Chinese Remainder Theorem (CRT) Chinese Remainder Theorem (CRT) n r - - PowerPoint PPT Presentation

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Chinese Remainder Theorem (CRT) Chinese Remainder Theorem (CRT) n r - - PowerPoint PPT Presentation

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Chinese Remainder Theorem (CRT) Chinese Remainder Theorem (CRT) n r 1 (mod m 1 ) gcd(m i , m j ) = 1 r (mod m ) r 2 (mod m 2 ) r k (mod m k ) 1 Chinese Remainder

slide-1
SLIDE 1

Chinese Remainder Theorem (CRT) Chinese Remainder Theorem (CRT)

n  r1 (mod m1)  r (mod m ) gcd(mi, mj) = 1

 

 r2 (mod m2)

  • • •

 rk (mod mk)

     

1

 

slide-2
SLIDE 2

Chinese Remainder Theorem (CRT) Chinese Remainder Theorem (CRT)

 solution:

n  r1 (mod m1)  r (mod m ) gcd(mi, mj) = 1 m = m1 m2 ꞏ ꞏ ꞏ mk zi = m / mi  r2 (mod m2)

  • • •

 rk (mod mk)  zi

  • 1Zmi s.t. zi ꞏ zi
  • 1  1 (mod mi) (since gcd(zi, mi) = 1)

k

* n   zi ꞏ zi

  • 1 ꞏ ri (mod m)

i=1 k

   

1

 

slide-3
SLIDE 3

Chinese Remainder Theorem (CRT) Chinese Remainder Theorem (CRT)

 solution:

n  r1 (mod m1)  r (mod m ) gcd(mi, mj) = 1 m = m1 m2 ꞏ ꞏ ꞏ mk zi = m / mi  r2 (mod m2)

  • • •

 rk (mod mk)  zi

  • 1Zmi s.t. zi ꞏ zi
  • 1  1 (mod mi) (since gcd(zi, mi) = 1)

k

* n   zi ꞏ zi

  • 1 ꞏ ri (mod m)

i=1 k

1 2 3

 ex: r1=1, r2=2, r3=3

m1=3, m2=5, m3=7 m = 3 ꞏ 5 ꞏ 7

 

1

 

slide-4
SLIDE 4

Chinese Remainder Theorem (CRT) Chinese Remainder Theorem (CRT)

 solution:

n  r1 (mod m1)  r (mod m ) gcd(mi, mj) = 1 m = m1 m2 ꞏ ꞏ ꞏ mk zi = m / mi  r2 (mod m2)

  • • •

 rk (mod mk)  zi

  • 1Zmi s.t. zi ꞏ zi
  • 1  1 (mod mi) (since gcd(zi, mi) = 1)

k

* n   zi ꞏ zi

  • 1 ꞏ ri (mod m)

i=1 k

1 2 3

 ex: r1=1, r2=2, r3=3

m1=3, m2=5, m3=7 m = 3 ꞏ 5 ꞏ 7 z1=35, z2=21, z3=15

1

 

slide-5
SLIDE 5

Chinese Remainder Theorem (CRT) Chinese Remainder Theorem (CRT)

 solution:

n  r1 (mod m1)  r (mod m ) gcd(mi, mj) = 1 m = m1 m2 ꞏ ꞏ ꞏ mk zi = m / mi  r2 (mod m2)

  • • •

 rk (mod mk)  zi

  • 1Zmi s.t. zi ꞏ zi
  • 1  1 (mod mi) (since gcd(zi, mi) = 1)

k

* n   zi ꞏ zi

  • 1 ꞏ ri (mod m)

i=1 k

1 2 3

 ex: r1=1, r2=2, r3=3

m1=3, m2=5, m3=7 m = 3 ꞏ 5 ꞏ 7 z1=35, z2=21, z3=15 z -1=2 z -1=1 z -1=1 35 ꞏ 2 + 3 (-23) = 1

1

z1 2, z2 1, z3 1

35 2 + 3 ( 23) 1

slide-6
SLIDE 6

Chinese Remainder Theorem (CRT) Chinese Remainder Theorem (CRT)

 solution:

n  r1 (mod m1)  r (mod m ) gcd(mi, mj) = 1 m = m1 m2 ꞏ ꞏ ꞏ mk zi = m / mi  r2 (mod m2)

