mod(m)
• bbn-1 mod(m) b mod(m) bn-1 mod(m)
• b 14
• n 165
• m 9
• bn 1.2916e189

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Modular Exponentiation

• Goal Find bn mod(m)
• Express n in binary

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Modular Exponentiation

• Goal Find bn mod(m)
• bn mod(m) bn/2 mod(m) bn/2 mod(m) (if n
  is even)
• or b mod(m)bn/2 mod(m) bn/2 mod(m) (if n
  is odd)
• Recursively apply technique
• What is O(f(n)) ?

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Expression of GCD

• GCD(a,b) can be expressed as a linear combination
  of a and b.
• GCD(a,b) sa tb
• GCD(14, 49)
• GCD (252, 198) 18 (in class)

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Congruence

• If ac ? bc (mod m)
• and GCD(c,m) 1
• Then a ? b (mod m)
• 37 ? 117 (mod 8) (8,7 are pairwise prime)
• 21 ? 77 (mod 8)
• therefore, 3 ? 11 (mod m)

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Linear Congruence

• ax ? b (mod m) is a linear congruence
• If a,m are pairwise prime, we can find a-1 of a
  mod m
• find x, y such that xa ym 1

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Solving for Inverses

• Find the inverse of 4 mod 9
• Find the inverse of 19 mod 141

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Using the Inverse

• What is the solution for the linear congruence
• 4x ? 7 (mod 9)

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Sun-Tzu

• A number is unknown
• When divided by 3, the remainder is 2
• When divided by 5, the remainder is 3
• When divided by 7, the remainder is 2
• What is the number?

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System for Chinese Remainder Problem

• x ? 2 (mod 3)
• x ? 3 (mod 3)
• x ? 2 (mod 7)
• This has a unique solution modulo m m1m2m3...mn

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Expression of Integers

• A positive integer a less than m can be
  represented by
• (a mod m1, a mod m2, ... , a mod mn)
• where m1m2...mn m and are pairwise prime
• Ex m 20 m1 4, m2 5
• 3 (3,3) 5 (1, 0) 14 (2, 4)

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Expression of Integers

• m 95 97 99 912,285
• 344,911
• 291, 022

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Arithmetic of Large Numbers

• Do addition on components, keeping modulo
• 344,911 291, 022

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Conversion to large Integers

• Re-conversion involves solving Chinese Remainder
  Problem