2022-04-15
m_1 = 42 / 2 = 21 m_2 = 42 / 3 = 14 m_3 = 42 / 7 = 6 compute t_i=(m_i)^(−1) mod n_i: When i = 1: EE(21, 2) -> EE(2, 1) -> EE(1, 0) = (1, 1, 0) EE(2, 1) = (1, 0, 1) EE(21, 2) = (1, 1, -10) 21*a - 10*2 = 1 where a = 1 c_1 = 21*1 = 21 When i = 2: EE(14, 3) -> EE(3, 2) -> EE(2, 1) -> EE(1, 0) = (1, 1, 0) EE(2, 1) = (1, 0, 1) EE(3, 2) = (1, 1, -1) EE(14, 3) = (1, -1, 5) 14*a + 5*3 = 1 where a = -1 c_2 = 14*-1 = -14 When i = 3: EE(6, 7) -> EE(7, 6) -> EE(6, 1) -> EE(1, 0) = return (1, 1, 0) EE(6, 1) = (1, 0, 1) EE(7, 6) = (1, 1, -1) EE(6, 7) = (1, -1, 1) 6*a + 1*7 = 1 where a = -1 c_3 = 6*-1 = -6 a = (a_1*c_1)+(a_2*c_2)+(a_3*c_3) a = 1*21 + 2*-14 + 5*-6 = -37 a = -37 mod 42 = 5 mod 42 Check and confirm all the following are true: 5 = 1 mod 2, 5 = 2 mod 3, 5 = 5 mod 7(Edited: 2022-04-15)
<pre>
m_1 = 42 / 2 = 21
m_2 = 42 / 3 = 14
m_3 = 42 / 7 = 6
compute t_i=(m_i)^(−1) mod n_i:
When i = 1:
EE(21, 2) -> EE(2, 1) -> EE(1, 0) = (1, 1, 0)
EE(2, 1) = (1, 0, 1)
EE(21, 2) = (1, 1, -10)
21*a - 10*2 = 1 where a = 1
c_1 = 21*1 = 21
When i = 2:
EE(14, 3) -> EE(3, 2) -> EE(2, 1) -> EE(1, 0) = (1, 1, 0)
EE(2, 1) = (1, 0, 1)
EE(3, 2) = (1, 1, -1)
EE(14, 3) = (1, -1, 5)
14*a + 5*3 = 1 where a = -1
c_2 = 14*-1 = -14
When i = 3:
EE(6, 7) -> EE(7, 6) -> EE(6, 1) -> EE(1, 0) = return (1, 1, 0)
EE(6, 1) = (1, 0, 1)
EE(7, 6) = (1, 1, -1)
EE(6, 7) = (1, -1, 1)
6*a + 1*7 = 1 where a = -1
c_3 = 6*-1 = -6
a = (a_1*c_1)+(a_2*c_2)+(a_3*c_3)
a = 1*21 + 2*-14 + 5*-6 = -37
a = -37 mod 42 = 5 mod 42
Check and confirm all the following are true:
5 = 1 mod 2,
5 = 2 mod 3,
5 = 5 mod 7
</pre>
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2022-04-17
m1 = 42/2 = 21 m2 = 42/3 = 14 m3 = 42/7 = 6 EE(21, 2) call EE(2, 1) call EE(1, 0) return (1, 1, 0) return (1, 0, 1) return (1, 1, -10)t1 = 1 mod 2 EE(14, 3)
call EE(3, 2) call EE(2, 1) call EE(1, 0) return (1, 1, 0) return (1, 0, 1) return (1, 1, -1) return (1, -1, 5)t2 = 1 mod 3 EE(6, 7)
call EE(7, 6) call EE(6, 1) call EE(1, 0) return (1, 1, 0) return (1, 0, 1) return (1, 1, -1) return (1, -1, 2)t3 = 1 mod 7 c1 = 21*1 = 21 c2 = 14*-1 = -14 C3 = 6*-1 = -6 a = 1 * 21 - 2 * 14 - 5 * 6 = 21 - 28 - 30 = -37 mod 42 = 5 mod 42 check 5 = 1 mod 2, 5 = 2 mod 3, 5 = 5 mod 7
<pre>
m1 = 42/2 = 21
m2 = 42/3 = 14
m3 = 42/7 = 6
EE(21, 2)
call EE(2, 1)
call EE(1, 0)
return (1, 1, 0)
return (1, 0, 1)
return (1, 1, -10)
t1 = 1 mod 2
EE(14, 3)
call EE(3, 2)
call EE(2, 1)
call EE(1, 0)
return (1, 1, 0)
return (1, 0, 1)
return (1, 1, -1)
return (1, -1, 5)
t2 = 1 mod 3
EE(6, 7)
call EE(7, 6)
call EE(6, 1)
call EE(1, 0)
return (1, 1, 0)
return (1, 0, 1)
return (1, 1, -1)
return (1, -1, 2)
t3 = 1 mod 7
c1 = 21*1 = 21
c2 = 14*-1 = -14
C3 = 6*-1 = -6
a = 1 * 21 - 2 * 14 - 5 * 6 = 21 - 28 - 30 = -37 mod 42 = 5 mod 42
check 5 = 1 mod 2, 5 = 2 mod 3, 5 = 5 mod 7
</pre>
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M1 = 42/2 = 21