  • • •

 rk (mod mk)  zi

  • 1Zmi s.t. zi ꞏ zi
  • 1  1 (mod mi) (since gcd(zi, mi) = 1)

k

* n   zi ꞏ zi

  • 1 ꞏ ri (mod m)

i=1 k

1 2 3

 ex: r1=1, r2=2, r3=3

m1=3, m2=5, m3=7 m = 3 ꞏ 5 ꞏ 7 z1=35, z2=21, z3=15 z -1=2 z -1=1 z -1=1

1

z1 2, z2 1, z3 1 n  35ꞏ2ꞏ1 + 21ꞏ1ꞏ2 + 15ꞏ1ꞏ3  157  52 (mod 105)

slide-7
SLIDE 7

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

2

slide-8
SLIDE 8

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

2

slide-9
SLIDE 9

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

2

slide-10
SLIDE 10

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

(mod m2)

2

slide-11
SLIDE 11

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

(mod m2) (mod m1)

2

slide-12
SLIDE 12

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

 n  r1 (m2 m2

  • 1) + r2 (m1 m1
  • 1)

(mod m1 m2)

2

slide-13
SLIDE 13

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

 n  r1 (m2 m2

  • 1) + r2 (m1 m1
  • 1)

(mod m1 m2)

2

slide-14
SLIDE 14

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

 n  r1 (m2 m2

  • 1) + r2 (m1 m1
  • 1)

(mod m1 m2)

2

Verification

slide-15
SLIDE 15

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

 n  r1 (m2 m2

  • 1) + r2 (m1 m1
  • 1)

(mod m1 m2) n mod m1 =

2

1

Verification

slide-16
SLIDE 16

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

 n  r1 (m2 m2

  • 1) + r2 (m1 m1
  • 1)

(mod m1 m2) n mod m1 =

2

1

n mod m2 =

Verification

slide-17
SLIDE 17

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

mod m1

 n  r1 (m2 m2

  • 1) + r2 (m1 m1
  • 1)

(mod m1 m2) r1 n mod m1 =

2

1 1

Verification

slide-18
SLIDE 18

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

 n  r1 (m2 m2

  • 1) + r2 (m1 m1
  • 1)

(mod m1 m2) r1 n mod m1 =

2

1

+

1

Verification

slide-19
SLIDE 19

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

mod m2

 n  r1 (m2 m2

  • 1) + r2 (m1 m1
  • 1)

(mod m1 m2) r1 n mod m1 =

2

1

r2

+

1

n mod m2 =

Verification

slide-20
SLIDE 20

CRT, gcd(m1, m2)=1 , g (

1, 2)

 n  r1 (mod m1)

gcd(m1, m2) = 1

 n

r1 (mod m1) gcd(m1, m2) 1  r2 (mod m2)

  s, t such that m1 s + m2 t = 1

i.e. m1 m1

  • 1 + m2 m2
  • 1 = 1

1 1 2 2

 n  r1 (m2 m2

  • 1) + r2 (m1 m1
  • 1)

(mod m1 m2) r1 n mod m1 =

2

1

r2

+

1

n mod m2 =

Verification

slide-21
SLIDE 21

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7)  3 (mod 7)

21

slide-22
SLIDE 22

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  n1  1 (mod 3) … satisfying the 1 eq. r1

22

slide-23
SLIDE 23

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  1 (mod 3)  2 (mod 5)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  n1  1 (mod 3) … satisfying the 1 eq.

23

slide-24
SLIDE 24

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  1 (mod 3)  2 (mod 5)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.

24

slide-25
SLIDE 25

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  1 (mod 3)  2 (mod 5)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.

inverse of 3 (mod 5)

25

slide-26
SLIDE 26

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  1 (mod 3)  2 (mod 5)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.

inverse of 3 (mod 5) inverse of 5 (mod 3)

26

slide-27
SLIDE 27

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  1 (mod 3)  2 (mod 5)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.

inverse of 3 (mod 5) inverse of 5 (mod 3)

 n2  2 ꞏ 3 ꞏ (-3) + 1 ꞏ 5 ꞏ 2 ^ n1 ^

27

slide-28
SLIDE 28

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  1 (mod 3)  2 (mod 5)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.

inverse of 3 (mod 5) inverse of 5 (mod 3)

 n2  2 ꞏ 3 ꞏ (-3) + 1 ꞏ 5 ꞏ 2 ^ n1 r2 ^

28

slide-29
SLIDE 29

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  1 (mod 3)  2 (mod 5)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.  n2  2 ꞏ 3 ꞏ (-3) + 1 ꞏ 5 ꞏ 2 ^  -8  7 (mod 15) …. satisfying first 2 eqs.

29

slide-30
SLIDE 30

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  7 (mod 15)  3 (mod 7)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.  n2  2 ꞏ 3 ꞏ (-3) + 1 ꞏ 5 ꞏ 2 ^  -8  7 (mod 15) …. satisfying first 2 eqs.

30

slide-31
SLIDE 31

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  7 (mod 15)  3 (mod 7)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.  n2  2 ꞏ 3 ꞏ (-3) + 1 ꞏ 5 ꞏ 2  15 1 7 ( 2) 1 ^  -8  7 (mod 15) …. satisfying first 2 eqs.  15 ꞏ 1 + 7 ꞏ (-2) = 1

31

slide-32
SLIDE 32

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  7 (mod 15)  3 (mod 7)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.  n2  2 ꞏ 3 ꞏ (-3) + 1 ꞏ 5 ꞏ 2  15 1 7 ( 2) 1 ^

inverse of 15 (mod 7)

 -8  7 (mod 15) …. satisfying first 2 eqs.  15 ꞏ 1 + 7 ꞏ (-2) = 1

inverse of 7 (mod 15)

32

slide-33
SLIDE 33

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  7 (mod 15)  3 (mod 7)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.  n2  2 ꞏ 3 ꞏ (-3) + 1 ꞏ 5 ꞏ 2  15 1 7 ( 2) 1 ^

inverse of 15 (mod 7)

 -8  7 (mod 15) …. satisfying first 2 eqs.  15 ꞏ 1 + 7 ꞏ (-2) = 1  n3  3 ꞏ 15 ꞏ 1 + 7 ꞏ 7 ꞏ (-2) ^

inverse of 7 (mod 15)

3

( )

33

n2 ^

slide-34
SLIDE 34

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  7 (mod 15)  3 (mod 7)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.  n2  2 ꞏ 3 ꞏ (-3) + 1 ꞏ 5 ꞏ 2  15 1 7 ( 2) 1 ^

inverse of 15 (mod 7)

 -8  7 (mod 15) …. satisfying first 2 eqs.  15 ꞏ 1 + 7 ꞏ (-2) = 1  n3  3 ꞏ 15 ꞏ 1 + 7 ꞏ 7 ꞏ (-2) ^

inverse of 7 (mod 15)

3

( )

34

n2 r3 ^

slide-35
SLIDE 35

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7) n  7 (mod 15)  3 (mod 7)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.  n2  2 ꞏ 3 ꞏ (-3) + 1 ꞏ 5 ꞏ 2  15 1 7 ( 2) 1 ^  -8  7 (mod 15) …. satisfying first 2 eqs.  15 ꞏ 1 + 7 ꞏ (-2) = 1  n3  3 ꞏ 15 ꞏ 1 + 7 ꞏ 7 ꞏ (-2) ^  -53  52 (mod 105)

3

( )

35

( ) … satisfying all 3 eqs.

slide-36
SLIDE 36

Manually Incremental Calculation Manually Incremental Calculation

n  1 (mod 3)  2 (mod 5)  3 (mod 7)  3 (mod 7)  n1  1 (mod 3) satisfying the 1st eq ^  3 ꞏ (-3) + 5 ꞏ 2 = 1  n1  1 (mod 3) … satisfying the 1 eq.  n2  2 ꞏ 3 ꞏ (-3) + 1 ꞏ 5 ꞏ 2  15 1 7 ( 2) 1 ^  -8  7 (mod 15) …. satisfying first 2 eqs.  15 ꞏ 1 + 7 ꞏ (-2) = 1  n3  3 ꞏ 15 ꞏ 1 + 7 ꞏ 7 ꞏ (-2) ^  -53  52 (mod 105)