M2 = 42/3 = 14
M3 = 42/7 = 6
EE(21, 2) -> x = 1, y = -10
21 * 1 - 10 * 2 = 1 => 21 * 1 equivalent 1 mod 2
t_1 = inv(21) = 1 mod 2
EE(14, 3) -> x = -1, y = 5
14 * -1 + 3 * 5 = 1 => 14 * -1 equivalent 2 mod 3
t_2 = inv(-14) = 2 mod 3
EE(6, 7) -> x = -1, y = 5
6 * -1 + 7 * 1 = 1 => 6 * -1 equivalent 5 mod 7
t_2 = inv(-6) = 5 mod 7
C_i = m_i * t_i
So
c_1 = 21, c_2 = -14, c_3 = -6
a = a_1 * c_ 1 + a_2 * c_2 + a_3 * c_3
a = 1 * 21 + 2 * -14 + 5 * -6
a = 21 - 28 - 30
a = 21 - 58
a = -37
Using this value of a,
-37 mod 42 = 5 mod 42
Which satisfies
5 equivalent 1 mod 2, 5 equivalent 2 mod 3, 5 equivalent 5 mod 7
(Edited: 2022-04-17) <nowiki>
M1 = 42/2 = 21
M2 = 42/3 = 14
M3 = 42/7 = 6
EE(21, 2) -> x = 1, y = -10
21 * 1 - 10 * 2 = 1 => 21 * 1 equivalent 1 mod 2
t_1 = inv(21) = 1 mod 2
EE(14, 3) -> x = -1, y = 5
14 * -1 + 3 * 5 = 1 => 14 * -1 equivalent 2 mod 3
t_2 = inv(-14) = 2 mod 3
EE(6, 7) -> x = -1, y = 5
6 * -1 + 7 * 1 = 1 => 6 * -1 equivalent 5 mod 7
t_2 = inv(-6) = 5 mod 7
C_i = m_i * t_i
So
c_1 = 21, c_2 = -14, c_3 = -6
a = a_1 * c_ 1 + a_2 * c_2 + a_3 * c_3
a = 1 * 21 + 2 * -14 + 5 * -6
a = 21 - 28 - 30
a = 21 - 58
a = -37
Using this value of a,
-37 mod 42 = 5 mod 42
Which satisfies
5 equivalent 1 mod 2, 5 equivalent 2 mod 3, 5 equivalent 5 mod 7
</nowiki>
m1= 21, m2=14, m3= 6 Let's find ti finding t1: 21.t1= 1 mod 2 t1=1 t2: 14.t2= 1 mod 3 t2=2 t3: 6.t3= 1 mod 7 t3=6 calculating ci: c1= 21 c2=28 c3=36 finding a: a= 1.21+2.28+ 5. 36 a= 257 257 mod 42= 5
check: 5mod2=1, 5 mod3=2, 5mod 7= 5
m1= 21, m2=14, m3= 6 Let's find ti finding t1: 21.t1= 1 mod 2 t1=1 t2: 14.t2= 1 mod 3 t2=2 t3: 6.t3= 1 mod 7 t3=6 calculating ci: c1= 21 c2=28 c3=36 finding a: a= 1.21+2.28+ 5. 36 a= 257 257 mod 42= 5
check: 5mod2=1, 5 mod3=2, 5mod 7= 5
2022-04-18
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a = 1 mod 2 a = 2 mod 3 a = 5 mod 7 m1 = 42/2 = 21 m2 = 42/3 = 14 m3 = 42/7 = 6 EE(21,2) = EE(2, 1) = EE(1, 0) -> (1, 1, 0) -> (1, 0, 1) -> (1, 1, -10) EE(14,3) = EE(3, 2) = EE(2, 1) = EE(1, 0) -> (1, 1, 0) -> (1, 0, 1) -> (1, 1, -1) -> (1, -1, 5) EE(7,6) = EE(6, 1) = EE(1, 0) -> (1, 1, 0) -> (1, 0, 1) -> (1, 1, -1) c1 = 21*1 = 21 c2 = 14*-1 = -14 c3 = 6*-1 = -6 a = 1*21 + 2*-14 + 5*-6 = -37 a = -37 mod 42 = 5 mod 42 we have 5 = 1 mod 2, 5 = 2 mod 3, 5 = 5 mod 7(Edited: 2022-04-18)
<pre>
a = 1 mod 2
a = 2 mod 3
a = 5 mod 7
m1 = 42/2 = 21
m2 = 42/3 = 14
m3 = 42/7 = 6
EE(21,2) = EE(2, 1) = EE(1, 0) -> (1, 1, 0) -> (1, 0, 1) -> (1, 1, -10)
EE(14,3) = EE(3, 2) = EE(2, 1) = EE(1, 0) -> (1, 1, 0) -> (1, 0, 1) -> (1, 1, -1) -> (1, -1, 5)
EE(7,6) = EE(6, 1) = EE(1, 0) -> (1, 1, 0) -> (1, 0, 1) -> (1, 1, -1)
c1 = 21*1 = 21
c2 = 14*-1 = -14
c3 = 6*-1 = -6
a = 1*21 + 2*-14 + 5*-6 = -37
a = -37 mod 42 = 5 mod 42
we have 5 = 1 mod 2, 5 = 2 mod 3, 5 = 5 mod 7
</pre>
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