3

( )

36

( ) … satisfying all 3 eqs.

slide-37
SLIDE 37

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

moduli are not relative prime

 r2 (mod m2)

37

slide-38
SLIDE 38

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

38

slide-39
SLIDE 39

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

( )  3 (mod 10)

39

slide-40
SLIDE 40

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1 ( )  3 (mod 10) ( )

40

slide-41
SLIDE 41

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( )

41

slide-42
SLIDE 42

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( )

42

slide-43
SLIDE 43

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2

43

slide-44
SLIDE 44

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

44

slide-45
SLIDE 45

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60) Verification: 26 mod 6 = 2, 26 mod 10 = 6

45

slide-46
SLIDE 46

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

46

slide-47
SLIDE 47

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

 n  1 (mod 6)  3 (mod 10), gcd(6,10)=2

47

slide-48
SLIDE 48

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

 n  1 (mod 6)  3 (mod 10), gcd(6,10)=2

n  1 (mod 6)  n  1 (mod 2)  1 (mod 3) CRT

48

slide-49
SLIDE 49

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

 n  1 (mod 6)  3 (mod 10), gcd(6,10)=2

d(2 3) 1 n  1 (mod 6)  n  1 (mod 2)  1 (mod 3) CRT gcd(2,3)=1

49

slide-50
SLIDE 50

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

 n  1 (mod 6)  3 (mod 10), gcd(6,10)=2

d(2 3) 1 n  1 (mod 6)  n  1 (mod 2)  1 (mod 3) CRT n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) gcd(2,3)=1 n  3 (mod 10)  n  1 (mod 2)  3 (mod 5)

50

slide-51
SLIDE 51

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

 n  1 (mod 6)  3 (mod 10), gcd(6,10)=2

d(2 3) 1 n  1 (mod 6)  n  1 (mod 2)  1 (mod 3) CRT n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) gcd(2,3)=1 n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) gcd(2,5)=1

51

slide-52
SLIDE 52

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

 n  1 (mod 6)  3 (mod 10), gcd(6,10)=2

d(2 3) 1 n  1 (mod 6)  n  1 (mod 2)  1 (mod 3) CRT n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) gcd(2,3)=1 n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) gcd(2,5)=1

consistent

52

slide-53
SLIDE 53

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

 n  1 (mod 6)  3 (mod 10), gcd(6,10)=2

n  1 (mod 6)  n  1 (mod 2)  1 (mod 3) CRT n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) n  1 (mod 2) 1 ( d 3)  1 (mod 3)  3 (mod 5)

53

slide-54
SLIDE 54

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

 n  1 (mod 6)  3 (mod 10), gcd(6,10)=2

n  1 (mod 6)  n  1 (mod 2)  1 (mod 3) CRT n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) n  1 (mod 2) 1 ( d 3) n  1 (mod 6)  1 (mod 3)  3 (mod 5) n 1 (mod 6)  3 (mod 5)

54

slide-55
SLIDE 55

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

 n  1 (mod 6)  3 (mod 10), gcd(6,10)=2

n  1 (mod 6)  n  1 (mod 2)  1 (mod 3) CRT n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) n  1 (mod 2) 1 ( d 3) n  1 (mod 6) n  r (mod m )  1 (mod 3)  3 (mod 5) n 1 (mod 6)  3 (mod 5) n  r1 (mod m1)  r2 (mod m2/d)

55

i.e.

slide-56
SLIDE 56

CRT, gcd(m1, m2)=d , g (

1, 2)

 n  r1 (mod m1)

 r (mod m )

gcd(m m ) = d > 1 moduli are not relative prime

 r2 (mod m2)

gcd(m1, m2) = d > 1

 n  1 (mod 6)

3ꞏ(-3) + 5ꞏ2 = 1

3-1-3 (mod 5), 5-12 (mod 3)

( )  3 (mod 10) ( ) n  3 ꞏ 6ꞏ(-3) + 1 ꞏ 10ꞏ2  -34  26 (mod 60)

Incorrect!!!

Verification: 26 mod 6 = 2, 26 mod 10 = 6

Incorrect!!!

 n  1 (mod 6)  3 (mod 10), gcd(6,10)=2

n  1 (mod 6)  n  1 (mod 2)  1 (mod 3) CRT n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) n  3 (mod 10)  n  1 (mod 2)  3 (mod 5) n  1 (mod 2) 1 ( d 3) n  1 (mod 6) n  r (mod m ) note: CRT works only when gcd(d,m2/d)=1  1 (mod 3)  3 (mod 5) n 1 (mod 6)  3 (mod 5) n  r1 (mod m1)  r2 (mod m2/d)

56

i.e.

slide-57
SLIDE 57

CAVEAT CAVEAT

 n  3 (mod 10)  n

3 (mod 10)  11 (mod 12)

5

slide-58
SLIDE 58

CAVEAT CAVEAT

 n  3 (mod 10)

10=2ꞏ5, 12=22ꞏ3

 n

3 (mod 10)  11 (mod 12)

5

slide-59
SLIDE 59

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 10=2ꞏ5, 12=22ꞏ3

 n

3 (mod 10)  11 (mod 12)

5

slide-60
SLIDE 60

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6)

5

slide-61
SLIDE 61

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3)

5

slide-62
SLIDE 62

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) n  23 (mod 30) n 23 (mod 30)

5

slide-63
SLIDE 63

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) n  23 (mod 30) 53 n 23 (mod 30)

5

slide-64
SLIDE 64

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) n  23 (mod 30) 53 n 23 (mod 30)

5

slide-65
SLIDE 65

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) gcd(4,3)=1

5

slide-66
SLIDE 66

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) gcd(4,3)=1 n  11 (mod 12)  n  1 (mod 2)  5 (mod 6) gcd(2,6)1

5

slide-67
SLIDE 67

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) n  11 (mod 12)  n  1 (mod 2)  5 (mod 6) n  1 (mod 2)  5 (mod 6) ( ) ( )

5

slide-68
SLIDE 68

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) n  11 (mod 12)  n  1 (mod 2)  5 (mod 6) n  1 (mod 2)  5 (mod 6)  n  1 (mod 2)  1 (mod 2)  2 (mod 3) ( ) ( )  ( od ) ( od ) ( od 3)

5

slide-69
SLIDE 69

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) n  11 (mod 12)  n  1 (mod 2)  5 (mod 6) n  1 (mod 2)  5 (mod 6)  n  1 (mod 2)  1 (mod 2)  2 (mod 3) ( ) ( )

 n  1 (mod 2)  2 (mod 3) 

( od ) ( od ) ( od 3)

5

slide-70
SLIDE 70

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) n  11 (mod 12)  n  1 (mod 2)  5 (mod 6) n  1 (mod 2)  5 (mod 6)  n  1 (mod 2)  1 (mod 2)  2 (mod 3) ( ) ( )

 n  1 (mod 2)  2 (mod 3)  n  5 (mod 6) 

( od ) ( od ) ( od 3)

5

slide-71
SLIDE 71

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) n  11 (mod 12)  n  1 (mod 2)  5 (mod 6) n  1 (mod 2)  5 (mod 6)  n  1 (mod 2)  1 (mod 2)  2 (mod 3) ( ) ( )

 n  1 (mod 2)  2 (mod 3)  n  5 (mod 6) 

( od ) ( od ) ( od 3)

 n  3 (mod 10)

5

 n  3 (mod 10)

 11 (mod 12)

slide-72
SLIDE 72

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) n  11 (mod 12)  n  1 (mod 2)  5 (mod 6) n  1 (mod 2)  5 (mod 6)  n  1 (mod 2)  1 (mod 2)  2 (mod 3) ( ) ( )

 n  1 (mod 2)  2 (mod 3)  n  5 (mod 6) 

( od ) ( od ) ( od 3)

 n  3 (mod 10)

n  1 (mod 2)  3 (mod 5)

5

 n  3 (mod 10)

 11 (mod 12) ( )  3 (mod 4)  2 (mod 3)

slide-73
SLIDE 73

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) n  11 (mod 12)  n  1 (mod 2)  5 (mod 6) n  1 (mod 2)  5 (mod 6)  n  1 (mod 2)  1 (mod 2)  2 (mod 3) ( ) ( )

 n  1 (mod 2)  2 (mod 3)  n  5 (mod 6) 

( od ) ( od ) ( od 3)

 n  3 (mod 10)

n  1 (mod 2)  3 (mod 5) n  3 (mod 5)  3 (mod 4)

5

 n  3 (mod 10)

 11 (mod 12) ( )  3 (mod 4)  2 (mod 3) ( )  2 (mod 3)

slide-74
SLIDE 74

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) n  11 (mod 12)  n  1 (mod 2)  5 (mod 6) n  1 (mod 2)  5 (mod 6)  n  1 (mod 2)  1 (mod 2)  2 (mod 3) ( ) ( )

 n  1 (mod 2)  2 (mod 3)  n  5 (mod 6) 

( od ) ( od ) ( od 3)

 n  3 (mod 10)

n  1 (mod 2)  3 (mod 5) n  3 (mod 5)  3 (mod 4) n  3 (mod 20)  2 (mod 3)

5

 n  3 (mod 10)

 11 (mod 12) ( )  3 (mod 4)  2 (mod 3) ( )  2 (mod 3) ( )

slide-75
SLIDE 75

CAVEAT CAVEAT

 n  3 (mod 10)

gcd(10,12)=2 n  3 (mod 10) gcd(10,6)=2 n  3 (mod 10) 10=2ꞏ5, 12=22ꞏ3 12 22 3

 n

3 (mod 10)  11 (mod 12) n  3 (mod 10)  5 (mod 6) n  3 (mod 10)  2 (mod 3) 12 22 3 n  23 (mod 30) 53 12=22ꞏ3 n  11 (mod 12)  n  3 (mod 4)  2 (mod 3) CRT 12=22ꞏ3 n 23 (mod 30) n  11 (mod 12)  n  1 (mod 2)  5 (mod 6) n  1 (mod 2)  5 (mod 6)  n  1 (mod 2)  1 (mod 2)  2 (mod 3) ( ) ( )

 n  1 (mod 2)  2 (mod 3)  n  5 (mod 6) 

( od ) ( od ) ( od 3)

 n  3 (mod 10)

n  1 (mod 2)  3 (mod 5) n  3 (mod 5)  3 (mod 4) n  3 (mod 20)  2 (mod 3)

5

 n  3 (mod 10)

 11 (mod 12) ( )  3 (mod 4)  2 (mod 3) ( )  2 (mod 3) ( ) n  23 (mod 60)

slide-76
SLIDE 76

CRT w/ Moluli not Relative Prime

 Chinese Remainder Theorem: 

6

slide-77
SLIDE 77

CRT w/ Moluli not Relative Prime

n  r1 (mod m1)  r (mod m )

 Chinese Remainder Theorem:

 r2 (mod m2)

  • • •

 rk (mod mk)

6

slide-78
SLIDE 78

CRT w/ Moluli not Relative Prime

n  r1 (mod m1)  r (mod m )

 Chinese Remainder Theorem:

 r2 (mod m2)

  • • •

 rk (mod mk) gcd(mi, mj) = 1

6

slide-79
SLIDE 79

CRT w/ Moluli not Relative Prime

n  r1 (mod m1)  r (mod m )

 Chinese Remainder Theorem:

 r2 (mod m2)

  • • •

 rk (mod mk) there exists a unique integer n  satisfying the

Zm1ꞏꞏꞏmk

set of k congruence equations

1 k

gcd(mi, mj) = 1

6

slide-80
SLIDE 80

CRT w/ Moluli not Relative Prime

n  r1 (mod m1)  r (mod m )

 Chinese Remainder Theorem:

 r2 (mod m2)

  • • •

 rk (mod mk) there exists a unique integer n  satisfying the

Zm1ꞏꞏꞏmk

set of k congruence equations

1 k

note: each tuple (r1, r2ꞏꞏꞏ, rk) maps to one of m1m2ꞏꞏꞏmk distinct gcd(mi, mj) = 1 p ( 1,

2

,

k)

p

1 2 k

integers, which are members of the field Zm1ꞏꞏꞏmk

6

slide-81
SLIDE 81

CRT w/ Moluli not Relative Prime

n  r1 (mod m1)  r (mod m )

 Chinese Remainder Theorem:

 r2 (mod m2)

  • • •

 rk (mod mk) there exists a unique integer n  satisfying the

Zm1ꞏꞏꞏmk

set of k congruence equations

1 k

note: each tuple (r1, r2ꞏꞏꞏ, rk) maps to one of m1m2ꞏꞏꞏmk distinct gcd(mi, mj) = 1 p ( 1,

2

,

k)

p

1 2 k

integers, which are members of the field Zm1ꞏꞏꞏmk

 Prime power moduli: n  r (mod pc)  Prime power moduli: n  r (mod p ) 

6

slide-82
SLIDE 82

CRT w/ Moluli not Relative Prime

n  r1 (mod m1)  r (mod m )

 Chinese Remainder Theorem:

 r2 (mod m2)

  • • •

 rk (mod mk) there exists a unique integer n  satisfying the

Zm1ꞏꞏꞏmk

set of k congruence equations

1 k

note: each tuple (r1, r2ꞏꞏꞏ, rk) maps to one of m1m2ꞏꞏꞏmk distinct gcd(mi, mj) = 1 p ( 1,

2

,

k)

p

1 2 k

integers, which are members of the field Zm1ꞏꞏꞏmk

 Prime power moduli: n  r (mod pc)

 n  r' (mod pc'), c'<c, r'  r (mod pc')

 Prime power moduli: n  r (mod p ) 

6

slide-83
SLIDE 83

CRT w/ Moluli not Relative Prime

n  r1 (mod m1)  r (mod m )

 Chinese Remainder Theorem:

 r2 (mod m2)

  • • •

 rk (mod mk) there exists a unique integer n  satisfying the

Zm1ꞏꞏꞏmk

set of k congruence equations

1 k

note: each tuple (r1, r2ꞏꞏꞏ, rk) maps to one of m1m2ꞏꞏꞏmk distinct gcd(mi, mj) = 1 p ( 1,

2

,

k)

p

1 2 k

integers, which are members of the field Zm1ꞏꞏꞏmk

 Prime power moduli: n  r (mod pc)

 n  r' (mod pc'), c'<c, r'  r (mod pc')

 Prime power moduli: n  r (mod p )   CRT with prime modulus: n  r (mod m)

6

slide-84
SLIDE 84

CRT w/ Moluli not Relative Prime

n  r1 (mod m1)  r (mod m )

 Chinese Remainder Theorem:

 r2 (mod m2)

  • • •

 rk (mod mk) there exists a unique integer n  satisfying the

Zm1ꞏꞏꞏmk

set of k congruence equations

1 k

note: each tuple (r1, r2ꞏꞏꞏ, rk) maps to one of m1m2ꞏꞏꞏmk distinct gcd(mi, mj) = 1 p ( 1,

2

,

k)

p

1 2 k

integers, which are members of the field Zm1ꞏꞏꞏmk

 Prime power moduli: n  r (mod pc)

 n  r' (mod pc'), c'<c, r'  r (mod pc')

 Prime power moduli: n  r (mod p )

m = p1

c1p2 c2ꞏꞏꞏpk ck

 CRT with prime modulus: n  r (mod m)

6

Unique Prime Factorization Theorem

slide-85
SLIDE 85

CRT w/ Moluli not Relative Prime

n  r1 (mod m1)  r (mod m )

 Chinese Remainder Theorem:

 r2 (mod m2)

  • • •

 rk (mod mk) there exists a unique integer n  satisfying the

Zm1ꞏꞏꞏmk

set of k congruence equations

1 k

note: each tuple (r1, r2ꞏꞏꞏ, rk) maps to one of m1m2ꞏꞏꞏmk distinct gcd(mi, mj) = 1 p ( 1,

2

,

k)

p

1 2 k

integers, which are members of the field Zm1ꞏꞏꞏmk

 Prime power moduli: n  r (mod pc)

 n  r' (mod pc'), c'<c, r'  r (mod pc')

 Prime power moduli: n  r (mod p )

m = p1

c1p2 c2ꞏꞏꞏpk ck

 CRT with prime modulus: n  r (mod m)

n  r1 (mod p1

c1)

 r2 (mod p2

c2)

6

2 2

  • • •

 rk (mod pk

ck)

Unique Prime Factorization Theorem

slide-86
SLIDE 86

CRT w/ Moluli not Relative Prime CRT w/ Moluli not Relative Prime

7

slide-87
SLIDE 87

CRT w/ Moluli not Relative Prime CRT w/ Moluli not Relative Prime

 CRT with moduli not relative prime:

7

slide-88
SLIDE 88

CRT w/ Moluli not Relative Prime CRT w/ Moluli not Relative Prime

 CRT with moduli not relative prime:

n  r1 (mod m1)

7

slide-89
SLIDE 89

CRT w/ Moluli not Relative Prime CRT w/ Moluli not Relative Prime

 CRT with moduli not relative prime:

m1 = p1

c1p2 c2ꞏꞏꞏps cs

n  r1 (mod m1)

7

slide-90
SLIDE 90

CRT w/ Moluli not Relative Prime CRT w/ Moluli not Relative Prime

 CRT with moduli not relative prime:

m1 = p1

c1p2 c2ꞏꞏꞏps cs

n  r1 (mod m1) n  r2 (mod m2)

7

slide-91
SLIDE 91

CRT w/ Moluli not Relative Prime CRT w/ Moluli not Relative Prime

 CRT with moduli not relative prime:

m1 = p1

c1p2 c2ꞏꞏꞏps cs

n  r1 (mod m1) m2 = q1

d1q2 d2ꞏꞏꞏꞏꞏ qt dt

n  r2 (mod m2)

7

slide-92
SLIDE 92

CRT w/ Moluli not Relative Prime CRT w/ Moluli not Relative Prime

 CRT with moduli not relative prime:

m1 = p1

c1p2 c2ꞏꞏꞏps cs

n  r1 (mod m1) m2 = q1

d1q2 d2ꞏꞏꞏꞏꞏ qt dt

n  r2 (mod m2)  i, j, such that pi = qj i.e. mululi share common factors

7

slide-93
SLIDE 93

CRT w/ Moluli not Relative Prime CRT w/ Moluli not Relative Prime

n  r11 (mod p1

c1) c

 CRT with moduli not relative prime:

 r12 (mod p2

c2)

  • • •

 r1s (mod ps

cs)

m1 = p1

c1p2 c2ꞏꞏꞏps cs

n  r1 (mod m1)

1s (

ps ) m2 = q1

d1q2 d2ꞏꞏꞏꞏꞏ qt dt

n  r2 (mod m2)  i, j, such that pi = qj i.e. mululi share common factors

7

slide-94
SLIDE 94

CRT w/ Moluli not Relative Prime CRT w/ Moluli not Relative Prime

n  r11 (mod p1

c1) c

 CRT with moduli not relative prime:

 r12 (mod p2

c2)

  • • •

 r1s (mod ps

cs)

m1 = p1

c1p2 c2ꞏꞏꞏps cs

n  r1 (mod m1)

1s (

ps )

d

m2 = q1

d1q2 d2ꞏꞏꞏꞏꞏ qt dt

n  r2 (mod m2) n  r21 (mod q1

d1)

 r22 (mod q2

d2)

  • • •

 r2t (mod qt

dt)

 i, j, such that pi = qj i.e. mululi share common factors

7

slide-95
SLIDE 95

CRT w/ Moluli not Relative Prime CRT w/ Moluli not Relative Prime

n  r11 (mod p1

c1) c

 CRT with moduli not relative prime:

 r12 (mod p2

c2)

  • • •

 r1s (mod ps

cs)

m1 = p1

c1p2 c2ꞏꞏꞏps cs

n  r1 (mod m1)

1s (

ps )

d

m2 = q1

d1q2 d2ꞏꞏꞏꞏꞏ qt dt

n  r2 (mod m2) n  r21 (mod q1

d1)

 r22 (mod q2

d2)

  • • •

 r2t (mod qt

dt)

 i, j, such that pi = qj i.e. mululi share common factors l ti i t if ( d

k) f

k i ( d )

7

solution exists if r1i  r2j (mod pi

k), for pi = qj, k=min(ci,dj)